On a conjecture about the quasirecognition by prime graph of some finite simple groups

It is proved that some finite simple groups are quasirecognizable by prime graph. In [A. Mahmoudifar and B. Khosravi, On quasirecognition by prime graph of the simple groups A(n)(+) (p) and A(n)(-) (p), J. Algebra Appl. 14(1) (2015) 12pp], the authors proved that if p is a prime number and p < 1000, then there exists a natural number m such that for all n >= m, the simple group A(n-1)(+/-) (q) (where A(n-1)(+/-) (q) is a linear or unitary simple group) is quasirecognizable by prime graph. Also, in that paper, the author posed the following conjecture: Conjecture. For every prime power q, there exists a natural number m such that for all n = m, the simple group A(n-1)(+/-) (q) is quasirecognizable by prime graph. In this paper, as the main theorem we prove that if q is a prime power and satisfies some especial conditions, then there exists a number l(q) associated to q such that for all n >= l(q), the finite linear simple group A(n-1)(+) (q) is quasirecognizable by prime graph. Finally, by a calculation via a computer program, we conclude that the above conjecture is valid for the simple group A(n-1)(+) (q), where q = p(alpha), alpha is an odd number and q < 1000.