Eccentric distance sum and adjacent eccentric distance sum index of complement of subgroup graphs of dihedral group

Abstract

Let G = (V(G),E(G)) is a connected simple graph. Let ec(v) is the eccentricity of vertex v, D(v) = Σu∈V(G) d(u,v) is the sum of all distances from vertex v and deg(v) is the degree of vertex v in G. The eccentric distance sum index of G is defined as ξd (G) = Σv∈V(G) ec(v)D(v) and the adjacent eccentric distance sum index of G is defined as ξsv(G)=Σv∈Vec(v)D(v)/deg(v). For positive integer m and m ≥ 3, let D 2m be dihedral group of order 2m and N is a normal subgroup of D 2m. The subgroup graph ΓN(D 2m) of dihedral group D 2m is a simple graph with vertex set D 2m and two distinct vertices x and y are adjacent if and only if xy ∈ N. In the present paper, we compute eccentric distance sum and adjacent eccentric distance sum index of complement of subgroup graph of dihedral group D 2m. Total eccentricity, eccentric connectivity index, first Zagreb index, and second Zagreb index of these graphs are also determined.