The Connectivity and Wiener Index of Order Graph in Symmetric Group

Abstract

Let G be a finite group and x is an element of G. Then, the order graph of a finite group denoted by r_OG, is a digraph and for any two distinct vertices x and y, there is an edge from x to y if and only if x divide y. The Wiener index is defined as the summation of distances between all pairs of vertices in a graph. It is one of the topological indices which can be used for analyzing intrinsic properties of molecule structure in chemistry. In this paper, the connectivity and Wiener index of r_OG are evaluated from the order graph of symmetric groups of degree up to10.