Let G be a finite abelian group, written additively, and H subgroup of G. The sum graph $$\varGamma _{G,H}$$ is the with vertex set G, in which two distinct vertices x y are joined if $$x+y\in H{\setminus }\{0\}$$ . These graphs form fairly large class Cayley graphs. Among cases have been considered previously prime graphs, case where $$H=pG$$ for some number p. In this paper we present their s...

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...