Centralizers in finite groups and Domination number of their commuting graphs

The proper commuting graph $\mathcal{C}^{**}(G)$ of a finite group $G$ is the simple graph whose vertices are the noncentral elements of $G$ and two distinct vertices are adjacent if they commute. In this paper, we study the domination number and total domination number of proper commuting graphs of finite groups. We first obtain general bounds for the domination number of proper commuting graphs. For finite nilpotent groups, we exploit a strong product decomposition of commuting graphs to derive exact formulas for the domination number. We further determine the exact domination number and total domination number for proper commuting graphs of several well-known families of finite groups, connecting with the centralizers of those groups.