Endo-Regularity of Generalized Wheel Graphs

A graph $G$ is endo-regular (endo-orthodox, endo-completely-regular) if the monoid of all endomorphisms on $G$ is regular (orthodox, completely regular respectively). In this paper, we characterize endo-regular (endo-orthodox, endo-completely-regular) of generalized wheel graphs $W_n(m)$. For each $m \ge 2,$ we found that the $W_n(m)$ is endo-regular (endo-orthodox resp.) if and only if $n$ is odd and $m = 2$ and $W_n(m)$ is endo-completely-regular if and only if it is $W_3(2).$