Recognition of finite simple groups whose first prime graph components are r-regular

Let G be a finite group and ?(G)={p1,p2,?,ps}. For p??(G), we put deg(p): =|{q??(G)|p?q in the prime graph of G}|, which is called the degree of p. We also define D(G):=(deg(p1),deg(p2),…,deg(ps)), where p1 < p2 < ? < ps, 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 U4(2) are OD-characterizable, where 0?r?2. In particular, U4(2) is exactly 2-fold OD-characterizable, which improves an earlier obtained result.