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