A New Characterization of Some Simple Groups by Order and Degree Pattern of Solvable Graph

Abstract

The solvable graph of a finite group $G$, denoted by ${\Gamma}_{\rm s}(G)$, is a simple graph whose vertices are the prime divisors of $|G|$ and two distinct primes $p$ and $q$ are joined by an edge if and only if there exists a solvable subgroup of $G$ such that its order is divisible by $pq$. Let $p_1<p_2<\cdots<p_k$ be all prime divisors of $|G|$ and let ${\rm D}_{\rm s}(G)=(d_{\rm s}(p_1), d_{\rm s}(p_2), \ldots, d_{\rm s}(p_k))$, where $d_{\rm s}(p)$ signifies the degree of the vertex $p$ in ${\Gamma}_{\rm s}(G)$. We will simply call ${\rm D}_{\rm s}(G)$ the degree pattern of solvable graph of $G$. In this paper, we determine the structure of any finite group $G$ (up to isomorphism) for which ${\Gamma}_{\rm s}(G)$ is star or bipartite. It is also shown that the sporadic simple groups and some of projective special linear groups $L_2(q)$ are characterized via order and degree pattern of solvable graph.