let $r$ be a ring with unity. the undirected nilpotent graph of $r$, denoted by $gamma_n(r)$, is a graph with vertex set ~$z_n(r)^* = {0neq x in r | xy in n(r) for some y in r^*}$, and two distinct vertices $x$ and $y$ are adjacent if and only if $xy in n(r)$, or equivalently, $yx in n(r)$, where $n(r)$ denoted the nilpotent elements of $r$. recently, it has been proved that if $r$ is a left ar...

given a non-abelian finite group $g$, let $pi(g)$ denote the set of prime divisors of the order of $g$ and denote by $z(g)$ the center of $g$. thetextit{ prime graph} of $g$ is the graph with vertex set $pi(g)$ where two distinct primes $p$ and $q$ are joined by an edge if and only if $g$ contains an element of order $pq$ and the textit{non-commuting graph} of $g$ is the graph with the vertex s...

let $g$ be a non-abelian finite group. in this paper, we prove that $gamma(g)$ is $k_4$-free if and only if $g cong a times p$, where $a$ is an abelian group, $p$ is a $2$-group and $g/z(g) cong mathbb{ z}_2 times mathbb{z}_2$. also, we show that $gamma(g)$ is $k_{1,3}$-free if and only if $g cong {mathbb{s}}_3,~d_8$ or $q_8$.