Characterizing finite groups whose enhanced power graphs have universal vertices

Let G be a finite group and construct a graph (G) by taking G\1\ as the vertex set of (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 (G) along the identity element. For a solvable group G, we present a necessary and sufficient conditon for K (G) to be nontrivial. We also develop a connection between (G) and K (G) when |G| is divisible by two distinct primes and the diameter of (G) is 2.