Maximum nullity of some Cayley graphs

Abstract: Recently, the nullity, the algebraic multiplicity of the number zero in the spectrum of the adjacency matrix, of a molecular graph has received a lot of attention as it has a number of direct applications in organic chemistry. In this regard, many researchers have been trying to find an upper or lower bound for the maximum nullity (minimum rank), M() (mr()), for a graph . In this paper, using a wellknown result which presents the spectrum of a Cayley graph in terms of irreducible characters of the underlying group, and using representation and character of groups, we give a lower bound for the maximum nullity of Cayley graph, XS(G), where G = ⟨a⟩ is a cyclic group, or G = G 1 ×⋯ × Gt such that G1 = ⟨a⟩ is a cyclic group and Gi is an arbitrary finite group, for 2 ≤ i ≤ t, with determine the spectrum of Cayley graphs.


Introduction
Motivation for founding the theory of graph spectra has come from applications in Chemistry and Physics. In theoretical chemistry, the -electron energy of a conjugated carbon molecule computed using the Hückel theory. In the Hückel Molecular orbital method, for conjugated hydrocarbons, the energy of the j-th molecular orbital of the so-called -electrons is related to the graph spectra. The nullity of a molecular graph, denoted by N(), is the algebraic multiplicity of the number zero in the spectrum of the adjacency matrix of the molecular graph. Recently, the nullity of a graph has received a lot of attention as it has a number of direct applications in organic chemistry.
For a positive integer n, let S n (ℝ) be the set of all symmetric matrices of order n over the real number. Suppose that A ∈ S n (ℝ). Then the graph of A which is denoted by (A) is a graph with the vertex set {u 1 , … , u n } and the edge set {u i ∼ u j :a ij ≠ 0, 0 ≤ i < j ≤ n}. It should be noted that the diagonal of A has no role in the determining of (A).

PUBLIC INTEREST STATEMENT
Maximum nullity and its related parameter zero forcing number are one of the interesting research areas in graph theory. The notion of maximum nullity was introduced by the wellknown mathematicians AIM Minimum Rank-Special Graphs Work Group. Motivation for founding the theory of graph spectra has come from applications in Chemistry and Physics. Recently, the nullity of a graph has received a lot of attention as it has a number of direct applications in organic chemistry.
The set of symmetric matrices of graph  is the set S() = {A ∈ S n (ℝ):(A) = }. The minimum rank of a graph was first introduced by AIM Minimum Rank-Special Graphs Work Group (2008) and is defined to be the minimum cardinality between the rank of symmetric matrices in S() and denoted by mr(). Similarly, the maximum nullity of  is defined to be the maximum cardinality between the nullity of symmetric matrices in S() and is denoted by M(). Clearly, mr() + M() = n.
One of the most interesting problems on minimum rank is to characterize mr() for graphs. In this regard, many researchers have been trying to find an upper or lower bound for the minimum rank.
The adjacency matrix of a graph  is the matrix A  whose the entry a ij = 1 if and only if vertices u i and u j are adjacent, and a ij = 0 otherwise. The eigenvalues of  are the eigenvalues of A  , and the spectrum of  is the collection of its eigenvalues together with multiplicities. If 1 … , t are distinct eigenvalues of a graph  with respective multiplicity n 1 , … n k , then we denote the spectrum of  by Let G be a group, and let S be a subset of G that is closed under taking inverse and does not contain the identity, e. Then the Cayley graph, X S (G), is the graph with vertex set G and edge set Since S is inverse closed and does not contain the identity, it is a simple fact that X S (G) is undirected and has no loop.
In Babai (1979) presented the spectrum of a Cayley graph in terms of irreducible characters of the underlying group G. The following important theorem was the result of this paper. Theorem 1.1 (Babai, 1979) Let G be a finite group of order n whose irreducible characters (over ℂ) are 1 , … , h with respective degree n 1 , … , n h . Then the spectrum of the Cayley graph X S (G) can be arranged as Λ = { ijk : i = 1, … , h; j, k = 1, … , n i } such that ij1 = … = ijn i (this common value will be denoted by ij ), and for any natural number t.
In this paper, using a well-known result of Babai (1979), we give a lower bound for the maximum nullity of Cayley graph, X S (G), where G = ⟨a⟩ is a cyclic group, or G = G 1 × ⋯ × G t such that G 1 = ⟨a⟩ is a cyclic group and G i is an arbitrary finite group, for some 2 ≤ i ≤ t, with determine the spectrum of Cayley graphs.
Suppose that k is a positive integer. The number of solutions of y 1 + ⋯ + y r ≡ t ( mod k), where y 1 , … , y r and t are belonged to the least non-negative residue system modulo k, is obtained in terms of the von sterneck function, Φ(n, k). In particular, von Sterneck studied the case where the polynomial resulting from the expansion is reduced modulo a positive integer. This function is used in several equivalent forms and in the form used by Hölder (1936), where k and n are positive integers, (n, k) is the greatest common divisor of k and n, (n) is the Eüler totient, and (n) is the Möbius number. In the sequel, the following fundamental result is obtained by Hölder.
Suppose that B(k, n) = t ∈ ℕ : t ≤ n , (t, n) = k , and let = exp(2 i∕n). Then the following function is called Ramanujan sum and is denoted by C(r, n).
In Ramanujan (2000), it was obtained that C(r, n) have only integral values, for some positive integers r and n. Also, (5) and (6) state that Φ(r, n) = C(r, n).
Lemma 2.1 Suppose that n > 1 and d > 1 are two positive integers such that d | n. Also, let B(d, n) = {t ∈ ℕ : t ≤ n , (t, n) = d}. Then If t ∈ B(d, n), then C(t, n) = C(d, n), Proof. The proof is straightforward.
Lemma 2.2 (James & Liebeck, 1993) The irreducible character of G × H is × such that and are the irreducible characters of G and H, respectively. the value of × for any g ∈ G and h ∈ H is ( × )(g, h) = (g) (h).

