A New Characterization of Some Simple Groups by Order and Degree Pattern of Solvable Graph

The solvable graph of a finite group G, denoted by Gamma(s)(G), is a simple graph whose vertices are the prime divisors of vertical bar G vertical bar 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 p(1) < p(2) < ... < p(k) be all prime divisors of vertical bar G vertical bar and let D-s(G) = (ds(p(1)), d(s)(P-2), ... , ds(p(k))), where d(s)(p) signifies the degree of the vertex p in Gamma(s) (G). We will simply call D-s(G) the degree pattern of solvable graph of G. In this paper, we determine the structure of any finite group G (up to isomorphism) for which rs(G) is star or bipartite. It is also shown that the sporadic simple groups and some of projective special linear groups L-2(q) are characterized via order and degree pattern of solvable graph.