The augmented eccentric connectivity index is defined as the summation of the quotients of the product of adjacent vertex degrees and eccentricity of the concerned vertex of a graph which is a generalization of eccentric connectivity index. In this paper we present explicit expressions for the values of augmented eccentric connectivity indices of some particular thorn graphs like thorn path, th...

let g be a simple connected graph and {v1, v2, …, vk} be the set of pendent (vertices ofdegree one) vertices of g. the reduced distance matrix of g is a square matrix whose (i,j)–entry is the topological distance between vi and vj of g. in this paper, we obtain the spectrumof the reduced distance matrix of thorn graph of g, a graph which obtained by attaching somenew vertices to pendent vertice...

For a graph G and any two vertices u and v in G, let d(u, v) denote the distance between u and v and let diam(G) be the diameter of G. A multilevel distance labeling (or radio labeling) for G is a function f that assigns to each vertex of G a positive integer such that for any two distinct vertices u and v, d(u, v)+ | f(u) − f(v) |≥ diam(G) + 1. The largest integer in the range of f is called t...

The Narumi-Katayama index of a graph G is equal to the product of degrees of all the vertices of G. In this paper, we examine the NarumiKatayama index of some derived graphs such as a Mycielski graph, subdivision graphs, double graph, extended double cover graph, thorn graph, subdivision vertex join and edge join graphs.

A path in an edge colored graph is said to be a rainbow path if every edge in this path is colored with the same color. The rainbow connection number of G, denoted by rc(G), is the smallest number of colors needed to color its edges, so that every pair of its vertices is connected by at least one rainbow path. A rainbow u − v geodesic in G is a rainbow path of length d(u, v), where d(u, v) is t...

In this note we introduce Thorn classes of microbundles. We determine the Thorn class of the Whitney sum as the cup product of Thorn classes and state two applications ; one to Gysin sequences of Whitney sums and one to the Atiyah-Bott-Shapiro duality theorem for Thorn spaces (cf. Atiyah [2]). Thus our main result states tha t for microbundles /xi, JU2 over a compact (topological) manifold X, i...

The eccentric connectivity index based on degree and eccentricity of the vertices of a graph is a widely used graph invariant in mathematics. In this paper we present the explicit generalized expressions for the eccentric connectivity index and polynomial of the thorn graphs, and then consider some particular cases.