Endomorphism spectra of bipartite graphs with diameter three and girth six
The author proves the following: If \(X\) is a bipartite graph with diameter 3 and girth 6, then \(X\) is of endotype 6. This means one has the following relations between the different sets of endomorphisms of \(X\): \(\operatorname{End} X = \operatorname{HEnd} X \neq \operatorname{LEnd} X\neq \operatorname{QEnd} X = \operatorname{SEnd}X = \operatorname{Aut} X\) for the graph \(X\). Definitions of these sets of endomorphisms can be found, for example, in [\textit{M. Böttcher} and \textit{U. Knauer}, Endomorphism spectra of graphs, Discrete Math. 109, No. 1-3, 45-57 (1992; Zbl 0792.05135)]. Note: An example in [\textit{U. Knauer}, Endomorphism types of bipartite graphs, in: Words, languages and combinatorics II, World Scientific, Singapore, 234-251 (1994; Zbl 0877.05051)] shows that the above result does not characterize bipartite graphs with endotype 6.
scientific article; zbMATH DE number 1747992