The power index of a graph

The {\em power index} $\Theta(\Gamma)$ of a graph $\Gamma$ is the least order of a group $G$ such that $\Gamma$ can embed into the power graph of $G$. Furthermore, this group $G$ is {\em $\Gamma$-optimal} if $G$ has order $\Theta(\Gamma)$. We say that $\Gamma$ is {\em power-critical} if its order equals to $\Theta(\Gamma)$. This paper focuses on the power indices of complete graphs, complete bipartite graphs and $1$-factors. We classify all power-critical graphs $\Gamma'$ in these three families, and give a necessary and sufficient condition for $\Gamma'$-optimal groups.