The Cyclic Graph of a Z-group

For a group $G$, we define a graph $Delta(G)$ by letting $G^{#} = G setminus { 1 }$ be the set of vertices and by drawing an edge between distinct elements $x,yin G^{#}$ if and only if the subgroup $langle x,yrangle$ is cyclic. Recall that a $Z$-group is a group where every Sylow subgroup is cyclic. In this short note, we investigate $Delta(G)$ for a $Z$-group $G$.