Abstract: Let G be a finite group. In this paper we introduce the generalized conjugate graph Γc (G,n) which is a graph whose vertices are all the non-central subsets of G with n elements and two distinct vertices X and Y joined by an edge if X = Y g for some g ∈ G. General properties of the graph such as the number of edges, clique, chromatic, dominating, independence numbers, automorphism and energy of the graph are discussed. We also present a condition under which two generalized conjugate graphs are isomorphic. Moreover, the generalized conjugate graph is a key to define the probability that two subsets of the group G with the same cardinality are conjugate. We obtain some upper and lower bounds for this probability.

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