Properties and families of minimal prime graphs

In this paper we continue the study of prime graphs of finite solvable groups. The prime graph, or Gruenberg-Kegel graph, of a finite group G has as its vertices the prime divisors of the order of G, and there is an edge between two such primes p and q if and only if G contains an element of order pq. Since the discovery of a simple, purely graph theoretical characterization of the prime graphs of solvable groups in 2015 these graphs have been studied in more detail from a graph theoretic angle. In this paper we explore several new aspects of these graphs. We prove that every vertex in a minimal prime graph is contained in a 5-cycle thereby correctingand extending an earlier result from 2015. Moreover, we establish a sharp lower bound for the number of edges of minimal prime graphs. We also study families of minimal prime graphs using the extremal property of superminimality, positively answering a conjecture by Huang et al. that one such family consists of base graphs, and we construct a new family of uniquely colorable superminimal prime graphs.