The commuting graph and a graph associated with centralizers

Let $G$ be a $p$-group. We begin to consider the relationship between the structure of the commuting graph and $\mathrm{\mid }G:Z(G)\mathrm{\mid }$. We also build a family of groups whose commuting graphs have more than one connected component whose diameter is at least $2$. For this, we introduce another graph related to the commuting graph that is associated with centralizers.