Recognition of PSL(2; p) and PGL(2; p) by orders and degree of vertex p in vanishing prime graph

For a finite group G, an element g is called a vanishing element of G whenever there is an irreducible character Χ in Irr(G) such that Χ(g) = 0. We denote by Vo(G) the set of orders of vanishing elements of G. Th e vanishing prime graph of G is denoted by Γ(G). Th e vertex set of Γ(G) is the set of all prime divisors of elements in Vo(G) and two distinct primes q and r are adjacent if and only if qr divides an element of Vo(G). In this paper, we determine the structure of a finite group G with |G| = |L|, such that L is isomorphic to either the simple group PSL(2; p) or the almost simple group PGL(2; p), where p ≥ 5 is a prime number and p is an isolated vertex in the vanishing prime graph of G.