摘要
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 P1 〈 P2 〈 … 〈 Pk be all prime divisors of ｜G｜- Then the degree pattern of G is defined as D（G） = （degG（pl）, degG（p2）,..., degG（pk））, 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 v（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）.
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 P1 〈 P2 〈 … 〈 Pk be all prime divisors of ｜G｜- Then the degree pattern of G is defined as D（G） = （degG（pl）, degG（p2）,..., degG（pk））, 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 v（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）.
