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 \(\mathrm{End}(X)\) 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 \(Y\).