Some Families of Fixed Points for the Eccentric Digraph Operator

Abstract

We investigate the existence of fixed point families for the eccentric digraph ($\text{ED}$) operator, which was introduced in $[1]$. In $[2]$, the notion of the period $\rho (G)$ of a digraph $G$ (under the $\text{ED}$ operator) was defined, and it was observed, but not proved, that for any odd positive integer $m$, $C_m\times C_m$ is periodic, and that $\rho (\text{ED}(C_m\times C_m))=2\rho (\text{ED}(C_m))$. Also in $[2]$, the following question was posed: which digraphs are fixed points under the digraph operator? We provide a proof for the observations about $C_m\times C_m$, and in the process show that these products comprise a family of fixed points under $\text{ED}$. We then provide a number of other interesting examples of fixed point families.