Recent developments on the power graph of finite groups – a survey

Algebraic graph theory is the study of the interplay between algebraic structures (both abstract as well as linear structures) and graph theory. Many concepts of abstract algebra have facilitated through the construction of graphs which are used as tools in computer science. Conversely, graph theory has also helped to characterize certain algebraic properties of abstract algebraic structures. In this survey, we highlight the rich interplay between the two topics viz groups and power graphs from groups. In the last decade, extensive contribution has been made towards the investigation of power graphs. Our main motive is to provide a complete survey on the connectedness of power graphs and proper power graphs, the Laplacian and adjacency spectrum of power graph, isomorphism, and automorphism of power graphs, characterization of power graphs in terms of groups. Apart from the survey of results, this paper also contains some new material such as the contents of Section 2 (which describes the interesting case of the power graph of the Mathieu group M11) and Section 6.1 (where conditions are discussed for the reduced power graph to be not connected). We conclude this paper by presenting a set of open problems and conjectures on power graphs.


Introduction
The study of graphical representation of an algebraic structure, especially a semigroup or a group become an energizing research topic over the recent couple of decades, prompting many intriguing outcomes and questions. In this context, the most well-known class of graphs is the Cayley graph. Cayley graphs were firstly presented in 1878, very much considered, and has numerous applications and wellstudied. In particular, Cayley graphs of finite groups are used as routing network in parallel computing due to the basic properties that Cayley graph are regular and vertextransitive. The notion of the power graph of a group is a very recent development in the domain of graphs from groups. The concept of directed power graphPðGÞ of a group G, introduced by Kelarev and Quinn [51], is a digraph with vertex set G and for any a, b 2 G, there is a directed edge from a to b inPðGÞ if and only if a k ¼ b, where k 2 N: For a semi-group, it was first considered in [53] and further studied in [52]. All of these papers used the brief term 'power graph' to refer to the directed power graph, with the understanding that the undirected power graph is the underlying undirected graph of the directed power graph. Motivated by this, Chakrabarty et al. [20] introduced the concept of an undirected power graph PðGÞ of a group G, which was defined as follows: Given a group G, the power graph PðGÞ of G is the simple undirected graph with vertex set G and two vertices a, b 2 G are adjacent in PðGÞ if and only if b 6 ¼ a and b k ¼ a or a k ¼ b, k 2 N: After that the undirected power graph became the main focus of study by several authors in [2,13,15,31,60,61].
As a simple example, we show the power graph of the group shown in Figure 1.
Many researchers have contributed towards the understanding of power graphs of groups, especially after 2010. In 2013, Abawajy et al. [2], made a survey about the power graphs in which they provided results about Eulerian, Hamiltonian, and complete characterizations of power graphs. Also, they collected and provided information about the number of edges, chromatic number, clique number, planarity, and isomorphism of power graphs. However, the authors did not explore properties like the spectrum, connectivity, automorphisms of power graphs. Motivated by this, we review both the classical as well as recent results on the power graphs from finite groups. We cover almost every known result about power graphs published after 2013 and also those results which are not available in the previous survey paper [2].

A case study: M 11
We begin by considering an example in some detail, the Mathieu group M 11 , a simple group of order 7920 ¼ 2 4 :3 2 :5:11; this group is small enough to be manageable but large enough to illustrate some interesting phenomena. Information about M 11 can be found in the Atlas of Finite Groups [30] or discovered using the computer algebra system GAP [43]. We will obtain information about the power graph, and also a construction of an interesting bipartite graph of large girth.
For this section, we only need the definition of the power graph given in the preceding section: the vertex set is the group G; two vertices a and b are joined if one is a power of the other. Let C ¼ PðM 11 Þ: The identity is joined to all other vertices in C (this is true in any finite group). This also means that the identity is an isolated vertex in the complement of C. To analyse further, we remove the identity, giving the so-called reduced power graph P 0 ðM 11 Þ: Of the remaining vertices, elements of order 11 and 5 are joined only to their powers; these form 144 complete graphs of size 10 and 396 complete graphs of size 4. (Now we observe that the non-identity elements form a single connected component in the complement of the power graph, since for any two such elements x and y, there is an element z of order 11 such that x, y 6 2 hziÞ. We remove the vertices of orders 5 and 11, and consider the remaining 4895 vertices, corresponding to elements whose orders are divisible by the primes 2 or 3 only. In detail, there are 165 of order 2, 440 of order 3, 990 of order 4, 1320 of order 6 and 1980 of order 8. Next we notice that vertices which generate the same cyclic subgroup have the same closed neighbourhood in the graph. (We will define x $ y to mean that x and y generate the same cyclic group.) Computation shows that the converse is false; the relation $ has 2035 equivalence classes, while the relation "same closed neighbourhood" has 1540. If we collapse each equivalence class of the second relation to a single vertex, we obtain a graph with 1540 vertices. This graph contains pairs of vertices with the same open neighbourhoods; collapsing such pairs yields a graph with 1210 vertices, in which no further such reduction is possible. These two reductions preserve connectedness and some other graph-theoretic properties.
We find that the automorphism group of this 1210-vertex graph is the Mathieu group M 11 , acting with four orbits, of sizes 165 (twice), 220 and 660. Numbering these orbits

Outline of the survey
This article has been carefully divided into 14 sections. In Section 4, we present the required definitions and notations. Section 5 investigates connectedness of power graphs, including minimal separating sets, disconnecting sets, and results on the vertex connectivity, the edge connectivity along with the relationship between the vertex connectivity and edge connectivity of the power graph of various groups. Section 6 elaborates the results on the connectivity of proper power graphs in which the number of components and the diameter of proper power graphs are also considered. Sections 7 and 8 deal with independence number and perfectness of power graphs, respectively. Section 9 has been devoted to the spectrum of power graphs, which includes the Laplacian spectrum and the adjacency spectrum of power graphs of certain finite groups. The relationship between the vertex connectivity and algebraic connectivity of the power graphs of some finite groups are also presented in this section. Section 10 presents results related to the isomorphism of power graphs, which includes power graphs of infinite groups also. Section 11 contains results on the automorphism of power graphs of finite groups. Followed by this, in Section 12, we present those results which provide the direct connection between the power graphs and their corresponding groups. Section 13 contains other properties of power graphs that cannot be classified into various sections mentioned above. Apart from survey of results, this paper also contains some new material such as Sections 2 and 6.1. We conclude this paper by giving certain open problems and conjectures in Section 14.

Definitions and notations
In this section, we present some definitions and notations from group theory and number theory as well as graph theory in order to make this paper self-contained. We use standard definitions and results from [38,42,76] for group theory and [8][9][10]34] for graph theory which we restate here along with our notations. N denotes the set of all natural numbers. For a positive integer n, Euler's phi function /ðnÞ, denotes the number of non-negative integers less than n that are relatively prime to n. When we consider the prime factorization of a positive integer n ¼ p a 1 1 p a 2 2 Á Á Á p a m m , it is assumed that m ! 2, p 1 < p 2 < Á Á Á < p m are primes and a i 2 N for all i with 1 i m:

Group theory
Throughout this paper, G denotes a group that may be of finite or infinite order, with identity e. Let Z(G) denote the center of the group G. For a group G, let p e ðGÞ ¼ foðaÞ : a 2 Gg, where o(a) is the order of the element a. The exponent of a finite group G, denoted by exp ðGÞ, is the least common multiple of orders of all its elements. Let pðGÞ be the set of all prime numbers p dividing the order of G, equivalently primes p such that G has an element of order p. A group G is called torsion-free if oðaÞ ¼ 1 for all a 2 G n feg: A group G is said to be of bounded exponent, if there exists n 2 N such that a n ¼ e for all a 2 G: A group G is said to be an EPO-group if every non-identity element of G is of prime order. A finite Abelian group G with identity e is called CP group if the order of every non-identity element is a power of a prime number. A group G is locally finite if every finitely generated subgroup H ¼ ha 1 , a 2 , :::, a k i of G, is of finite order. Further, G is called locally center-by-finite, if every finitely generated subgroup H of G has centre of finite index in H. We use hSi for the subgroup of G generated by the subset S. Let rðGÞ be the number of cyclic subgroups and sðGÞ denote the order of the smallest cyclic subgroup of G.
Define a relation $ on G by a $ b if hai ¼ hbi, where hai is the cyclic subgroup of G generated by a 2 G: It can be seen that $ is an equivalence relation on G. We denote the equivalence class containing a 2 G under $ by ½a: We note here that if a $ b, then a and b are joined in the power graph of G, and they have the same neighbours (except for one another).
Z n ¼ f0, 1, :::, n À 1g denotes the finite cyclic group of order n. The notation Z n m means that the direct product of n copies of Z m : SðZ n Þ denotes the set of all generators together with the identity element of the group Z n : That is, SðZ n Þ ¼ fa : 1 a < n, gcdða, nÞ ¼ 1g [ f0g: We use the following: D 2n ¼ ha, b j a n ¼ b 2 ¼ e, ab ¼ ba À1 i denotes the dihedral group of order 2n; Q 4n ¼ ha, b j a 2n ¼ e, a n ¼ b 2 , ab ¼ ba À1 i denotes the dicyclic group of order 4n. If n is a power of 2, this group is the generalized quaternion group. S n and A n denote the symmetric group and the alternating group on the set of n symbols, respectively.
For r 2 A n , the support of r is denoted by SuppðrÞ and is defined by SuppðrÞ ¼ fi : rðiÞ 6 ¼ ig: We recall here a theorem of Burnside (see [44,Theorem 12.5

.2]):
Theorem 4.1. [44,Theorem 12.5.2] Let G be a finite group whose order is a power of a prime p. Suppose that G has a unique subgroup of order p. Then either i. G is cyclic; or ii. p ¼ 2 and G is a generalized quaternion group.

