Some Simple Groups Which are Determined by Their Orders and Degree-Patterns

Let G be a finite group and Irr(G) the set of all irreducible complex characters of G. Let cd(G) be the set of all irreducible complex character degrees of G and denote by rho(G) the set of all primes which divide some character degree of G. The character-prime graph Gamma (G) associated to G is a simple undirected graph whose vertex set is rho(G) and there is an edge between two distinct primes p and q if and only if the product pq divides some character degree of G. We show that the finite nonabelian simple groups A(7) , J(1) , J(3) , J(4) , L-3 (3) and U-3 (4) are uniquely determined by their degree-patterns and orders.