Prime graph of a ring R is a graph whose vertex set is the whole set R any any two elements x and y of R are adjacent in the graph if and only if xRy = 0 or yRx = 0. Prime graph of a ring is denoted by PG(R). Directed Prime graphs for non-commutative rings and connectivity in the graph are studied in the present paper. The diameter and girth of this graph are also studied in the paper.

Let G be a finite group. The Prime graph of G is a graph  G (G) with vertex set p (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 G (G)=G (G2 (5)), then G has a normal subgroupN such that p (N)Í{2; 3; 5} and G/N@G2 (5).

Let G be a finite group. We construct the Prime graph of G, which is denoted by  G (G) as follows: the vertex set of this graph is the Prime divisors of |G| and two distinct vertices p and qare joined by an edge if and only if G contains an element of orderpq. In this paper, we determine finite groups G with G (G) =G (L3 (q)), 2£q<100 and prove that if q 6¹2, 3, then L3 (q) is quasirecognizable by the Prime graph, i.e., if G is a finite group with the same Prime graph as the finite simple group L3 (q), then G has a unique non-Abelian composition factor isomorphic to L3 (q).As a consequence of our results we prove that the simple group L3 (4) is recognizable and the simple groups L3 (7) and L3 (9) are 2-recognizable by the Prime graph.

In this study‎, ‎we generalized the co-Prime graph of a finite group called the co-Prime power order graph of a finite group‎. ‎It is denoted by $\beta_G$‎, ‎and its vertex set is $G$‎, ‎such that two distinct vertices $x$ and $y$ are adjacent if and only if $\operatorname{gcd}(|x|,|y|)=p^n$‎, ‎where $p$ is a Prime number‎, ‎and $n \in \mathbb{Z}^{+} \cup\{0\}$‎. ‎We characterized complete graphs and planar graphs on the co-Prime power order graphs‎, ‎and investigated some properties of graph $\beta_G$ for some groups such as cyclic groups‎, ‎dihedral groups‎, ‎and the generalized quaternion groups‎, ‎and obtained the vertex-connectivity among them‎. ‎Finally‎, ‎we characterized some induced subgraphs of co-Prime power order graph for some finite groups.

Throughout this paper, every groups are finite. The Prime graph of a group G is denoted by G(G). Also G is called recognizable by Prime graph if for every finite group H with G (H) = G (G), we conclude that G@H. Until now, it is proved that if k is an odd number and p is an odd Prime number, then PGL (2; pk) is recognizable by Prime graph. So if k is even, the recognition by Prime graph of PGL (2; pk), where p is an odd Prime number, is an open problem. In this paper, we generalize this result and we prove that the almost simple group PGL (2; 25) is recognizable by Prime graph.