Characterizing finite groups whose enhanced power graphs have universal vertices

Abstract

Let G be a finite group and construct a graph Δ(G) by taking G {1} as the vertex set of Δ(G) and by drawing an edge between two vertices x and y if 〈x, y〉 is cyclic. Let K(G) be the set consisting of the universal vertices of Δ(G) along the identity element. For a solvable group G, we present a necessary and sufficient condition for K(G) to be nontrivial. We also develop a connection between Δ(G) and K(G) when ∣G∣ is divisible by two distinct primes and the diameter of Δ(G) is 2.