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Let G be a finite group. In this paper we introduce the generalized conjugate graph Γ c (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 graphsdoi:10.1515/gmj-2014-0062 fatcat:7snpsq66t5gjpoey2zlnb7vbba