For a simple connected graph G, Eccentric connectivity index denoted by ( ) c G 7 is defined as: ( ) ( ) ( ( ) ( )) c i V G G E i d i 7 , where E(i) is the eccentricity of vertex i, d(i) is the degree of vertex i and V(G) is the set of vertices of G. Eccentricity of vertex i in a graph G is the path length from verte i to vertex j that is farthest from i. In this paper, an algorithm is presente...

The eccentric connectivity index ξ is a distance–based molecular structure descriptor that was recently used for mathematical modeling of biological activities of diverse nature. We prove that the broom has maximum ξ among trees with a fixed maximum vertex degree, and characterize such trees with minimum ξ . In addition, we propose a simple linear algorithm for calculating ξ of trees.

Let G be a connected graph with vertex set V (G) and edge set E(G). The eccentric connectivity index of G, denoted by ξc(G), is defined as ∑ v∈V (G) deg(v)ec(v), where deg(v) is the degree of a vertex v and ec(v) is its eccentricity. In this paper, we propose the edge version of the above index, the edge eccentric connectivity index of G, denoted by ξc e(G), which is defined as ξc e(G) = ∑ f∈E(...

Let $G$ be a non-abelian group. The non-commuting graph $Gamma_G$ of $G$ is defined as the graph whose vertex set is the non-central elements of $G$ and two vertices are joined if and only if they do not commute.In this paper we study some properties of $Gamma_G$ and introduce $n$-regular $AC$-groups. Also we then obtain a formula for Szeged index of $Gamma_G$ in terms of $n$, $|Z(G)|$ and $|G|...