Some Families of Fixed Points for the Eccentric Digraph Operator

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.