Let G be a finite group with |G| = p 1 1 p 2 2 · · · p h h , where p1 < p2 < · · · < ph are prime numbers and 1, 2, . . . , h, h are natural numbers. The prime graph (G) of G is a simple graph whose vertex set is {p1, p2, . . . , ph} and two distinct primes pi and pj are joined by an edge if and only if G has an element of order pipj . The degree degG(pi) of a vertex pi is the number of edges incident on pi, and the h-tuple (degG(p1), degG(p2), . . . , degG(ph)) is called the degree pattern of G. We say that the problem of OD-characterization is solved for a finite group G if we determine the number of pairwise non-isomorphic finite groups with the same order and degree pattern as G. The purpose of this paper is twofold. First, it completely solves the OD-characterization problem for every finite non-Abelian simple groups their orders having prime divisors at most 17. Second, it provides a list of finite (simple) groups for which the problem of OD-characterization have been already solved
