Topological Indices of Relative g-noncommuting Graph of Dihedral Groups

Abstract

Let G be a finite group, H be a subgroup of G and g be a fixed element of G. The relative g-noncommuting graph Γ(g,H,G) of G is defined as a graph with vertex set is G and two distinct vertices x and y are adjacent if [x, y] ̸= g or [x, y] ̸= g−1, where at least x or y belong to H. In this paper, we will discuss the relative g-non-commuting graph of the dihedral groups D(2n), in particular case when n is an odd number. We give several topological indices of the relative g-noncommuting graph of the dihedral groups D2n including the first Zagreb index, Wiener index, Edge-Wiener index, Hyper-Wiener index, and Harary index.