Characterizing finite groups whose enhanced power graphs have universal vertices

Let $G$ be a finite group and construct a graph $\mathrm{\Delta }(G)$ by taking $G\setminus \{1\}$ as the vertex set of $\mathrm{\Delta }(G)$ 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 $\mathrm{\Delta }(G)$ 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 $\mathrm{\Delta }(G)$ and $K(G)$ when $\mathrm{\mid }G\mathrm{\mid }$ is divisible by two distinct primes and the diameter of $\mathrm{\Delta }(G)$ is $2$.