On Recognition of Finite Simple Groups by Order and Degree Pattern of Solvable Graph

The solvable graph of a finite group G, denoted by s(G), is a simple graph whose vertices are the prime divisors of jGj and two distinct primes p and q are joined by an edge if and only if there exists a solvable subgroup of G such that its order is divisible by pq. Let p1 p2 pk be all prime divisors of jGj and let Ds(G) = (ds(p1); ds(p2); : : : ; ds(pk)), where ds(p) signifies the degree of the vertex p in s(G). We will simply call Ds(G) the degree pattern of solvable graph of G. The purpose of this article is twofold. First, it provides the structure of any finite group G (up to isomorphism) for which s(G) is star or bipartite. Second, it is proved that the sporadic simple groups and some of projective special linear groups L2(q) are characterized via order and degree pattern of solvable graphs.