End-Regular and End-Orthodox Joins of Split Graphs

Abstract

A graph $X$ is said to be End-regular (resp., End-orthodox, End-inverse) if its endomorphism monoid $\mathrm{E}\mathrm{n}\mathrm{d}(X)$ is a regular (resp., orthodox, inverse) semigroup. In this paper, End-regular (resp., End-orthodox, End-inverse) graphs which are the join of split graphs $X$ and $Y$ are characterized. It is also proved that $X+Y$ is never End-inverse for any split graphs $X$ and $Y$.