Characterizing finite groups whose enhanced power graphs have universal vertices

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 conditon 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 $2$.