The thorn graph of a given graph is obtained by attaching pendent vertices to each vertex of . The pendent edges, called thorns of , can be treated as or , so that a thorn graph is generalized by replacing by and by and the respective generalizations are denoted by and . The modified eccentric connectivity index of a graph is defined as the sum of the products of eccentricity with the total degree of neighboring vertices, over all vertices of the graph in a hydrogen suppressed molecular structure. In this paper, we give the modified eccentric connectivity index and the concerned polynomial for the thorn graph and the generalized thorn graphs and . 1. Introduction Let be a simple connected graph with vertex set and edge set , so that and . Let the vertices of be labeled as . For any vertex the number of neighbors of is defined as the degree of the vertex and is denoted by . Let denote the set of vertices which are the neighbors of the vertex , so that . Also let , that is, sum of degrees of the neighboring vertices of . The distance between the vertices and is equal to the length of the shortest path connecting and . Also for a given vertex , the eccentricity is the largest distance from to any other vertices of and the sum of eccentricities of all the vertices of is denoted by [1]. The eccentric connectivity index of a graph was proposed by Sharma et al. [2]. A lot of results related to chemical and mathematical study on eccentric connectivity index have taken place in the literature [3–5]. There are numerous modifications of eccentric connectivity index reported in the literature till date. These include edge versions of eccentric connectivity index [6], eccentric connectivity topochemical index [7], augmented eccentric connectivity index [8], superaugmented eccentric connectivity index [9], and connective eccentricity index [10]. A modified version of eccentric connectivity index was proposed by Ashrafi and Ghorbani [11]. Similar to other topological polynomials, the corresponding polynomial, that is, the modified eccentric connectivity polynomial of a graph, is defined as so that the modified eccentric connectivity index is the first derivative of this polynomial for . Several studies on this modified eccentric connectivity index are also found in the literature. In [11], the modified eccentric connectivity polynomials for three infinite classes of fullerenes were computed. In [12], a numerical method for computing modified eccentric connectivity polynomial and modified eccentric connectivity index of one-pentagonal carbon nanocones was presented. In
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