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 non-trivial group, $S\subseteq G\setminus \{1\}$ and $S=S^{-1}:=\{s^{-1} \;|\; s\in S\}$. The Cayley graph of $G$ denoted by $\Gamma(S:G)$ is a graph with vertex set $G$ and two vertices $a$ and $b$ are adjacent if $ab^{-1}\in S$. A graph is called integral, if its adjacency eigenvalues are integers. In this paper we determine all connected cubic integral Cayley graphs. We also introduce some infinite families of connected integral Cayley graphs.

We associate a graph N G with a group

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