Graph theory
Throughout this paper C ¼ ðV, EÞ denotes a graph with vertex set V and edge set E. dðCÞ denotes the minimum among degrees of vertices in C. For a subset A V of vertices in a graph C ¼ ðV, EÞ, the induced subgraph hAi is the subgraph of C with vertices in A and edges with both ends in A. A set of vertices T of a graph C is said to be a separating set or cut-set, if its removal increases the number of connected components of C. T is called a minimal separating set or minimal cut-set if none of its non-empty proper subset is a separating set. If T is of least cardinality, then it is called a minimum separating set or minimum cut-set of C. The cardinality of a minimum separating set is called the vertex connectivity of C and it is denoted by jðCÞ: A disconnecting set of C is a set of edges whose removal increases the number of connected components of C. A disconnecting set is said to be minimal if none of its proper subsets disconnects C. A minimum disconnecting set of C is a disconnecting set of C with least cardinality. If A, B VðCÞ, then the set of all edges having one end in A and the other in B is denoted by E½A, B: If A ¼ fvg, we write E½v, B instead of E½A, B: The diameter, diamðCÞ, of a graph C is the maximum distance between two vertices of C. If C is not connected, the diameter is defined to be 1: The girth of C, denoted by gðCÞ, is the length of a shortest cycle in C: A subset X & V of C ¼ ðV, EÞ is called an independent set, if there does not exist any edge in C whose both end vertices are in X. The cardinality of a largest independent set, denoted by bðCÞ is called independence number of C. A complete subgraph of C is called a clique, and the supremum of size of cliques in C, denoted by xðCÞ is called the clique number of C. A subset S & V of C is called a dominating set, if for any v 2 V, either v 2 S or there exists a vertex w 2 S such that v is adjacent to w. The cardinality of a minimum dominating set is denoted by cðCÞ and is called the dominating number of C.
For C ¼ ðV, EÞ and a 2 V, the neighbourhood of a is denoted by N(a) and its defined as NðaÞ ¼ fb 2 V j b is adjacent to ag: We sometimes call this the open neighbourhood of a, as opposed to the closed neighbourhood NðaÞ [ fag: The chromatic number of C is denoted by vðCÞ is the smallest number of colors needed to color the vertices of C so that no two adjacent vertices receive the same color.
A graph C is called perfect if the chromatic number of any finite induced subgraph of C is equal to its clique number. We recall the Strong Perfect Graph Theorem of Chudnovsky et al. [29], which characterizes perfect graphs by forbidden subgraphs and is given in Theorem 8.1(ii) of this survey.
Other interesting classes of graphs such as cographs, chordal graphs, split graphs, and threshold graphs, and the concepts of open and closed twins and twin reduction, will be introduced later.

Connectivity of power graphs
This section is divided into six subsections, which are devoted to the results based on vertex connectivity of power graphs, edge connectivity of power graphs and equality of these two parameters. Recall that for a given group G, the power graph PðGÞ of G is the simple undirected graph with vertex set G and two vertices a, b 2 G are adjacent in PðGÞ if and only if b 6 ¼ a and b k ¼ a or a k ¼ b, for some k 2 N: The power graph of any finite group is connected with diameter at most 2, since there is an edge from any nonidentity group element to the identity. In other words, {e} is a dominating set in PðGÞ:

Vertex connectivity of power graphs of finite cyclic groups
The finite cyclic group Z n , the dihedral group D 2n , and the dicyclic group Q 4n play an important role in the deeper parts of finite group theory, and invariably they appear as subgroups of a given group. The connectivity of the power graph of certain finite cyclic groups of particular order and their generalization was dealt in [21,22,26]. In continuation of these results, Panda and Krishna [70] focused on the power graph of finite cyclic groups in general and obtained minimal separating sets of the power graph PðZ n Þ, which in turn gives the vertex connectivity of the power graph PðZ n Þ: For a given X & Z n , X c ¼ Z n n X and hX c i is the induced subgraph of PðZ n Þ induced by X c . Recall that SðZ n Þ denotes the set of all generators together with the identity of the group Z n : For an arbitrary element a 2 Z n , a is some power of each of the generators of Z n and some power of a is the identity. Due to this, every element in SðZ n Þ is adjacent to every other element in PðZ n Þ: Hence the induced subgraph hSðZ n Þ c i plays some vital role in the connectivity of PðZ n Þ: For a subset A & Z n , A Ã ¼ A n feg: Chattopadhyay and Panigrahi [21,22], determined the tight lower bound for the vertex connectivity of power graphs corresponding to cyclic groups Z n and gave exact value of jðPðZ n ÞÞ when n is a power of some prime number. After that, many researchers extended these results and gave an upper bound of jðPðZ n ÞÞ for different values of n. Also, the vertex connectivity of the dihedral group D 2n , dicyclic group Q 4n , non-cyclic finite nilpotent group and non-cyclic abelian group of finite order were computed. The results in this regard are given below: Theorem 3]. The vertex connectivity jðPðZ n ÞÞ of the power graph of the finite cyclic group Z n , can be computed as follows: i. jðPðZ n ÞÞ ¼ n À 1 when n ¼ 1 or p a , where p is a prime number and a is an non negative integer; ii. jðPðZ n ÞÞÞ ! /ðnÞ þ 1, when n 6 ¼ p a : Further equality holds when n ¼ p 1 p 2 for distinct primes p 1 and p 2 .
In 2015, Chattopadhyay and Panigrahi [22], obtained another lower bound for the vertex connectivity of power graphs of certain finite cyclic groups and the same is given below.
where a 1 , a 2 2 N and p 1 , p 2 are distinct primes. Then the vertex connectivity jðPðZ n ÞÞ of PðZ n Þ satisfies the inequality jðPðZ n ÞÞ /ðnÞ þ p a 1 À1 1 p a 2 À1 2 : Theorem 5.3. [22, Theorem 2.9] For n ¼ p 1 p 2 p 3 , where p 1 < p 2 < p 3 are primes, the vertex connectivity jðPðZ n ÞÞ of PðZ n Þ satisfies the inequality jðPðZ n ÞÞ /ðnÞ þ p 1 þp 2 À 1: A natural question that arises is whether the converse of Theorem 5.1 is true. Since no information was provided by the authors in [21], Panda and Krishna [70], gave the answer for the above question in affirmative. i. If n is not prime power then every separating set of PðZ n Þ contains SðZ n Þ: i. n ¼ p 1 p 2 , where p 1 6 ¼ p 2 are primes; ii. SðZ n Þ is a separating set of PðZ n Þ; iii. jðPðZ n ÞÞ ¼ /ðnÞ þ 1: In [70], Panda and Krishna improved Theorems 5.2 and 5.3 by giving exact expression of jðPðZ n ÞÞ where n is the product of powers of two distinct primes and the same is given below.
Proposition 5.6. [70, Theorem 2.38] Assume that n ¼ p a 1 1 p a 2 2 , where p 1 , p 2 are distinct primes and a 1 , a 2 2 N: In fact, for n 6 ¼ p 1 p 2 , hp 1 p 2 Ã i is a minimum separating set of hSðZ n Þ c i: , a, b 2 N and p is an odd prime, then jðPðZ n ÞÞ ¼ n 2 : In the following theorem, we see the exact expression for jðPðZ n ÞÞ where n is the product of three distinct primes.
The following result is a consequence of Theorem 5.12 (i) and (ii), when the total number of distinct prime divisors of n less than or equal to the smallest prime divisor of n.
By proving the following theorem, the authors exhibited that the bound is sharp for many values of n.
Theorem 5.14. [26, Theorem 1.5] Let n ¼ p a 1 1 p a 2 2 p a 3 3 , where a i 2 N, for each 1 i 3 and p 1 < p 2 < p 3 are primes. If 2/ðp 1 p 2 Þ < p 1 p 2 , then p 1 ¼ 2 and Further, there is only one subset X of Z n with j X j ¼ jðPðZ n ÞÞ such that the induced subgraph hX c i of PðZ n Þ is disconnected.
In view of the fact proved in Theorem 5.14, the vertex connectivity of PðZ n Þ is completely determined for m 3: A natural question arises: can we find vertex connectivity of PðZ n Þ when n has more than three prime factors? that is m > 3. Chattopadhyay et al. [27] gave partial affirmative answer to this question. Let In the following theorem, it is observed that h 3 ðnÞ is an upper bound for the vertex connectivity of the power graph in certain cases of n. 1 p a 2 2 Á Á Á p a m m , m ! 3, p 1 < p 2 < Á Á Á < p m are primes and a i 2 N for 1 i m. If a m ! 2, then jðPðZ n ÞÞ ¼ minfh 1 ðnÞ, h j ðnÞ : 1 j m, a j ! 2g, where In this subsection, we are concerned with the vertex connectivity of power graphs of other groups such as the dihedral and dicyclic groups. Chattopadhyay et al. [21] obtained results on the vertex connectivity of these groups: In [28], the authors computed the vertex connectivity of power graphs of some special classes of groups which includes finite non-cyclic nilpotent groups, finite noncyclic abelian groups and non-cyclic groups of finite orders.
In this and following results, we denote by MðGÞ the collection of all maximal cyclic subgroups of G. If H is a cyclic subgroup of G, then e H denotes the set of non-generators in H.
Theorem 5.21. [28, Theorem 1.2] Let G be a finite non-cyclic nilpotent group of order n ¼ p a 1 1 p a 2 2 Á Á Á p a m m , m ! 2. Let P i be the Sylow p i -subgroup of G, for each 1 i m and assume that each of them is cyclic except P r for some r 2 f1, 2, :::, mg and that P r is not a generalized quaternion group if r ¼ 1 and 1 p a 2 2 and P 1 , P 2 denote its Sylow p i -subgroups for i ¼ 1, 2. Then following statements hold good.
i. Suppose that one of the Sylow subgroups is non-cyclic. If P i is non-cyclic, then P j is a minimum separating set of PðGÞ and so jðPðGÞÞ ¼ P j ¼ p a j j , where fi, jg ¼ f1, 2g. In fact, if p 1 ! 3, or p 1 ¼ 2 and P 2 is non-cyclic, then there exists only one minimum separating set of PðGÞ: ii. Suppose both Sylow subgroups are non-cyclic. If p 1 ! 3 and G has a maximal cyclic subgroup M of order p 1 p 2 , then jðPðGÞÞ ¼ min P 1 j j, /ð j M j Þg: È iii. Suppose both P 1 and P 2 are non-cyclic and P 1 is elementary abelian. Then jððPðGÞÞÞ ¼ min , where M is maximal cyclic subgroup of G of least possible order.
Remark 5.23. If p 1 ¼ 2 and exactly one of the Sylow subgroups G is non-cyclic in Theorem 5.22(i), then we can have more than one minimum separating set of PðGÞ (see [28,Example 4.3]).
Theorem 5.24. [28, Theorem 1.4] Let G be a non-cyclic abelian group of order p a 1 1 p a 2 2 p a 3 3 and P i be a Sylow p i -subgroup of G for i 2 f1, 2, 3g. Suppose that exactly two Sylow subgroups of G are cyclic. Then the following statements hold.
i. If p 1 ¼ 2 and P 1 is non-cyclic, then jðPðGÞÞ ¼ min P 1 P 2 j j, jðPðMÞÞg È , where M is a maximal cyclic subgroup of G of least possible order. More precisely, if j M j ¼ 2 a p a 1 1 p a 2 2 for some a 2 N, then ii. If p 1 ¼ 2 and P k is non-cyclic, then P 1 P j is the only minimum separating set of PðGÞ and so jðPðGÞÞ ¼ P 1 P j ¼ p a 1 1 p a j j , where fj, kg ¼ f2, 3g: iii. If p 1 ! 3 and P k is non-cyclic, then P i P j is the only minimum separating set of PðGÞ and so jðPðGÞÞ ¼ P i P j ¼ p a i i p a j j , where fi, j, kg ¼ f1, 2, 3g: Remark 5.25. [28] One can see that jðPðGÞÞ in Theorem 5.24(i) can also be obtained using the expression in Theorem 5.14.

