THE TOPOLOGICAL INDICES OF THE CAYLEY GRAPHS OF DIHEDRAL GROUP \(D_{2n}\) AND THE GENERALIZED QUATERNION GROUP \(Q_{2^n}\)

Abstract

A topological index of a simple connected graph Γ is a numeric quantity related to the structure of the graph Γ.  The set of all automorphisms of Γ under the composition of mapping forms a group which is denoted by Aut(Γ).  Let G be a group, and let S ⊂ G be a set of group elements such that the identity element 1 ∉ S.  The Cayley graph associated with (G, S) is defined as the directed graph with vertex set G and edge set E such that e = xy is an edge of E if (x⁻¹y) ∈ S for every vertices x, y in G.  In this paper we define the Cayley graph of the Dihedral group \(D_{2n}\) and the Cayley graph of the generalized quaternion group \(Q_{2^n}\) on the specified subsets of these groups, and compute the Wiener, Szeged and PI indices of these graphs.