OD-Characterization of almost simple groups related to $L_{3}(25)$

Abstract

Let $G$ be a finite group and $pi(G)$ be the set of all the prime‎ ‎divisors of $|G|$‎. ‎The prime graph of $G$ is a simple graph‎ ‎$Gamma(G)$ whose vertex set is $pi(G)$ and two distinct vertices‎ ‎$p$ and $q$ are joined by an edge if and only if $G$ has an‎ ‎element of order $pq$‎, ‎and in this case we will write $psim q$‎. ‎The degree of $p$ is the number of vertices adjacent to $p$ and is‎ ‎denoted by $deg(p)$‎. ‎If‎ ‎$|G|=p^{alpha_{1}}_{1}p^{alpha_{2}}_{2}...p^{alpha_{k}}_{k}$‎, ‎$p_{i}^{,}$s different primes‎, ‎$p_{1}<p_{2}<...<p_{k}$‎, ‎then‎ ‎$D(G)=(deg(p_{1}),deg(p_{2}),...,deg(p_{k}))$ is called the degree‎ ‎pattern of $G$‎. ‎A finite group $G$ is called $k$-fold‎ ‎OD-characterizable if there exist exactly $k$ non-isomorphic‎ ‎groups $S$ with $|G|=|S|$ and $D(G)=D(S)$‎. ‎In this paper‎, ‎we‎ ‎characterize groups with the same order and degree‎ ‎pattern as an almost simple groups related to $L_{3}(25)$‎.