Separating sets of power graphs
Recall that $ on G is defined by a $ b if hai ¼ hbi, where hai is the cyclic subgroup of G generated by a 2 G: It can be seen that $ is an equivalence relation on G. We denote the equivalence class of containing a 2 G under $ by ½a: The quotient graph of PðGÞ is called the quotient power graph of G and is denoted by e PðGÞ: Now a natural question arises: what will happen with power graphs of non-cyclic finite groups? Motivated by this, Chattopadhyay et al. [28], proved some results on power graphs by considering some special classes of groups including non-cyclic finite nilpotent groups and non-cyclic Abelian groups of finite order which are corresponding to their maximal cyclic subgroups. First, let us see the case of a non-cyclic group. i. If T has no element which will generate a member of MðGÞ, then PðM n TÞ is connected for every M 2 M ðGÞ: ii. If T is a minimal cut-set of PðGÞ and PðM n TÞ is connected for every M 2 MðGÞ, then T has no element which will generate a member of MðGÞ: Let G be a nilpotent group of order n ¼ p a 1 1 Á Á Á p a m m , m ! 2 and P r is neither cyclic nor a generalized quaternion group for some r 2 f1, 2, :::, mg. Then Q ¼ P 1 P 2 Á Á Á P rÀ1 P rþ1 Á Á Á P m is a minimal separating set of PðGÞ:

Disconnecting Sets in power graphs
In [71], Panda and Krishna determined minimum disconnecting sets of power graphs of finite cyclic groups, dihedral groups, dicyclic groups and abelian p-groups of finite order.

Equality of vertex and edge connectivity of power graphs
Now, we can ask the question: is there any relationship between graph invariants like vertex degree and diameter of power graph and its vertex connectivity and edge connectivity? The answer to this question is affirmative; this was proved in [71]. In fact, it was proved that j 0 ðPðGÞÞ ¼ dðPðGÞÞ, since diamðPðGÞÞ ¼ 2: But, this result is not true for jðPðGÞÞ in general. However, the authors of [71] examine the relationship between vertex connectivity and the minimum degree of power graphs of finite groups. They first explain some necessary conditions under which the vertex connectivity and the minimum degree of power graphs of finite groups coincide and computed the minimum degree when the equality holds for cyclic groups. Also, they gave a necessary and sufficient condition for jðPðZ n ÞÞ ¼ dðPðZ n ÞÞ: Theorem 5.40. [71, Theorem 6.2] Let G be a group of finite order at least 2 and jðPðGÞÞ ¼ dðPðGÞÞ. If G 6 ¼ Z p a , for some prime p is prime, a 2 N and dðPðGÞÞ ¼ degðaÞ, then following hold: i. N(a) is a minimum separating set of PðGÞ; ii. a is an element of order 2 in G. Consequently, G is a group of even order. i. For n ! 3, jðPðD 2n ÞÞ ¼ dðPðD 2n ÞÞ; ii. For n ! 2, jðPðQ 4n ÞÞ 6 ¼ dðPðQ 4n ÞÞ: In 2018, Panda and Krishna [71], calculated the minimum degree of power graphs of finite cyclic groups Z n , for some particular values of n. Also, they gave sharp upper bound for dðPðZ n ÞÞ for any n 2 N: Following this, Panda et al. [72], generalized these results for several other values of n. i. If n is not a power of a prime number, then dðPðZ n ÞÞ ¼ /ðnÞ þ 1 þ dðhSðZ n Þ c iÞ.
and p 1 , p 2 , :::, p m are prime numbers with p 1 < p 2 < Á Á Á < p m . Then dðPðZ n ÞÞ ¼ minfdegðp mÀ1 p m Þ, degðp m Þg. Further, In particular, if /ðp m Þ ! ðm À 2Þ/ðp mÀ1 Þ, then dðPðZ n ÞÞ ¼ degðp m Þ: For an arbitrary integer n, under certain conditions involving its prime divisors, the following theorem is proved on the minimum degree of PðZ n Þ: 1 p a 2 2 Á Á Á p a m m , m ! 2, p 1 < p 2 < Á Á Á < p m are primes and a i 2 N for 1 i m. Suppose that any of the following two conditions holds: ii. /ðp iþ1 Þ ! m/ðp i Þ for each i 2 f1, 2, 3, :::, m À 1g: If t 2 f2, 3, :::, mg is the largest integer such that a t ! a j for 2 j m, then dðPðZ n ÞÞ ¼ minfdegðp a s s Þ : t s mg: Using the above Theorem 5.51, the authors proved the following Corollary by which minimum degree of the power graph of a finite cyclic group can be calculated.
primes and a i 2 N for 1 i m. Suppose that any of the following two conditions holds: i. p 1 ! m þ 1 and p m > mp mÀ1 , ii. p iþ1 > mp i for each i 2 f1, 2, :::, m À 1g: Remark 5.53. Theorem 5.48 can be obtained from Theorems 5.50 and 5.51 and it was proved in [72].
If n has exactly three prime divisors, then following theorem shows that the minimum degree of the power graph of Z n can be calculated without any condition as stipulated in Theorem 5.51.
Theorem 5.54. [72, Theorem 1.5] Let n ¼ p a 1 1 p a 2 2 p a 3 3 where p 1 < p 2 < p 3 are primes and a i 2 N for 1 i 3. Then, It is already known that any abelian group of finite order is isomorphic to an unique product of cyclic groups of prime power order [15]. If the group G has an element x of order greater than 2, then x is joined to x 2 in P 0 ðGÞ, so the complement of the power graph is not complete. Thus, in the above theorem, the diameter is 2 except in the case when G is an elementary abelian 2-group.
Other connectivity questions relating to the complement of the power graph have not been studied yet.

Connectivity in proper power graphs
It is known that PðGÞ is connected for any group G, since {e} is a dominating set. A natural question is: what will be the effect on connectivity properties of PðGÞ if we remove the identity element from the vertex set of PðGÞ? This section is dedicated to all results based on the connectivity of proper power graph P 0 ðGÞ (power graph without identity element) of a group G.
Before beginning, we should address the question whether there may be vertices other than identity which are joined to all other vertices in the group. In other words, which elements a 2 G have the property that, for all b 2 G, either a is a power of b or b is a power of a? ii. the set of generators of G together with the identity, if G is cyclic but not of prime power order; iii. Z(G), if G is a generalized quaternion group; iv. {e}, in any other case.
To investigate connectivity, it makes sense to delete all such vertices; but, in all cases except cyclic and generalized quaternion groups, this just requires us to delete the identity, giving P 0 ðGÞ: The remaining cases can be dealt with separately.

Conditions for non-connectedness
We begin with a general condition for the reduced power graph not to be connected. Theorem 6.2. Let G be a finite group which is not of prime power order. Let p be a prime dividing j G j . Suppose that, for all primes q 6 ¼ p, there is no element of order pq in G. Then P 0 ðGÞ is not connected.
The hypothesis implies that there is no edge of the power graph between an element whose order is a power of p and one whose order is not a power of p. We saw an example of this in our discussion of the Mathieu group M 11 , where the primes 5 and 11 have this property, and elements of orders 5 and 11 form complete graphs not connected to anything else in the reduced power graph. The property of the theorem is not uncommon: many (but not all) finite simple groups have such a prime. See Conjecture 6.7 below.
A finite group is called a CP-group or EPPO group if every non-trivial element of the group has prime power order. For example, a p-group is also a CP-group. Following a lot of earlier research, the CP-groups have been determined in [19,Theorem 1.7]. It follows from the preceding theorem that, in a CP-group G, the set of elements of ppower order is a union of connected components of P 0 ðGÞ: A more general condition uses the Gruenberg-Kegel graph. The Gruenberg-Kegel graph, or prime graph, of a finite group G is the graph whose vertex set is the set of prime divisors of j G j , with an edge joining primes p and q whenever G contains an element of order pq. This graph has been the subject of a lot of research: see [19] for a summary. Theorem 6.3. Let G be a finite group whose Gruenberg-Kegel graph is disconnected. Then P 0 ðGÞ is disconnected.
For suppose that p is a connected component of the Gruenberg-Kegel graph. Then there can be no edge in P 0 ðGÞ joining an element whose order is a p-number to one whose order is not a p-number. For suppose that there were such an edge {a, b}. Then b is not a power of a, so a is a power of b. But then the order of b is divisible by both a prime p 2 p and a prime q 6 2 p; so some power of a has order pq, a contradiction.
We note that these graphs were introduced by Gruenberg and Kegel to study the integral group ring of G, in particular the decomposability of its augmentation ideal, in an unpublished manuscript in the 1970s. One of their main results was a structure theorem for groups with disconnected Gruenberg-Kegel graph; this was published by Williams (a student of Gruenberg) in 1981 [80]: Theorem 6.4. Let G be a finite group whose Gruenberg-Kegel graph is disconnected. Then one of the following holds: i. G is a Frobenius or 2-Frobenius group; ii. G is an extension of a nilpotent p-group by a simple group by a p-group, where p is the set of primes in the connected component containing 2.
Here, a 2-Frobenius group is a group G with normal subgroups H and K with H K such that K is a Frobenius group with Frobenius kernel H; G/H is a Frobenius group with Frobenius kernel K/H.
The simple groups in Case (ii) have been determined in several papers by Williams, Kondrat'ev and Mazurov.

