On the graph connectivity and the variable sum exdeg index

Abstract

Topological indices are important descriptors which can be used to characterize the structural properties of organic molecules from different aspects. The variable sum exdeg index $SEI_{a}(G)$ of a graph $G$ is defined as $\sum _{u\in V(G)}d_{G}(u)a^{d_{G}(u)}$, where $d_{G}(u)$ is the degree of vertex $u$ and $a$ is an arbitrary positive real number different from 1. In this paper, we obtain the extremal values of the variable sum exdeg indices (for $a>1$) in terms of the number of cut edges, or the number of cut vertices, or the vertex connectivity, or the edge connectivity of a graph. Furthermore, the corresponding extremal graphs are characterized.