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).

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.

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.

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.