Components in proper power graph
In 2014, Moghaddamfar et al. [65], computed some properties of proper power graphs P 0 ðGÞ, which are summarized below. Here, for any group G, pðGÞ denotes the set of all prime divisors of j G j : i. If oðGÞ ¼ p a , for some prime p and positive integer a, then P 0 ðGÞ is connected if and only if G has a unique minimal subgroup. and only if G is either a cyclic group or a generalized quaternion group. ii. If pðZðGÞÞ ! 2, then P 0 ðGÞ is connected.
iii. If pðGÞ ! 2 and center of G is a p-group for some p 2 pðGÞ, then the proper power graph P 0 ðGÞ is connected if and only if every non-central element a of order p there exists a non p-element b such that a $ b in P 0 ðGÞ: In connection with the first part, Theorem 4.1 shows that these groups are cyclic or generalized quaternion.
In 2015, Pourgholi et al. [74], proved that the number of edges in the power graphs of a simple group of order n is at most the number of edges in the power graph of the cyclic group of order n. They also proposed the following question on non-Abelian simple groups with 2-connected power graphs.  Having proposed the above conjecture, Narges Akbari et al. [3] proved that this conjecture true for some classes of finite simple groups. The relevant result is given below. It seems that the following result is the best for connectivity of power graphs without identity of periodic groups. Lemma 6.9. [49, Lemma 2.1] Let G be a periodic group. Then P 0 ðGÞ is connected if and only if for any two elements a, b of prime orders with a $ b, there exist elements a ¼ a 0 , :::, a m ¼ b such that oða 2i Þ is prime, oða 2iþ1 Þ ¼ oða 2i Þoða 2iþ2 Þ for i 2 f0, 1, 2, :::, m=2g and a i is adjacent to a iþ1 for i 2 f0, 1, :::, m À 1g: In [37], the authors calculated the number of connected components of the power graph of a special class of finite groups including nilpotent groups, Hughes-Thompson group, Suzuki group SzðmÞ, symmetric group S n and alternating group A n on n symbols. The results in this regard have been clubbed and presented below: If G is a finite group with exactly one element of order 2, then P 0 ðGÞ is connected. Theorem 6.11. [37, Theorem 2.5] Let G be a finite p-group. Then there exists a one-to-one correspondence between the connected components of P 0 ðGÞ and the minimal cyclic subgroups of G.
Theorem 6.12. [37, Theorem 2.6(i)] Let G be a finite pgroup. Then the number of connected components of P 0 ðGÞ is same as the number of subgroups of G of order p. In particular P 0 ðGÞ is connected if and only if G is a cyclic p-group or a generalized quaternion 2-group.
For a group G and a prime number p, the Hughes subgroup H p ðGÞ of G is defined as the subgroup generated by all elements of G whose orders are different from p. A finite group G is called a Hughes-Thompson group if G is not a p-group and H p ðGÞ & G for some prime divisor p of j G j : Recall that cðCÞ denotes the number of connected components of a graph C: vi. If n ¼ 2p þ 2 ! 12, then P 0 ðGÞ is connected when 2p þ 1 is not prime, and it is disconnected with 2ðp þ 1Þð2p À 1Þ! þ 1 connected components if 2p þ 1 is prime. vii. If n ¼ 3, 4, 5, 6, 7, 8, 9, 10, then the number of connected components of P 0 ðGÞ is equal to 1, 7, 31, 121, 421, 962, 5442, and 29345 respectively. Now, we present some results on the connectivity of the proper graph of a finite p-group proved by Panda et al. [70].   It follows from Theorem 3.3 [70] that the proper power graph of a non-cyclic abelian p-group has more than one component. This leads to the fact stated in Corollary 5.20. Cameron et. al [17], considered the question of connectivity of the proper power graph of infinite groups. Lemma 6.23. [17, Lemma 1] If P 0 ðGÞ is connected, then G is a torsion-free or a periodic group.
Theorem 6.24. [17, Theorem 7] Let G be a locally center-byfinite group which is torsion free. Then P 0 ðGÞ is connected if and only if G is isomorphic to a subgroup of Q: If G is a finite p-group, then P 0 ðGÞ is connected if and only if G is a cyclic group or a generalized quaternion group Q 2 n : For infinite case, we have the following result.
Theorem 6.25. [17, Theorem 9] Let G be infinite locally finite p-group. Then P 0 ðGÞ is connected if and only if G ffi C p 1 for some prime number p, or G ffi Q 2 1 :

Distance in proper power graphs
Recall that the diameter of a graph C is the maximum distance between pairs of vertices in C. Thus the diameter of a complete graph is precisely 1. It can be seen that not every proper power graph is connected. For example, the proper power graph of any dihedral group is disconnected since the involutions are isolated vertices. In [32], Curtin et al. focused on the groups with low diameter proper power graphs and proved the following results.
Lemma 6.26. [32, Lemma 12] For a finite group G, suppose that P 0 ðGÞ has a diameter at most 3. Then any Sylow subgroup of G either a cyclic group or a generalized quaternion 2-group.
We remark that groups satisfying the conclusion of those lemma can be determined by using group-theoretic characterization theorems including Glauberman's Z Ã -Theorem and the Gorenstein -Walter Theorem.   In 2015, Alireza et al. [37], proved some results on the proper power graph of finite groups and among other results, they proved that the connected proper power graph P 0 ðGÞ has diameter at most 4, 26, or 22 when G is a nilpotent group, symmetric group, or alternating group, respectively. These results lead to a conjecture which claims that connected proper power graphs of finite groups must have bounded diameter. i. If G is a p-group, then the number of connected components of P 0 ðGÞ is the same as the number of subgroups of G of order p. In particular, P 0 ðGÞ is connected if and only if G is a cyclic p-group or a generalized quaternion 2-group. ii. If G is not a p-group and each of the Sylow p-subgroup of G is a cyclic p-group or a generalized quaternion 2group, then P 0 ðGÞ is connected and diamðP 0 ðGÞÞ ¼ 2: iii. If G is not a p-group and G has a Sylow p-subgroup, which neither a cyclic pÀgroup nor a generalized quaternion 2-group, then P 0 ðGÞ is connected and diamðP 0 ðGÞÞ ¼ 4: Utilizing the above theorem, one can classify all finite groups for which the proper power graph is of diameter at most three. The characterization in this regard is given below. are not primes, then P 0 ðA n Þ is connected and diamðP 0 ðA n ÞÞ 22: Recently, in [73], the authors have improved the upper bound of diameter of proper power graphs of alternating groups to 11, for n > 51.

Independence Number of power graphs
In [79], Tamizh Chelvam et al. proved some results on the power graph of a finite abelian group in which they provided a lower bound for the independence number of the power graph of a finite group, computed the independence number of an elementary abelian p-group and characterized all finite abelian groups whose power graph has independence number 2. In 2018, Ma and Lu [59] provided sharp lower and upper bounds for the independence number of PðGÞ and characterized the groups achieving the bounds. Also, they determined the independence number PðGÞ of certain finite groups. Finally, they classified all finite groups G, whose power graphs have independence number 3 or oðGÞ À 2: For a group G, we have X P ¼ fP S : P is a subgroup of G of prime orderg: A maximal cyclic subgroup of G is a cyclic subgroup, which is not a proper subgroup of some proper cyclic subgroup of G. Denote by M G the set of all maximal cyclic subgroups of G. Recall that a finite group is called a CP-group or EPPO group if every non-trivial element of the group has prime power order.   In the next few theorems, we present results on the independence number of the power graph of an infinite group G. In the same article, the authors posed the question, does above theorem hold without assuming nilpotence? Cameron et al. [17] gave an affirmative answer to this question.
Theorem 7.19. [17, Theorem 3] Let G be a group satisfying bðPðGÞÞ < 1. Then either G is finite, or G ffi C p 1 Â H, where H is a finite group and p-oðHÞ: As a corollary of this result, Cameron et al. [17] proved the following corollary.
Corollary 7.20. [17, Corollary 1] Let G be a group whose power graph PðGÞ has finite independence number. Then the independence number and clique cover number of PðGÞ are equal.

Perfectness of the power graph
Recall that a finite graph is perfect if every induced subgraph has clique number equal to chromatic number. In the next theorem, we recall several facts about perfect graphs. The comparability graph WðPÞ of a partially ordered set ðP, Þ is the simple graph with the vertex set P and two distinct vertices x and y adjacent if and only if either x y or y x: i. If a graph is perfect, then its complement is perfect (The Weak Perfect Graph Theorem, Lov asz [56]). ii. A graph is perfect if and only if it contains no odd cycle or complement of an odd cycle of length at least 5 as an induced subgraph (The Strong Perfect Graph Theorem, Chudnovsky et al. [29]). iii. The comparability graph of a partial order, and its complement, are perfect (Dilworth's Theorem, [35]).
Theorem 8.2. [1,4,40] The power graph of a finite group is the comparability graph of a partial order, and hence is a perfect graph. In particular, its clique number and chromatic number are equal, and the clique number and chromatic number of the complement are also equal.
For consider the directed power graphPðGÞ, with a loop at each vertex. This is a partial preorder, a reflexive and symmetric relation. Writing x y if each of x and y precedes the other in the partial preorder (that is, if each is a power of the other); this is an equivalence relation, and the equivalence classes are partially ordered. Refining this relation by a total order on each equivalence class, we obtain a partial order whose comparability graph is PðGÞ:

Induced subgraphs
Any induced subgraph of the comparability graph of a partial order is itself a comparability graph. Subject to this, power graphs are universal: Apart from perfect graphs, there are various other interesting classes of graphs which are defined by forbidden induced subgraphs. Let P n , C n and K n denote the path, cycle and complete graph with n vertices, and 2K 2 the graph consisting of two disjoint edges. Some other graph classes considered are i. cographs, with no induced P 4 ; ii. chordal graphs, with no induced C n for n > 3; iii. split graphs, with no induced P 4 , C 5 or 2K 2 [46]; iv. threshold graphs, with no induced P 4 , K 4 or 2K 2 : We refer to [18] for further discussion of these graph classes. i. Let G be a finite nilpotent group. Then PðGÞ is a cograph if and only if either G has prime power order, or G is cyclic with order the product of two distinct primes. ii. Let G be a finite nilpotent group. Then PðGÞ is a chordal graph if and only if either G has prime power order, or G has just two prime divisors p and q, the Sylow p-subgroup is cyclic, and the Sylow q-subgroup has exponent q. iii. Let G be an arbitrary finite group. Then the following are equivalent: a. PðGÞ is a split graph; b. PðGÞ is a threshold graph; c. G is cyclic of prime power order, or an elementary abelian or dihedral 2-group, or cyclic of order 2p, or dihedral of order 2p n or 4p, where p is an odd prime.
A preliminary result towards the characterisation of finite groups whose power graph is split was given in [57].
One of the most important questions about power graphs of finite groups is: Problem 8.6. For which finite groups G is PðGÞ a cograph?
We will see another reason for examining this question in Section 11. Theorem 8.5(i) gives useful information, since it shows that, if PðGÞ is a cograph, then any nilpotent subgroup of G is either of prime power order or a cyclic group whose order is the product of two distinct primes. This greatly restricts the possible groups: here is a sample result.
Theorem 8.7. [14,Proposition 8.7] Suppose that q is a prime power. If q is a power of 2, then let l ¼ q À 1 and m ¼ q þ 1; if q is odd, let l ¼ ðq À 1Þ=2 and m ¼ ðq þ 1Þ=2. Let G ¼ PSLð2, qÞ. Then PðGÞ is a cograph if and only if each of l and m is either a power of a prime number or the product of two distinct primes.
We note that the smallest non-Abelian simple group whose power graph is not a cograph is the alternating group A 7 [14, Table 1].

