For a group G, we define a graph $\Delta (G)$ by letting $G^{\scriptsize\#}=G{\setminus} \lbrace 1\rbrace $ be the set of vertices and by drawing an edge between distinct elements $x,y\in G^{\scriptsize\#}$ if and only if the subgroup $\langle x,y\rangle $ 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.

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.