On a Special Quotient of the Generating Graph of a Finite Group

Suppose G is a finite group with identity element 1. The generating graph (G) is defined as a graph with vertex set G in such a way that two distinct vertices are connected by an edge if and only if they generate G and the Q-generating graph (G) is defined as the quotient graph , where C(G) is the set of all non-identity conjugacy classes of G and (G)\{1} is a graph obtained from (G) by removing the vertex 1. In this paper, some structural properties of this graph are investigated. The structure of Q-generating graphs of dihedral, semidihedral, dicyclic and all sporadic groups other than M, B and Fi24 is also presented.