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The prime graph of a finite group $G$, which is denoted by ${\rm GK}(G)$, isa simple graph whose vertex set is comprised of the prime divisors of $|G|$ andtwo distinct prime divisors $p$ and $q$ are joined by an edge if and only ifthere exists an element of order $pq$ in $G$. Let $p_1<p_2<...<p_k$ be allprime divisors of $|G|$. Then the degree pattern of $G$ is defined as ${\rmD}(G)=(deg_G(p_1), deg_G(p_2),..., deg_G(p_k))$, where $deg_G(p)$ signifies thedegree of the vertex $p$ in ${\rm GK}(G)$. A finite group $H$ is said to beOD-characterizable if $G\cong H$ for every finite group $G$ such that $|G|=|H|$and ${\rm D}(G)={\rm D}(H)$. The purpose of this article is threefold. First,it finds sharp upper and lower bounds on $\vartheta(G)$, the sum of degrees ofall vertices in ${\rm GK}(G)$, for any finite group $G$ (Theorem 2.1). Second,it provides the degree of vertices 2 and the characteristic $p$ of the basefield of any finite simple group of Lie type in their prime graphs(Propositions 3.1-3.7). Third, it proves the linear groups $L_4(19)$,$L_4(23)$, $L_4(27)$, $L_4(29)$, $L_4(31)$, $L_4(32)$ and $L_4(37)$ areOD-characterizable (Theorem 4.2).

OD-Characterization of Certain Four Dimensional Linear Groups with Related Results Concerning Degree Patterns