Further results
In 2015, Alireza et al. [4] proved the following results on the clique number and the chromatic number of power graphs. The chromatic number was calculated earlier by Mirzagar et al. [64,Theorem 2]. We have reformulated their results somewhat. First we deal with cyclic groups. iii. The chromatic number of PðZ n Þ is equal to the clique number. For the first part, we notice that the /ðnÞ generators of Z n are dominating vertices, and so lie in every maximal clique; it can be shown that the remainder of a clique must lie in a proper subgroup, and the best we can do is to take the largest such subgroup. The second part follows by expanding the recurrence, and the third holds because the power graph is perfect. From this result it is possible to obtain an estimate for the clique number: Theorem 8.10. where the constant on the right has the analytic expression From these results we can give a formula, found by Mirzagar et al. and Alireza et al. [4,57] for xðPðGÞÞ for any group G. Recall that p e ðGÞ denotes the set of all orders of elements of G. Note also that, since a cyclic subgroup is a clique in the enhanced power graph, we have xðP e ðGÞÞ ¼ max p e ðGÞ: In the following theorem, the authors characterized all power graphs which are uniquely colorable. Aalipour et al. [1] proved that that the chromatic number of the power graph of G is finite if and only if the clique number of the power graph of G is finite and this statement is also equivalent to the finiteness of exponents of G. They also proved that the clique number of the power graph of G is at most countable. The fact that the chromatic number is also at most countable was subsequently proved in [78]. If there exists an integer n such that for all g 2 G, g n ¼ e, then G is said to be of bounded exponent. Utilizing Lemma 8.13 to colour the power graph with a finite set of colours, we require the group to be of a bounded exponent. It was proved that, for such groups, the resulting power graph is always perfect and can be finitely coloured. To prove this result, Aalipour et al. [1] use the concept of comparability graph. Let be a binary relation on the elements of a set P. If is reflexive and transitive, then ðP, Þ is called a pre-ordered set. All partially ordered sets are pre-ordered. The comparability graph WðPÞ of a pre-ordered set ðP, Þ is the simple graph with the vertex set P and two distinct vertices x and y are adjacent if and only if either x y or y x (or both). This is relevant to the power graph since the directed power graph of a group is a pre-ordered set and the power graph is its comparability graph.
Aalipour et al. [1] proved the following with regard to chromatic and clique numbers of power graphs of groups.  If G is abelian with exponent e, then vðPðGÞÞ ¼ xðPðGÞÞ ¼ f ðeÞ: Remark 8.17. Theorem 7.19 can be deduced using the fact that the power graph of the group of finite exponent is perfect, together with the weak Perfect Graph Theorem of Lov asz [56], asserting that the complement of a finite perfect graph is perfect. This argument also requires a compactness argument to show that the clique cover number of PðGÞ is equal to the maximum clique cover number of its finite subgroups. However in [17], Cameron  Since PðHÞ is a complete graph, bðHÞ ¼ 1: Clearly, the set f1g Â C 2 1 is an independent set and so bðPðGÞÞ ¼ 1:

Adjacency spectrum of power graphs
For any simple graph C with vertex set fv 1 , v 2 Á Á Á , v n g, the adjacency matrix AðCÞ ¼ ðx ij Þ is defined as the n Â n matrix, where x ij ¼ 1 if v i is adjacent to v j , and 0 otherwise. The adjacency characteristic polynomial of a graph C is given by UðC, aÞ ¼ detðaI À AðCÞÞ: The eigenvalues of AðCÞ are called eigenvalues of the graph C and denoted by l i ðCÞ, i ¼ 1, 2, 3 Á Á Á , n: Clearly, AðCÞ is a real symmetric matrix and so all its eigenvalues are real. Thus, they can be arranged in a non-decreasing order as l 1 ! l 2 Á Á Á ! l n : The multiset of all eigenvalues of C is called the spectrum of C denoted by rðCÞ and the largest eigenvalue l 1 is called the spectral radius of C.
Mehranian et al. [63] computed the spectrum of the power graph of cyclic groups, dihedral groups, elementary abelian groups of prime power order. In the following theorem, the authors calculated the characteristic polynomial of PðZ n Þ and P 0 ðZ n Þ: Then the characteristic polynomial of the power graph PðZ n Þ and the proper power graph P 0 ðZ n Þ can be computed as follows: i. UðPðZ n Þ, xÞ ¼ UðT, xÞðx þ 1Þ nÀtÀ1 ; ii. UðP 0 ðZ n Þ, sÞ ¼ UðT 0 , xÞðx þ 1Þ nÀtÀ2 , where the entries of T 0 equal to those of T in all columns but the first and each entry of the first column of T 0 is one less the corresponding entry of T. The following theorem gives us the characteristic polynomial of the power graph PðD 2n Þ and the proper power graph P 0 ðD 2n Þ of the dihedral group D 2n : Theorem 9.2. [63, Theorem 2.5] Suppose n is a prime power. Then the characteristic polynomial of the power graph PðD 2n Þ and proper power graph P 0 ðD 2n Þ of the dihedral group D 2n can be computed as: i. UðPðD 2n Þ,xÞ ¼ a nÀ1 ðx þ 1Þ nÀ2 ðx 3 À ðn À 2Þx 2 À ð2n À 1Þx þ n 2 À 2nÞ, ii. UðP 0 ðD 2n Þ, xÞ ¼ x n ðx þ 1Þ nÀ2 ðx À ðn À 2ÞÞ: In the following theorem, the authors obtained the characteristic polynomial and also computed the eigenvalues of the power graph of an elementary abelian group Eðp n Þ, where p is a prime number. pÀ1 . Then UðPðEðp n ÞÞ, xÞ ¼ ðx À ðp À 2Þ 'À1 Þðx þ 1Þ ðpÀ2Þ' ðx 2 À ðp À 2Þx À ðp n À 1ÞÞ: In particular, the eigenvalues of PðEðP n ÞÞ are -1 with multiplicity ðp À 2Þ', p À 2 with multiplicity ' À 1 and two simple eigenvalues x 1, 2 ¼ pÀ26 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðpÀ2Þ 2 þ4ðp n À1Þ p 2 : Hamzeh et al. [47] generalized some results proved in [63] through some more results on power graphs and they are presented below.
Theorem 9.4. [47, Theorem 3.9] The characteristic polynomial of PðD 2n Þ can be computed as follows: UðPðD 2n Þ,xÞ ¼ x nÀ1 ðxþ1Þ nÀtÀ2 xðxþ1ÞUðT,xÞÀnUðT 0 ,xÞ Â Ã : where d i ,1 i t, are all non-trivial divisors of n, Chattopadhyay et al. [25] obtained both upper and lower bounds for the spectral radius of the power graph of Z n and characterized the graphs for which these bounds are extremal. Further, they computed spectra of power graphs of the dihedral group D 2n and dicyclic group Q 4n partially and gave bounds for the spectral radii of these graphs.
In Theorem 9.1, the characteristic polynomial of PðZ n Þ has been obtained in terms of the characteristic polynomial of the quotient matrix T whose entries are some functions of the divisors of n.
Also, note that the spectral radius of PðZ n Þ is the same as that of the matrix T. Since the increase in the number of factors of n leads to a rapid increase of the degree of the polynomial of T, it is sometimes too complicated to find the exact value of the spectral radius of PðZ n Þ: Therefore, one can use some graph invariants like vertex degrees and diameter to approximate the spectral radius. The following theorem gives both upper and lower bounds for the spectral radius of PðZ n Þ in terms of the maximum and minimum degrees of the nonidentity non-generator elements of Z n : The next result provides the characteristic polynomial of PðD 2n Þ in terms of characteristic polynomials of PðZ n Þ and P 0 ðZ n Þ: Theorem 9.7. [25, Theorem 2.2] For any integer n ! 3, the characteristic polynomial of PðD 2n Þ is given by uðPðD 2n Þ, xÞ ¼ x nÀ1 xUðPðZ n Þ, xÞ À nUðP 0 ðZ n Þ, xÞ ½ : Remark 9.8. In the above theorem, the characteristic polynomial of PðD 2n Þ has been obtained for any natural number n ! 3 whereas in Theorem 9.2, the characteristic polynomial of PðD 2n Þ is given only when n is a prime power. In the following theorem, full spectrum of the power graph of the generalized quaternion group Q 4n is computed.

