OD-Characterization of certain four dimensional linear groups with related results concerning degree patterns

The prime graph of a finite group G, which is denoted by GK(G), is a simple graph whose vertex set is comprised of the prime divisors of |G| and two distinct prime divisors p and q are joined by an edge if and only if there exists an element of order pq in G. Let p₁2<?<p_k be all prime divisors of |G|. Then the degree pattern of G is defined as D(G) = (degG(p₁), degG(p₂), ? , degG(p_k)), where degG(p) signifies the degree of the vertex p in GK(G). A finite group H is said to be OD-characterizable if G? H for every finite group G such that |G| = |H| and D(G) = D(H). The purpose of this article is threefold. First, it finds sharp upper and lower bounds on ?(G), the sum of degrees of all vertices in GK(G), for any finite group G (Theorem 2.1). Second, it provides the degree of vertices 2 and the characteristic p of the base field of any finite simple group of Lie type in their prime graphs (Propositions 3.1-3.7). Third, it proves the linear groups L₄(q), q = 19, 23, 27, 29, 31, 32, and 37, are OD-characterizable (Theorem 4.2).