• Bulletin of the Korean Mathematical Society
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• v.51 no.4
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• pp.1145-1153
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• 2014

Let G be a finite group and X be a union of conjugacy classes of G. Define C(G,X) to be the graph with vertex set X and $x,y{\in}X$ ($x{\neq}y$) joined by an edge whenever they commute. In the case that X = G, this graph is named commuting graph of G, denoted by ${\Delta}(G)$. The aim of this paper is to study the automorphism group of the commuting graph. It is proved that Aut(${\Delta}(G)$) is abelian if and only if ${\mid}G{\mid}{\leq}2$; ${\mid}Aut({\Delta}(G)){\mid}$ is of prime power if and only if ${\mid}G{\mid}{\leq}2$, and ${\mid}Aut({\Delta}(G)){\mid}$ is square-free if and only if ${\mid}G{\mid}{\leq}3$. Some new graphs that are useful in studying the automorphism group of ${\Delta}(G)$ are presented and their main properties are investigated.

• Communications of the Korean Mathematical Society
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• v.27 no.1
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• pp.57-68
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• 2012

The commuting graph of an arbitrary ring R, denoted by ${\Gamma}(R)$, is a graph whose vertices are all non-central elements of R, and two distinct vertices a and b are adjacent if and only if ab = ba. In this paper, we investigate the connectivity, the diameter, the maximum degree and the minimum degree of the commuting graph of group ring $Z_nQ_8$. The main result is that $\Gamma(Z_nQ_8)$ is connected if and only if n is not a prime. If $\Gamma(Z_nQ_8)$ is connected, then diam($Z_nQ_8$)= 3, while $\Gamma(Z_nQ_8)$ is disconnected then every connected component of $\Gamma(Z_nQ_8)$ must be a complete graph with a same size. Further, we obtain the degree of every vertex in $\Gamma(Z_nQ_8)$, the maximum degree and the minimum degree of $\Gamma(Z_nQ_8)$.

• Bulletin of the Korean Mathematical Society
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• v.50 no.1
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• pp.117-123
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• 2013

Let G be a finite non-abelian group. We define the non-commuting graph ${\nabla}(G)$ of G as follows: the vertex set of ${\nabla}(G)$ is G-Z(G) and two vertices x and y are adjacent if and only if $xy{\neq}yx$. In this paper we prove that if G is a finite group with $${\nabla}(G){\simeq_-}{\nabla}(\mathbb{A}_{22})$$, then $$G{\simeq_-}\mathbb{A}_{22}$$where $\mathbb{A}_{22}$ is the alternating group of degree 22.