Let G be a non-abelian group and let Z(G) be the center of G. Associate a graph Γ G (called noncommuting graph of G) with G as follows: Take G\Z(G) as the vertices of Γ G and join two distinct vertices x and y, whenever xy = yx. We want to explore how the graph theoretical properties of Γ G can effect on the group theoretical properties of G. We conjecture that if G and H are two non-abelian finite groups such that Γ G ∼ = Γ H , then |G| = |H |. Among other results we show that if G is a finite non-abelian nilpotent group and H is a group such that Γ G ∼ = Γ H and |G| = |H |, then H is nilpotent.
Let G be a group. The power graph of G is a graph with the vertex set G, having an edge between two elements whenever one is a power of the other. We characterize nilpotent groups whose power graphs have finite independence number. For a bounded exponent group, we prove its power graph is a perfect graph and we determine its clique/chromatic number. Furthermore, it is proved that for every group G, the clique number of the power graph of G is at most countably infinite. We also measure how close the power graph is to the commuting graph by introducing a new graph which lies in between. We call this new graph as the enhanced power graph. For an arbitrary pair of these three graphs we characterize finite groups for which this pair of graphs are equal. *
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.