Tarbiat Modares University, Tehran
Abstract: The commuting graph of a group $G$, denoted by $\G(G)$, is a simple graph whose vertices are all non-central elements of $G$ and two distinct vertices $x,y$ are adjacent if $xy=yx$. In  it is conjectured that if $M$ is a simple group and $G$ is a group satisfying $\G(G)\cong\G(M)$, then $G\cong M$. In this paper we prove this conjecture for many simple groups.
Keywords: Simple group, Commuting graph, prime graph, order components
Classification (MSC2000): 20D05; 20D06, 05C25
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