Throughout this paper, every groups are finite. The prime graph of a group $G$ is denoted by $Gamma(G)$. Also $G$ is called recognizable by prime graph if for every finite group $H$ with $Gamma(H) = Gamma(G)$, we conclude that $Gcong H$. Until now, it is proved that if $k$ is an odd number and $p$ is an odd prime number, then $PGL(2,p^k)$ is recognizable by prime graph. So if $k$ is even, the r...

The prime graph of a finite group $G$ is denoted by $Gamma(G)$ whose vertex set is $pi(G)$ and two distinct primes $p$ and $q$ are adjacent in $Gamma(G)$, whenever $G$ contains an element with order $pq$. We say that $G$ is unrecognizable by prime graph if there is a finite group $H$ with $Gamma(H)=Gamma(G)$, in while $Hnotcong G$. In this paper, we consider finite groups with the same prime gr...

A graph is said to be symmetric if its automorphism group is transitive on its arcs. A complete classification is given of pentavalent symmetric graphs of order 24p for each prime p. It is shown that a connected pentavalent symmetric graph of order 24p exists if and only if p=2, 3, 5, 11 or 17, and up to isomorphism, there are only eleven such graphs.

let $g$ be a non-abelian group and let $z(g)$ be the center of $g$. associate with $g$ there is agraph $gamma_g$ as follows: take $gsetminus z(g)$ as vertices of$gamma_g$ and joint two distinct vertices $x$ and $y$ whenever$yxneq yx$. $gamma_g$ is called the non-commuting graph of $g$. in recent years many interesting works have been done in non-commutative graph of groups. computing the clique...

Let g be a fixed element of a finite group G. We introduce the g-noncommuting graph of G whose vertex set is whole elements of the group G and two vertices x,y are adjacent whenever [x,y] g  and  [y,x] g. We denote this graph by . In this paper, we present some graph theoretical properties of g-noncommuting graph. Specially, we investigate about its planarity and regularity, its clique number a...

Let $G$ be a finite group. The prime graph of $G$ is a graph $Gamma(G)$ with vertex set $pi(G)$, the set of all prime divisors of $|G|$, and two distinct vertices $p$ and $q$ are adjacent by an edge if $G$ has an element of order $pq$. In this paper we prove that if $Gamma(G)=Gamma(G_2(5))$, then $G$ has a normal subgroup $N$ such that $pi(N)subseteq{2,3,5}$ and $G/Nequiv G_2(5)$.