Volume 36 • Number 1 • 2013   • Recognition of Finite Simple Groups Whose First Prime Graph Components are $r$-Regular Liangcai Zhang, Wujie Shi, Dapeng Yu and Jin Wang Abstract. Let $G$ be a finite group and $\pi(G)=\{p_{1},p_{2},\cdots,p_{s}\}$. For $p\in \pi(G)$, we put $\deg(p):$ $=|\{q\in \pi(G) | p\sim q$ in the prime graph of $G$$\}|$, which is called the degree of $p$. We also define $D(G):=(\deg(p_{1}),\deg(p_{2}),\ldots,\deg(p_{s}))$, where $p_{1}$ < $p_{2}$ < $\cdots$ < $p_{s}$, which is called the degree pattern of $G$. We say $G$ is $k$-fold $OD$-characterizable if there exist exactly $k$ non-isomorphic finite groups having the same order and degree pattern as $G$. In particular, a $1$-fold $OD$-characterizable group is simply called an $OD$-characterizable group. In the present paper, we determine all finite simple groups whose first prime graph components are $1$-regular and prove that all finite simple groups whose first prime graph components are $r$-regular except $U_{4}(2)$ are $OD$-characterizable, where $0\leq r\leq2$. In particular, $U_{4}(2)$ is exactly $2$-fold $OD$-characterizable, which improves an earlier obtained result. 2010 Mathematics Subject Classification: 20D05, 20D06, 20D60 Full text: PDF