Laplacian Spectrum of power graphs
For any finite simple undirected graph C, the Laplacian matrix LðCÞ is given by LðCÞ ¼ DðCÞ À AðCÞ, where AðCÞ is the adjacency matrix of C and DðCÞ is the diagonal matrix of vertex degrees. Clearly LðCÞ is a real symmetric matrix and so all its eigenvalues are real. For a graph C on n vertices, we denote the Laplacian eigenvalues of C by k 1 ðCÞ ! k 2 ðCÞ ! Á Á Á ! k n ðCÞ always arranged in nonincreasing order and repeated according to their multiplicity. Since LðCÞ is symmetric, positive semi-definite and singular, and all its eigenvalues are non-negative and k n ðCÞ ¼ 0: To know, more interesting facts about Laplacian eigenvalues of a graph, we refer the survey paper [66]. Let k 1 , :::, k m be the distinct Laplacian eigenvalues with corresponding multiplicities t 1 , :::, t m : Then the Laplacian spectrum is denoted by k 1 ::: k m t 1 ::: t m : It is known that [33], the Laplacian eigenvalue with multiplicity 0 of a graph C is equal to the number of connected components of C. Thus, one gets that the second smallest Laplacian eigenvalue k nÀ1 ðCÞ > 0 if and only if C is connected. Fiedler [41], called k nÀ1 ðCÞ as the algebraic connectivity of C, viewing it as a measure of connectivity of C. The largest Laplacian eigenvalue k 1 ðCÞ is called the Laplacian spectral radius of C. A graph is called Laplacian integral if all its Laplacian eigenvalues are integers. In [41], Fiedler proved that the algebraic connectivity k nÀ1 ðCÞ of a noncomplete graph C does not exceed its vertex connectivity jðCÞ: The Laplacian spectrum of a graph has a number of applications, including random walks, expansion properties, and statistical efficiency and optimality properties. See [8] for some of these.
We write the characteristic polynomial detðxI À LðCÞÞ of LðCÞ by HðC, xÞ instead of HðLðCÞ, xÞ and called HðC, aÞ the Laplacian characteristic polynomial of C.
Let C be a graph with vertex set VðCÞ ¼ fv 1 , v 2 , :::, v n g: Then, for the vertices v 1 , v 2 , :::v i in C, L v 1 , v 2 , :::, v i ðCÞ is defined as the principal submatrix of LðCÞ formed by deleting rows and columns corresponding to the vertices v 1 , v 2 , :::, v i : In particular, if i ¼ n, then for convention it is taken as HðL v 1 , v 2 , :::, v n ðC, xÞÞ ¼ 1: Chattopadhyay [22] obtained the Laplacian spectrum of PðZ n Þ and PðD 2n Þ for particular values of n. In fact, the relationship between the spectrum of these two power graphs are discussed. Also, they gave sharp lower and upper bounds for algebraic connectivity of PðZ n Þ: Panda [68] considered various aspects of Laplacian spectra of power graphs of finite cyclic groups, dicyclic groups, and finite p-groups. More specifically, Panda [68] determined completely the Laplacian spectral radius of power graphs of all of these groups apart from the algebraic connectivity and its multiplicity. Then, the equality of the vertex connectivity and the algebraic connectivity is characterized for power graphs of all of the above classes of groups. Orders of dicyclic groups with Laplacian integral power graphs are determined. Moreover, it is proved that the notion of equality of the vertex connectivity and the algebraic connectivity and the notion of Laplacian integral are equivalent for power graphs of dicyclic groups. All possible values of Laplacian eigenvalues are obtained for power graphs of finite p-groups and hence it is proved that power graphs of finite p-groups are Laplacian integral.
In the following theorem, the authors gave an expression for Laplacian characteristic polynomial of PðZ n Þ in terms of the characteristic polynomial of L 0, g 1 , g 2 ÁÁÁ, g /ðnÞ ðPðZ n ÞÞ, where g i ði ¼ 1, 2, :::, /ðnÞÞ are the generators of Z n : The following result is a consequence of Theorem 5.2 in which a sharp upper bound of the algebraic connectivity is given. , where p 1 and p 2 are distinct primes and a 1 , a 2 2 N, the algebraic connectivity k nÀ1 ðPðZ n ÞÞ /ðnÞ þ p a 1 À1 Corollary 9.18. [22, Corollary 2.10] For n ¼ p 1 p 2 p 3 , where p 1 , p 2 and p 3 are distinct primes with p 1 < p 2 < p 3 , the algebraic connectivity k nÀ1 ðPðZ n ÞÞ /ðnÞ þ p 1 þ p 2 À 1: In the following theorem, the authors gave a lower bound for the algebraic connectivity of PðZ n Þ, for arbitrary positive integer n ! 2: Theorem 9.19. [22, Theorem 2.12] For each positive integer n ! 2, the algebraic connectivity of PðZ n Þ k nÀ1 ðPðZ n ÞÞ satisfies the inequality k nÀ1 ðPðZ n ÞÞ ! /ðnÞ þ 1. Equality holds if n is either a prime or a product of two distinct primes.
In the following theorem Panda [68] obtained the multiplicity of n as a Laplacian eigenvalue of PðZ n Þ for all n 2 N: & Recall that SðZ n Þ denotes the set of all generators together with the identity of the group Z n and hSðZ n Þ c i is the induced subgraph of the power graph of Z n : Observe that HðhSðZ n Þ c i, x À /ðnÞ À 1Þ equals with the characteristic polynomial of the submatrix of LðPðZ n ÞÞ obtained by deleting rows and columns corresponding to the elements of SðZ n Þ c : Thus, using Theorem 9.12, the authors proved the following lemma.
Hamzeh et al. [47] denoted the set of all cyclic subgroups of finite group G by CðGÞ ¼ fC 1 , C 2 , :::C k g be and D G (renamed for the sake of convenience) be the simple undirected graph with vertex set CðGÞ in which two cyclic subgroups are adjacent if one is contained in other. Let K a i be the complete graph of order a i ¼ /ð C i j jÞ: If K G ¼ fK a i j a i ¼ /ð C i j jÞ, C i 2 CðGÞg, then the power graph PðGÞ is isomorphic to D G -join of K a 1 , K a 2 , :::, K a k , see [39] for details.
Theorem 9.22. [47, Theorem 3.17] The Laplacian spectrum of PðGÞ ¼ D G ½K a 1 , :::, K a k can be calculated as follows: In Theorem 9.12, the Laplacian polynomial of PðZ n Þ was obtained. Hamzeh et al. [47] applied the Theorem 9.22 and provided a complete description of the Laplacian spectrum of PðZ n Þ in [47, Corollary 3.18]. Also in [47], the authors determined the Laplacian spectrum of PðZ n Þ when n is a prime power and a product of two distinct primes using Theorem 9.22.
In the following theorem, Laplacian characteristic polynomial of PðD 2n Þ is calculated in terms of Laplacian characteristic polynomial of PðZ n Þ: for /ðnÞ þ 2 i n À 1 1 for n i 2n À 1 0 for i ¼ 2n: In [22], Laplacian spectrum of PðD 2n Þ were also calculated and the authors proved that the Laplacian spectrum of PðD 2n Þ is the union of that of PðZ n Þ and f2n, 1g: If n is a product of two distinct primes, then by applying Theorems 9.15 and 9.24, we have the following result.
Corollary 9.27. [22, Corollary 3.5] If n ¼ p 1 p 2 , where p 1 and p 2 are distinct primes, then the Laplacian spectrum of PðD 2n Þ is given by In the following lemma, the authors gave bounds for the algebraic connectivity of power graph PðQ 4n Þ of the group Q 4n : Lemma 9.28. [68, Lemma 9] For any integer n ! 2, the algebraic connectivity of PðQ 4n Þ satisfies 1 < k 4nÀ1 ððQ 4n ÞÞ 2: The following theorem, provides the multiplicity of the Laplacian spectral radius of PðQ 4n Þ: Theorem 9.29. [68, Theorem 12] For any integer n ! 2, the Laplacian eigenvalue 4n of PðQ 4n Þ has multiplicity two if Q 4n is a generalized quaternion and one otherwise.
The following result determines when exactly a dicyclic group is a generalized quaternion in terms of its power graph. Let G be a group. For a 2 G, UðaÞ ¼ fh 2 G : a 2 hhig andÛ ðaÞ ¼ UðaÞ n ½a: Let CðaÞ be the subgraph of PðGÞ induced by U(a). We denote the component of P 0 ðGÞ containing a by C(a). For the above subsets and subgraphs, the underlying group will always be clear from context. For g, h 2 G, we say that ½h is a primitive class of g if ½g ¼ ½h p and h 6 ¼ e: We denote the number of primitive classes of any g 2 G by pðgÞ: It should be noted that if G is finite p-group, then for any a 2 G, we cannot have pðaÞ ¼ 0 and o(a) ¼ 1 simultaneously.

Equality of algebraic and vertex connectivity of power graphs
The following lemma proved by Kirkland [54] is useful for the characterization of graphs with equal vertex connectivity and algebraic connectivity.
Theorem 9.39. [54, Theorem 2.1] Let C be a non-complete and connected graph on n vertices. Then jðCÞ ¼ k nÀ1 ðCÞ if and only if it can be written as C 1 Ú C 2 , where C 1 is disconnected graph on n À jðCÞ vertices and C 2 is a graph on jðCÞ vertices with k nÀ1 ðC 2 Þ ! 2jðCÞ À n: In the following theorem, the authors determined all n for which vertex and algebraic connectivity of PðZ n Þ are equal. In the next theorem, the authors proved the equivalence of various properties of Laplacian spectra for power graphs of dicyclic groups. iii. The algebraic connectivity of PðQ 4n Þ is an integer; iv. PðQ 4n Þ is Laplacian integral; v. Q 4n is generalized quaternion.
Next theorem shows that the vertex connectivity and the algebraic connectivity of power graphs of finite p-groups are equal exactly when it is not cyclic. Ankir Raj et al. [75] obtained results on the Laplacian spectrum of PðZ n p m Þ:

Isomorphism of power graphs
In 2010, Cameron [13], proved that if undirected power graphs of two finite groups are isomorphic, then their directed power graphs are also isomorphic. However, it is not true that any from one power graph to another preserves the orientation of edges, and the mentioned result fails for an infinite group. For a counter example one can see [15,16]. In 2019, Cameron et al. [16], considered power graphs of torsion-free groups and proved the following theorems. In [16], some examples were provided to exhibit that even under the hypotheses of Theorems 10.1 and 10.2, PðGÞ ffi PðHÞ need not imply G ffi H: Further examples were also provided to show that some more hypothesis on G is needed for the above property. iii. Any isomorphism from PðGÞ to PðHÞ induces an isomorphism fromPðGÞ toPðHÞ: Moreover, all groups G satisfying the hypothesis have isomorphic power graphs. Zahirovi c [81] gave an affirmative answer to this problem: Theorem 10.6 (Theorem 21, [81]). Let G be a torsion-free group of nilpotency class 2. Let H be a group such that PðGÞ ffi PðHÞ. ThenPðHÞ ffiPðGÞ: Subsequently, Zahirovi c proved a stronger result, applying to all groups, and showing clearly the special role played by the Pr€ ufer groups C p 1: Theorem 10.7. [82, Theorem 3.18] Let G be a group with the following property: G has no subgroup H isomorphic to C p 1 which has trivial intersection with any subgroup K not contained in H. If G 1 is a group with PðGÞ ffi PðG 1 Þ, thenPðGÞ ffiPðG 1 Þ: Corollary 10.8. [82, Corollary 3.19] Any two torsion-free groups having isomorphic power graphs have isomorphic directed power graphs.

Automorphism groups of power graphs
The set of all automorphisms of a graph C forms a permutation group AutðCÞ, acting on the object set VðCÞ, called the automorphism group of C. One can refer to [12] for the terminology and main results of permutation group theory.
Let A and B be permutation groups acting on object sets X and Y respectively. Define is the wreath product of B and A, in its usual imprimitive action.

