Graphs with regular monoids
The author proves that the endomorphism monoid of \(\overline{C_n}\) \((n\geq 3)\) is regular where \(\overline{C_n}\) denotes the complement graph of a undirected cycle of \(n\) vertices. It is noted that \(\text{End}(\overline{C_6})\) is not an orthodox monoid. Moreover \(|\text{End}(\overline{C_{2m+1}})|= 4m +2\), \(|\text{End}(\overline{C_{2m}})|= 4(m+m!)\), and \(\text{End}(G)\) is a regular monoid iff \(\text{End}(G+K_n)\) is regular, \(n\geq 1\). In addition the author characterizes regular \(\text{End}(T_1+T_2)\) where \(T_1,T_2\) are trees, using known results on regularity of \(\text{End}(T)\), \(T\) a tree.
scientific article; zbMATH DE number 1915538