The Automorphism Groups of the Involution G-Graph and Cayley Graph

Let G be a finite group and Φ(G, S) is the G−graph of a group G with respect to a non-empty subset S. The aim of this paper is to study the structure and the automorphism group of a simple form of G−graph for some finite groups like alternating group, dihedral, semi-dihedral, dicyclic, Zm δ Z2n, where δ is inverse mapping and V8n = {a, b|a 2n = b 4 = 1, aba = b −1 , ab −1 a = b}. Then we compare it with the automorphism group of the corresponding Cayley graph. Also we study the structure of involution G−graphs when S = Inv is the set of all involutions of G.