The power index of a graph

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