General remarks
The first thing one notices about automorphism groups of power graphs is that they are extremely large, so that naive analysis with computer algebra software runs into difficulties. For example, the automorphism group of PðA 5 Þ has order 668594111536199848062615552000000: We begin this section by exploring the reason for this phenomenon.
There is one general fact about automorphism groups of power graphs: Theorem 11.1. [14, Section 10] Let G be a non-trivial group. Then AutðPðGÞÞ has a non-trivial normal subgroup which is a direct product of symmetric groups. Hence the direct product of symmetric groups on the twin classes is a normal subgroup of AutðCÞ: Now the theorem above follows from the observation that, in any non-trivial finite group G, the twin relation in the power graph is not the relation of equality. Indeed, recall the relation $ defined by x $ y if hxi ¼ hyi, see Section 5.3. If x $ y, then x and y are closed twins; so the set of generators of each cyclic subgroup is contained in a twin class, necessarily non-trivial if the order of the elements is greater than 2. This covers all cases except that when G is an elementary abelian 2-group. But in this case, the power graph is a star K 1, 2 d À1 , where j G j ¼ 2 d , and clearly all non-identity elements are open twins.
In order to describe further the automorphism group of PðGÞ, we need to be able to describe the quotient group AutðPðGÞÞ=N, where N is the direct product of symmetric groups on the closed twin classes in G. This problem was addressed by Feng et al. [40]; we will state their result later in this section. They describe the quotient as a permutation group on the set of cyclic subgroups of G which preserves order, inclusion and non-inclusion.
Continuing our analysis of PðA 5 Þ : the group A 5 has 15 cyclic subgroups of order 2, 10 of order 3 and 6 of order 5.
So the subgroup N is S 10 2 Â S 6 4 , and the quotient is S 15 Â S 10 Â S 6 ; the product of the orders of these groups is the number quoted earlier.
As we saw in our discussion of the Mathieu group M 11 , the closed twin relation does not necessarily coincide with the relation $, although it contains this relation. We can analyse further. If we collapse each twin class of C to a single vertex, we obtain a new graph D; and, if N is the direct product of symmetric groups on the twin classes, then AutðCÞ=N is a subgroup of AutðDÞ: It may be that D also has twins, in which case the reduction can be repeated until no twins remain. It is easy to show that the final result is independent of the order in which the twin reduction is done. Of course, the final result may be a graph with a single vertex, but it is known when this happens: i. A finite graph is a cograph if and only if the process of iterated twin reduction terminates with the one-vertex graph. ii. The automorphism group of a cograph can be built from the trivial group by the operations "direct product" and "wreath product with a symmetric group".
It is observed in [14] that the power graph of every finite simple group of order less than 2500 is a cograph. But we saw earlier that the power graph of M 11 is not a cograph.
The above results give added importance to Problem 8.6: for which finite groups G is PðGÞ a cograph?

Specific results
The first result about automorphism groups of power graphs was obtained by Cameron and Ghosh [15], where they proved that the only finite group whose automorphism group is the same as that of its power graph is the Klein four-group. Alireza et al. [4], obtained the following fact about the automorphism group of the power graph of the cyclic group Z n : This result can be interpreted in the light of Theorem 11.1, since elements of the same order in a cyclic group are twins (and the identity lies in the twin class of the generators).  In [40], same result is proved independently and they also asserted the following result on the automorphism group of the directed power graphPðD 2n Þ: For n ! 3, Moreover, in [62], the automorphism group of PðZ p 1 p 2 Þ, PðZ p 1 p 2 p 3 Þ and PðZ p 2 1 p 2 2 Þ are computed as follows.
Problem 11.8. [62, Question 3.1]What is the automorphism group of PðGÞ, where G is a sporadic group?
Subsequently A.R. Ashrafi et al. [6] computed the automorphism group of certain finite groups listed below along with their generating relations.
Note that the paper also describes a group U 6n of order 6n which, like V 8n of order 8n, taken from an exercise on page 178 of the book [50]. However, the relations for U 6n are stated incorrectly in [6]. The correct presentation is It seems likely that this is just a transcription error, and that the correct group is analysed in the paper; but since detailed proof is not included, we have not been able to check this.
The results concerning the automorphism group of above finite groups are listed below: ( is an integer such that 3-t. Then iii. For n ¼ 2 k t, with a nonnegative integer k and some positive odd integer t, Ashrafi et al. [6] also compute the automorphism groups of the power graphs of the sporadic simple groups M 11 and J 1 . However, we warn readers that their results are not correct. We have given a correct analysis for M 11 in Section 2 of the present paper. Theorem 11.9(i) was also proved by M. Feng et al. [40]. Also they asserted that the following result for the automorphism group of directed power graphPðQ 4n Þ: For n ! 3, We now turn to the general analysis of the automorphism group of the power graph by Feng et al. [40].
As mentioned earlier, let CðGÞ ¼ fC 1 , :::, C k g be the set of all cyclic subgroups of a group G. For C 2 CðGÞ, let S(C) be the set of all generators of C and SðC i Þ ¼ fSðC i Þ 1 , SðC i Þ 2 , :::, SðC i Þ k g: Define I(G) as the set of permutations r on CðGÞ preserving order, inclusion and non-inclusion, i:e:, C r i ¼ C i j j for each i 2 f1, :::, kg and C i & C j if and only if C r i & C r j : Note that I(G) is a permutation group on CðGÞ: This group induces the faithful action on the set G: We remark that it suffices to consider the action of I(G) on the set of maximal cyclic subgroups of G, since a cyclic group contains at most one cyclic subgroup of each possible order.
For X G, let S X denote the symmetric group on X: Since G is the disjoint union of SðC 1 Þ, :::, SðC k Þ, we get the faithful group action on G: ðSðC i Þ j , ðn 1 , :::, n k Þ ! ðSðC i Þ j Þ n i : With the above notations, we have the following theorem proved by M. Feng et al. [40].
Theorem 11.10. [40, Theorem 2.1] Let G be a finite group. Then where I(G) and Q k i¼1 S SðC i Þ act on G as in (1) and (2), respectively.
In the power graph PðGÞ, for a, b 2 G, define a b if N½a ¼ N½b: Observe that is an equivalence relation. Let cl(a) denote the equivalence class containing a. Write !ðGÞ ¼ fclðaÞ j a 2 Gg ¼ fclðu 1 Þ, :::, clðu ' Þg: Since G is the disjoint union of clðu 1 Þ, :::, clðu ' Þ, the following is a faithful group action on the set G: Similar to the last theorem, we have the following for the automorphism group of undirected power graph.
Theorem 11.11. [40, Theorem 2.2] Let G be a finite group. Then where I(G) and Q ' i¼1 S clðu i Þ act on G as in (1) and (3) respectively.
Our observation that it suffices to consider the action of I(G) on maximal cyclic subgroups now shows that Theorem 11.6 can be derived from this result. For, if G is cyclic, then it has onlly one maximal cyclic subgroup, and so I(G) is the trivial group; thus, AutðGÞ is just the direct product of cyclic groups on the closed twin equivalence classes.

Characterization of finite groups through power graphs
In this section, we present those results by which, one can characterize finite groups in terms of their power graphs and vice versa. Tamizh Chelvam et al. [79] proved the following characterizations for the power graph of an arbitrary finite group. A. Doostabadi et al. [36] characterized all finite groups G whose power graphs are claw-free, K 1, 4 free or C 4 free and they are given below. i. Z p 1 p 2 p 3 ; ii. P Â Q, H p 1 ðPÞ is cyclic and exp ðQÞ ¼ p 2 , in which P is a p 1 group and Q is a p 2 group; iii. p 1 group, where p 1 , p 2 , p 3 are distinct primes.
In the same paper, the authors proved further results for groups G where the center Z(G) is divisible by at least two primes.
Theorem 12.8. [36, Theorem 3.6] Let G be a group with C 4free power graph. If Z(G) is not a p-group, then Theorem 12.9. [36, Theorem 3.7] Let G be a finite group with C 4 -free power graph, which is not prime power group. If Z(G) is a p-group which is not an elementary abelian p-group, then Z(G) is cyclic and exp ðS p 2 ðGÞ Þ ¼ p 2 for every p 1 6 ¼ p 2 . Also, for every p 2 -element a, i. pðC G ðaÞÞ ¼ fp 1 , p 2 g; ii. S p 1 ðC G ðaÞÞ is a normal cyclic subgroup of C G ðaÞ; iii. C G ðaÞ ¼ hai Â S p 2 ðC G ðaÞÞ, or C G ðaÞ ¼ ðhbi Â S p 2 ðC C G ðaÞ ðbÞÞÞ 3 Z p 2 if p 1 is odd prime, where hbi ¼ S p 1 ðC G ðaÞÞ and pðGÞ is the set of all prime divisors of o(G). Mirzargar et al. [64] conjectured that the power graph PðZ n Þ has the maximum number of edges among all power graphs of finite groups of order n. In the following, Pourgholi et al. [74] proved this conjecture for finite simple groups. Amiri et al. [5,Theorem 2] proved that among all finite groups of any given order, the cyclic group of that order has the maximum number of edges in its power graph. Now, we present a characterization of finite groups whose power graph is a union of complete subgraphs which share the identity element of G.

Properties of power graphs
In this section, we collect all the miscellaneous properties of power graphs.

Relationship between power graph and Cayley graph
In 2015, Chattopadhyay [23], obtained some relationship between the power graph and the Cayley graph of a finite cyclic group motivated by an open problem given in survey [2]. For a group G and a subset S of G not containing the identity element e and satisfying S À1 ¼ fa À1 : a 2 Sg ¼ S, the Cayley graph of G with edge set S, Cay(G, S) is an undirected graph with vertex set G and two vertices a, b 2 G are adjacent in Cay(G, S) if and only if ab À1 2 S: Let U n ¼ fa 2 Z n : gcdða, nÞ ¼ 1g: In this subsection, we denote the vertex deleted subgraph PðZ n Þ n SðZ n Þ of PðZ n Þ by P S ðZ n Þ and similarly Cay S ðZ n , U n Þ ¼ CayðZ n , U n Þ n SðZ n Þ: Also, for any graph C, C is the complement of C. , where p 1 , p 2 are distinct primes and a 1 , a 2 are positive integer, then P S ðZ n Þ is a spanning graph of Cay S ðZ n , U n Þ. These two graphs are equal if and only if a 1 ¼ 1 ¼ a 2 : As mentioned in [48], Z n ffi Q m i¼1 Z p a i i as group (under addition) through the isomorphism g : Z n ! Q m i¼1 Z p a i i defined by gð½a n Þ ¼ ð½a p a 1 1 , :::, ½a p am m Þ: Note that the map- i. l 2 ðPðZ p 1 p 2 ÞÞ p 1 þp 2 2 À 2 and l j ðPðZ p 1 p 2 ÞÞ ¼ À1, j ¼ 3, 4, :::, p 1 p 2 À 1; ii. À p 1 þp 2 2 À 2 l p 1 p 2 ðPðZ p 1 p 2 ÞÞ À1; iii. E N ðPðZ p 1 p 2 ÞÞ 2p 1 p 2 þ p 1 þ p 2 À 6 where E N ðPðGÞÞ denotes the energy of the power graph PðGÞ: It is known [55] that, for any two graphs C 1 and

