The Vertex Degree of Relative G-Noncommuting Graph of the Dihedral Groups

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 \(\Gamma _{g, H, G}\) of G is defined as a graph with the vertex set G where two distinct vertices x and y are adjacent if \([x,y] \ne g\) and \([x,y] \ne g^{-1}\) , and at least one of x or y belongs to H. This paper will discuss the vertex degree of the relative g-noncommuting graph for the dihedral group \(D_{2n}\) , focusing specifically on cases where n is an even number. In this dihedral group, only two types of subgroups will be discussed, namely \(H=\langle a\rangle \) and \(H=\{e, a^j b \mid a,b \in G \}\) for some \(j=0,1, \dots , n-1\) . Additionally, we will examine several topological indices of the relative g-noncommuting graph, including the first Zagreb index, the Wiener index, the edge Wiener index, the Hyper Wiener index, and the Harary index.