The Gruenberg–Kegel graph GKG = VGEG of a finite group G is a simple graph with vertex set VG = G, the set of all primes dividing the order of G, and such that two distinct vertices p and q are joined by an edge, p q ∈ EG, if G contains an element of order pq. The degree degGp of a vertex p ∈ VG is the number of edges incident to p. In the case when G = p1 p2 ph with p1 < p2 < · · · < ph, we consider the h-tuple DG = degGp1 degGp2 degGph, which is called the degree pattern of G. The group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups H satisfying condition H DH = G DG. Especially, a 1-fold OD-characterizable group is simply called OD-characterizable. In this paper, we prove that the simple groups L102 and L112 are OD-characterizable. It is also shown that automorphism groups AutLp2 and AutLp+12, where 2p − 1 is a Mersenne prime, are OD-characterizable. Finally, a list of finite (simple) groups which are presently known to be k-fold OD-characterizable, for certain values of k, is presented.