Rainbow connection number of the power graph
A path P in a graph C is called rainbow, if any two edges in P are of different colors. If C has a rainbow path from a to b for each pair of vertices a and b, then C is called a rainbow connected under the coloring f, and f is called a rainbow k-coloring of C.
Ma et al. [58] studied the rainbow connection number of the power graph PðGÞ of a finite group G. In fact, they determined the rainbow connection number of PðGÞ if G has maximal involutions or G is nilpotent, and proved that the rainbow connection number of PðGÞ is at most three if G has no maximal involutions. The rainbow connection numbers of power graphs of some non-nilpotent groups are also determined. The results in this connection are given below. Sehgal and Singh [77] gave a precise formula to count the degree of a vertex in the directed power graphs of finite abelian groups of prime power order. We shall write d þ G ðaÞ, d À G ðaÞ and d 6 G ðaÞ respectively to denote out-degree of a, in-degree of a and the number of bidirectional edges on a in the digraphPðGÞ: By the definition of directed power graphPðGÞ, it is very easy to check that d þ G ðgÞ ¼ j hgi j À 1 ¼ oðgÞ À 1 and d 6 G ðgÞ ¼ /ðoðgÞÞ À 1: We thus precisely obtain d G ðgÞ ¼ oðgÞ À /ðoðgÞÞ þ d À G ðgÞ: To determine the degree d G ðgÞ of a non-identity group element g, it is therefore sufficient to count the in-degree d À G ðgÞ: However, it is easy see that d À G ðgÞ ¼ j fh 2 G : g 6 ¼ h and g 2 hhig j : We start our investigation on the in-degree d À G ðgÞ of a non-identity group element g of G.
Theorem 13.14. [77, Theorem 3.2] Let G ¼ ha 1 i Â ha 2 i Á Á Á Â ha n i be an abelian p-group where a r j j ¼ p m r and 1 m 1 m 2 Á Á Á m n . If a ¼ Q n a¼1 a i a a is a nonidentity element of G and oða i a a Þ ¼ p t a , then d À G ðaÞ ¼ À1 þ /ðoðaÞÞ Y minfm kþ1 Àt k þ1, :::, m n Àt n g b¼0 p P n j¼1 minfm j , bg : In the following, the authors gave a new proof to show that the power graph of a cyclic group of prime-power order is complete, using degree formula. If G is the internal direct product of its normal subgroups with the condition that their orders are mutually relatively prime, then we have the following theorem.
Theorem 13.16. [77, Theorem 3.5] Let G be a finite group and let H 1 , H 2 , :::, H n be normal subgroups of G such that ðoðH i Þ, oðH j ÞÞ ¼ 1 for i 6 ¼ j. If G is internal direct product of subgroups H 1 , H 2 , :::, H n , then, for an element a ¼ a 1 a 2 Á Á Á a n of the group G, with a i 2 H i we have the following:

Product of power graphs
In this subsection, we consider products of power graphs. Let C i ¼ ðV i , E i Þ for i ¼ 1, 2 be two graphs. Then the product graph is defined as the graph C 1 £ C 2 ¼ ðV 1 Â V 2 , EÞ, where ðða 1 , b 1 Þ, ða 2 , b 2 ÞÞ 2 E if one of the following three conditions is true: a 1 ¼ a 2 and ðb 1 , b 2 Þ 2 E 2 ; ða 1 , a 2 Þ 2 E 1 and b 1 ¼ b 2 ; ða 1 , a 2 Þ 2 E 1 and ðb 1 , b 2 Þ 2 E 2 : This graph is referred to in the literature as the strong product. There are many types of products of graphs that are defined between any two graphs. In particular, we will refer the following two products in this paper. See [45, pp. 37, 74, and 49] for further details.
Let C 1 and C 2 be two graphs.
i. The direct product C 1 Â C 2 (also called the categorical product) of C 1 and C 2 is the simple graph with vertex set VðC 1 Â C 2 Þ ¼ VðC 1 Þ Â VðC 2 Þ, in which ða 1 , b 1 Þ is adjacent to ða 2 , b 2 Þ if and only if a 1 is adjacent to a 2 in C 1 and b 1 is adjacent to b 2 in C 2 : ii. The Cartesian product C 1 w C 2 of C 1 and C 2 is the simple graph whose vertex set is VðC 1 w C 2 Þ ¼ VðC 1 Þ Â VðC 2 Þ, in which ða 1 , b 1 Þ is adjacent to ða 2 , b 2 Þ if and only if a 1 ¼ a 2 and b 1 is adjacent to b 2 or a 1 is adjacent to a 2 and b 1 ¼ b 2 : The notation for each of these three products is supposed to suggest the corresponding product of two copies of K 2 ; see Figure 2.
Mukherjee [67] defined the abstract power graph and product of abstract power graphs and called them "strong product". A graph C ¼ ðV, EÞ along with a map f : E ! Figure 2. Products of graphs and groups. PðNÞ, where PðNÞ is the power set of N, is called an abstract power graph if there exists a group H such that C ¼ PðHÞ and f ðg, hÞ ¼ fk 2 N : g k ¼ hg: Let C 1 and C 2 be two abstract power graphs with corresponding edge functions f 1 and f 2 : Then the strong product C 1 C 2 ¼ ðC 1 Â C 2 , E 1 E 2 Þ where ðða 1 , b 1 Þ, ða 2 , b 2 ÞÞ is an edge in E 1 E 2 if f 1 ða 1 , b 1 Þ \ f 2 ða 2 , b 2 Þ 6 ¼ ;: We note the risk of terminological confusion here: Mukherjee's strong product is defined in a different category, namely graphs with edges labelled by sets of natural numbers. We use the symbol for this product. Bhuniya et al. [7] proved that the power graph of the direct product of two groups is not in general isomorphic to the direct, Cartesian or strong product of the power graphs of the factors. Also, they introduced a new product of graphs which they called the generalized product; they proved that power graph of the direct product of groups is isomorphic to the generalized product of their power graphs.

Some more properties of power graphs
In this section, we collect some more important properties of power graphs.  i. PðGÞ is connected; ii. G is periodic; iii. cðPðGÞÞ ¼ 1; iv. diamðPðGÞÞ 2: Theorem 13.32. [1, Theorem 19] If degðaÞ < 1 for every a 2 G, then G is a finite group.
The next theorem is the analogue for infinite groups of Theorem 6.1.
Theorem 13.33. [17, Theorem 10] Let G be an infinite group. Suppose that x 2 G has the property that for all y 2 G, either x is a power of y or vice versa. Assume that x 6 ¼ e. Then the following hold: i. If G is not a torsion group, then G is infinite cyclic and x is a generator; ii. If G is locally finite, then either G ¼ C p 1 for some prime p, and x is arbitrary; or G ¼ Q 2 1 and x has order 2.
i. If a i 6 ¼ 0 for some ið1 i rÞ, then degðaÞ ¼ 1; ii. If a i ¼ 0 for all ið1 i rÞ, then degðaÞ ¼ 2 rþs þ 1: Let G be a non-trivial finite group. Then P 0 ðGÞ is strongly regular if and only if G is a p-group of order p a for which exp ðGÞ ¼ p or p a for some prime p and natural number a: In 2013, Tamizh Chelvam et al. [79] proved some results on the power graphs and they are listed below. i. Let G be a group with at least one non-self-inverse element. Then grðPðGÞÞ ¼ 3; ii. Let G be a finite group with n elements and Z(G) be its center. If degðaÞ ¼ n À 1 in PðGÞ, then a 2 ZðGÞ; iii. Let G be a finite group with n elements. Then PðGÞ is a graph with 1 2 P a6 ¼e oðaÞ edges if and only if every element other than identity of G is of prime order; iv. Let G be a finite group. Then PðGÞ is Eulerian if and only if G is a group of odd order; v. Let G be a finite abelian group. Then PðGÞ is a uni-cyclic graph if and only if G ffi Z 3 : i. Suppose G is a p-group. The power graph PðGÞ is 2 -connected if and only if G is a cyclic or generalized quaternion group. ii. Let G be a finite nilpotent group. If G is not a p-group, then the power graph PðGÞ is 2-connected. iii. Let G and H are groups of finite order such that ðoðGÞ, oðHÞÞ ¼ 1. If G is cyclic of prime order, then PðG Â HÞ is 2-connected. iv. Let G be a finite group with oðGÞ 6 ¼ p, where p is a prime number. If maxfoðaÞ : a 2 Gg ¼ p, then PðGÞ is not 2-connected.

Conclusion and avenues for future research
Algebraic graph theory was initially developed as an intersection of algebra (both abstract and linear) and graph theory. Many concepts of abstract algebra have facilitated the study of graphs from algebraic structures with applications in computer science. On the other hand, graph theory has also helped to characterize certain properties of algebraic structures. We reviewed both the classical as well as the recent results and presentation is centered around a single yet rich structure, namely the power graphs of groups. The literature on the topic of this paper is vast, and we gave almost all the significant results published on power graphs. We hope to have delivered a survey as seen from the perspective of algebraic graph theory, that brought the reader from the basics to the research frontier in power graphs of groups. We want to conclude by listing a few fundamental open problems. Some unsolved problems in this area, have already been presented within the sections, and we also redirect the interested reader to the survey article [2] for open questions that are still unsolved. Further, we compile and list some of the interesting problems. The list is by no means complete and is colored by our own interests and experiences. Problem 14.1. Determine the exact expression for jðPðZ n ÞÞ and dðPðZ n ÞÞ, for n ¼ p a 1 1 p a 2 2 Á Á Á p a m m , where m ! 2, p 1 < p 2 < Á Á Á < p m are primes a i 2 N for all 1 i m: Problem 14.2. [1] Characterize all groups G having the property that the power graph PðGÞ is connected even when the set of vertices in a dominating set is removed from PðGÞ: Ashrafi et al. [6] made the following conjecture. Conjecture 14.3. The automorphism group of the power graph of each group is a member of F, where F denotes the set of all groups that can be constructed from symmetric groups by direct or wreath product.
However, this conjecture is not correct. The computations reported in Section 2 show that the automorphism group of PðM 11 Þ has the group M 11 as a homomorphic image. However, there are many groups G for which PðGÞ is a cograph; and we observed in Theorem 11.2 that the automorphism group of a cograph belongs to the class F: So we regard Problem 8.6 as an important topic for research, and as a substitute for the conjecture of Ashrafi et al. i. The algebraic connectivity of PðZ n Þ is an integer; ii. PðZ n Þ is Laplacian integral; iii. n is a prime power or product of two primes.

Disclosure statement
No potential conflict of interest was reported by the author(s).