%0 Journal Article
%T Eccentric Connectivity and Zagreb Coindices of the Generalized Hierarchical Product of Graphs
%A M. Tavakoli
%A F. Rahbarnia
%A A. R. Ashrafi
%J Journal of Discrete Mathematics
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/292679
%X Formulas for calculations of the eccentric connectivity index and Zagreb coindices of graphs under generalized hierarchical product are presented. As an application, explicit formulas for eccentric connectivity index and Zagreb coindices of some chemical graphs are obtained. 1. Introduction All graphs considered here are undirected, simple, and connected. For two vertices and of a graph , the distance is equal to the length of a shortest path connecting and . Suppose that and are the set of vertices and edges of , respectively. For every vertex , the edge connecting and is denoted by and ( for short) denotes the degree of in . The diameter of , denoted by , is the maximum distance among all pairs of vertices in the graph. The first and second Zagreb indices are defined as respectively [1]. The applications of these graph invariants and their mathematical properties are reviewed in two important survey articles [2, 3]. When a topological index or a new graph operation is introduced, then the following mathematical questions are usually raised.(1)What are the extremal properties of this new topological index?(2)Is it possible to find exact formulas for this topological index under old and new graph operations?We refer to [4¨C7] for such questions about Zagreb group indices. The Zagreb indices can be viewed as the contributions of pairs of adjacent vertices to certain degree-weighted generalizations of Wiener polynomials. The first and second Zagreb coindices were first introduced by Do£¿li£¿ [8]. They are defined as follows: In [9], the authors computed exact formulas for these graph parameters under some graph operations. Now we define a vertex version of Zagreb indices as follows: The graph invariants and are called the first and second vertex Zagreb indices of . The eccentricity is the largest distance between and any other vertex of . The total connectivity index of a graph is defined as . Also, the eccentric connectivity index of is defined as [10]. We refer to [11] for a good survey on this topological index. A graph with a specified vertex subset is denoted by . Suppose and are graphs and . The generalized hierarchical product, denoted by , is the graph with vertex set and two vertices and are adjacent if and only if and or and ; see Figure 1. This graph operation was introduced recently by Barri¨¨re et al. [12, 13] and found some applications in computer science. We encourage the reader to consult [14¨C16] for the mathematical properties of the hierarchical product of graphs. The Cartesian product, , of graphs and has the vertex set and is an edge of
%U http://www.hindawi.com/journals/jdm/2014/292679/