The endomorphism monoids of ($n-3$)-regular graphs of order $n$

This paper is motivated by the result of W. Li, that presents an infinite family of graphs - complements of cycles — which possess a regular monoid. We show that these regular monoids are completely regular. Furthermore, we characterize the regular, orthodox and completely regular endomorphisms of the join of complements of cycles, i.e. ($n-3$)-regular graphs of order $n$.