The power index of a graph

Abstract

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

12 pages, 1 figure