Eccentric harmonic index of a graph

Abstract In this paper, we introduce the eccentric harmonic index of a graph G, so that it is the sum of the terms for the edges where ei is the eccentricity of the ith vertex of the graph G. We compute the exact values of He for some standard graphs. Bounds for He are established. Relationships between He and the eccentric connectivity index are derived.


Introduction
In this paper, all graphs are assumed to be finite simple connected graphs. A graph G ¼ ðV, EÞ is a simple graph, that is, having no loops, no multiple and directed edges. As usual, we denote n to be the order and m to be the size of the graph G. For a vertex v 2 V, the open neighborhood of v in a graph G, denoted N(v), is the set of all vertices that are adjacent to v and the closed neighborhood of v is N½v ¼ NðvÞ [ fvg: The degree of a vertex v i in G is d i ¼ dðv i Þ ¼ jNðv i Þj: A vertex of degree one is called pendant vertex. A graph G is said to be k-regular graph if d(v) ¼ k for every v 2 VðGÞ: The distance d (u, v) between any two vertices u and v in a graph G is the length of the shortest path connecting them. The eccentricity of a vertex v 2 VðGÞ is eðvÞ ¼ maxfdðu, vÞ : u 2 VðGÞg: The radius of G is r ¼ rðGÞ ¼ minfeðvÞ : v 2 VðGÞg and the diameter of G is diamðGÞ ¼ maxfeðvÞ : v 2 VðGÞg: Hence rðGÞ eðvÞ diamðGÞ, for every v 2 VðGÞ: A vertex v in a connected graph G is central vertex if eðvÞ ¼ rðGÞ, while a vertex v in a connected graph G is peripheral vertex if eðvÞ ¼ diamðGÞ: A graph G is called a selfcentered graph if eðvÞ ¼ rðGÞ ¼ diamðGÞ for all v 2 VðGÞ: If G is a regular graph with eðvÞ ¼ rðGÞ ¼ diamðGÞ for all v 2 VðGÞ, then G is a regular self-centered graph. We denote the eccentricity of a vertex v i by eðv i Þ ¼ e i : As usual we use the characters T, P n , C n , K a, b , K 1, nÀ1 , K n for the tree, path, cycle, complete bipartite, star and complete graph, respectively. All the definitions and terminologies about the graph in this paragraph is available in Harary, (1969).
A single number representing a chemical structure, by means of the corresponding molecular graph, is known as topological descriptor. Topological descriptors play a prominent role in mathematical chemistry, particularly in studies of quantitative structure property and quantitative structure activity relationships. Moreover, a topological descriptor is called a topological index if it has a mutual relationship with a molecular property. Thus, since topological indices encode some characteristics of a molecule in a single number, they can be used to study physicochemical properties of chemical compounds (Hernndez-Gmez, Mndez-Bermdez, Rodrguez, & Sigarreta, 2018).
After the seminal work of Wiener (Wiener, 1947), many topological indices have been defined and analyzed. Among all topological indices, probably the most studied is the Randi c connectivity index (R) (Randi c, 1975). Several hundred papers and, at least, two books report studies of R (see, e.g. Gutman & Furtula, 2008;Li & Gutman, 2006 and references therein). Moreover, with the aim of improving the predictive power of R, many additional topological descriptors (similar to R) have been proposed. In fact, the first and second Zagreb indices, M 1 and M 2 , respectively, can be considered as the main successors of R. They are defined as Both M 1 and M 2 have recently attracted much interest (see, e.g. Borovi canin & Furtula, 2016; Das, 2010;Das, Gutman, & Furtula, 2011; in particular, they are included in algorithms used to compute topological indices).
Sharma et al. introduced the eccentric connectivity index of a graph (Sharma, Goswami, & Madan, 1997), where they defined it for a graph with n vertices and m edges, as In analogy with the harmonic index and its applications, we introduce the eccentric harmonic index as an eccentric version of the harmonic index. Also, the relation between the eccentric harmonic index and the eccentric connectivity index motivates us to study the eccentric connectivity index and its applications in another way.

Eccentric harmonic index of a graph
In this section, we define the eccentric harmonic index H e ðGÞ of a graph G. The eccentric harmonic index of some well-known graphs are computed. The starting is with the definition of H e ðGÞ which is explained in the following definition.
Definition 2.1. Let G be a graph with n vertices and m edges. Then the eccentric harmonic index H e ðGÞ of G is defined as Theorem 2.2. Let G be a self-centered graph of order n and size m. Then H e ðGÞ ¼ m diamðGÞ : Proof. Let G be a self-centered graph of order n and size m.
Then Proof. Let P n be a path with vertex set fv 1 , v 2 , . . . , v n g, n ! 2 and assume that n is odd.
Then, for n ¼ 3, Assume that it is true for n ¼ k, k is odd, i.e.
where r is the radius of G. For n ¼ k þ 2, the radius is r ¼ kþ2À1 Thus, we show the part when n is odd. For the part, when n is even, the proof is similar with a little difference for the n 2 , n 2 þ 1 vertices; which are the central vertices of the path; with eccentricities n 2 , n 2 : By putting this term; which equals to 1 n 2 þ n 2 ¼ 1 n ; outside the summation, then the result follows.

Bounds for eccentric harmonic index of a graph
In this section, we derive upper and lower bounds for H e ðGÞ of a graph G. Relations between H e ðGÞ and the eccentric connectivity index n c ðGÞ are established. Hence the result follows.
To show the equality, it is clear that the equality holds if and only if e i þ e j ¼ 2r ¼ 2diamðGÞ, which holds if and only if G is a self-centered graph.
Theorem 3.2. Let G be a graph with n vertices and m edges. Then H e ðGÞ ! mð1 À ln ðdiamðGÞÞÞ, with equality holds if and only if G is a complete graph.
Proof. Let G be a graph of order n and size m. Assume the function f ðxÞ ¼ x À ln x À 1: Easy calculations gives f ðxÞ ! 0: Hence, for v i v j 2 E 2 e i þ e j À ln 2 e i þ e j À 1 ! 0: So, 2 e i þ e j ! 1 þ ln 2 e i þ e j : By taking the summation over the edges of the graph, we get Hence, H e ðGÞ ! mð1 À ln ðdiamðGÞÞÞ: To show the equality, let f(x) ¼ 0, then x ¼ 1. So 2 e i þe j ¼ 1, v i v j 2 E: Hence e i þ e j ¼ 2, which holds if and only if e i ¼ e j ¼ 1 for all i, j ¼ 1, . . . , n: Thus G is complete. Proof. Let G be a graph of order n, n ! 2 and size m. Assume the function f ðxÞ ¼ x þ 1 x À 2: Easy calculations gives f ðxÞ ! 0: Thus, for v i v j 2 E 2 e i þ e j þ e i þ e j 2 À 2 ! 0: By taking the summation over the edges of the graph G, we get H e ðGÞ þ n c ðGÞ 2 À X v i v j 2E 2 ! 0: Hence, H e ðGÞ ! 2m À n c ðGÞ 2 : To show the equality, let f(X) ¼ 0, then x ¼ 1. The rest of the proof is similar to that in Theorem 3.2.
Theorem 3.4. Let G be a graph with n ! 2 vertices and m edges. Then H e ðGÞ H e ðK n Þ: Proof. Let G be a graph of order n, n ! 2 and size m. Then for v i v j 2 E 2 e i þ e j 1: By taking the summation over the edges, we get The bound is sharp and the self-centered graph satisfies it.
Proof. Let G be a graph of order n ! 2, and size m. By Cauchy-Schwartz inequality we obtain,