Main theorems
In the following theorem, we determine the spectrum of Cayley graph X S (G) whose G is a cyclic group of order n. Here, we define F(n i ) = (−1) k i (n)∕ (n i ), where k i is the number of prime factors in the decomposition of n i .
Proof. First, suppose that n is a prime number. Thus X S (G) is isomorphic to the complete graph K n , and so Now, consider the case in that n is not prime. Let For other cases, n The following theorem, which is proven by Akbari & Vatandoost (2017), help us to make a connection between the multiplicity of the eigenvalues of a graph  and its maximum nullity M().
Theorem 3.2 (Akbari, Vatandoost & Golkhandy Pour, 2017) Let  be a graph of order n, and let i be its eigenvalue with respective multiplicity n i . Then M() ≥ n i .
As a result, Theorems 3.1 and 3.2, state the following corollary.
Corollary 3.2.1 Let n be a positive integer and D be its divisors set. Also, let G = ⟨ a ⟩ be a cyclic group of order n, and let S = {a i : i ∈ B(1, n)}. For some prime p and d i ∈ D, the followings are established.
(1) If n has a squared factor, then M � (2) If n is a square free, then M X S (G) ≥ (d i ).
Definition 3.1 Let G be a group, and let S be a subset of G. Also, let Λ = 1 , … , k be the set of irreducible characters with degree 1 of G. A character i ∈ Λ is defined to be an -index character of G, if has the same value on all letters in S; in other word, i ∈ Λ is an -index character of G if (s i ) = , for all s i ∈ S. In the sequel, An -index number of G is defined to be the number of -index characters of G and is denoted by N G ( ).
Theorem 3.3 Let n be a positive integer whose divisors set is denoted by D. Also, let G 1 = ⟨a⟩ be a cyclic group of order n, and let S � = a i : i ∈ B (1, n) . Suppose that G 2 , … , G t are some arbitrary finite groups, and let S k is a subset of G k , for some 2 ≤ k ≤ t. If S = (a i , 1 , … , t ) : a i ∈ S � , k ∈ S k , then for some prime p and d i ∈ D, the followings are established.
where n = 2m + 1 is odd. In this case, D n has m irreducible character of degree 2 and 2 characters of degree 1. See Table 1, for more details.
Example 1 Let G = ⟨ g ⟩ be a group of odd order n, and let S = g i , a j : g i ∈ G, a j ∈ D n , i, j ∈ B(1, n) . Obviously, m+1 and m+2 are two 1-index irreducible characters of D n , and so N D n (1) = 2. Hence, by Theorem 3.3, we have (1) If n has a square factor, then (2) If n is square free, then M X S (G × D n ) ≥ 2 (n) (d i ) .