LAUSR

There are various graphs associated with groups which have been studied in the literature, e.g., Cayley graph, commuting graph, generating graph, prime graph. An interesting problem in this respect is to investigate the relation between a group and the associated graph. For instance, an algebraic characterization was given in [6] for the generating sets of Cayley graphs with neighbor connectivity one for a class of groups including abelian groups. Whereas, in [1], the finite quasisimple groups with perfect commuting graphs were classified. The notion of the directed power graph of a group was introduced by Kelarev and Quinn in [7] (also, see [8] for a semigroup). The underlying undirected graph, simply termed as power graph, was first considered by Chakrabarty et al. in [3]. The power graph of a group G, denoted by P(G), is the simple graph with vertex set G and two vertices u and v are adjacent if u = v or v = u for some positive integers k and l. By considering the aforementioned problem for power graphs, Cameron [2] proved that if the power graphs of two finite groups are isomorphic, then they have the same numbers of elements of each order. Curtin and Pourgholi [5] showed that among all finite groups of a given order, the cyclic group of that order has the maximum number of edges in its power graph. Moreover, in [9], Ma and Feng classified all finite groups whose power graphs are uniquely colorable, split or unicyclic. In [11], Panda characterized the finite noncyclic groups of prime exponent in terms of the edge connectivity of their power graphs. In the present paper, working further on the relation between a group and its power graph, we characterize the finite nilpotent groups whose power graphs have equal vertex connectivity and minimum degree.