Commutativity pattern of finite non-abelian pp-groups determine their orders

Let $G$ be a non-abelian group and $Z(G)$ be the center of $G$. Associate a graph $\Gamma_G$ (called non-commuting graph of $G$) with $G$ as follows: take $G\setminus Z(G)$ as the vertices of $\Gamma_G$ and join two distinct vertices $x$ and $y$, whenever $xy\neq yx$. Here, we prove that "the commutativity pattern of a finite non-abelian $p$-group determine its order among the class of groups"; this means that if $P$ is a finite non-abelian $p$-group such that $\Gamma_P\cong \Gamma_H$ for some group $H$, then $|P|=|H|$.Comment: to appear in Communications in Algebr

A note on the coprime graph of a group

In this paper we study the coprime graph of a group $G$. The coprime graph of a group $G$, is a graph whose vertices are elements of $G$ and two distinct vertices $x$ and $y$ are adjacent iff $(|x|,|y|)=1$. In this paper we classify all the groups which the coprime graph is a complete r-partite graph or a planar graph. Also we study the automorphism group of the coprime graph

Some results on the comaximal ideal graph of a commutative ring

Let R R be a commutative ring with unity. The comaximal ideal graph of R R, denoted by C(R) C(R), is a graph whose vertices are the proper ideals of R R which are not contained in the Jacobson radical of R R, and two vertices I 1 I1 and I 2 I2 are adjacent if and only if I 1 +I 2 =R I1+I2=R. In this paper, we classify all comaximal ideal graphs with finite independence number and present a formula to calculate this number. Also, the domination number of C(R) C(R) for a ring R R is determined. In the last section, we introduce all planar and toroidal comaximal ideal graphs. Moreover, the commutative rings with isomorphic comaximal ideal graphs are characterized. In particular we show that every finite comaximal ideal graph is isomorphic to some C(Z n ) C(Zn)