### Unicyclic Graphs Whose Completely Regular Endomorphisms form a Monoid

Rui Cu, Hailong Hou
<span title="2020-02-13">2020</span> <i title="MDPI AG"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ye33srllvnanjouxn4tmrfgjsq" style="color: black;">Mathematics</a> </i> &nbsp;
In this paper, completely regular endomorphisms of unicyclic graphs are explored. Let G be a unicyclic graph and let c E n d ( G ) be the set of all completely regular endomorphisms of G. The necessary and sufficient conditions under which c E n d ( G ) forms a monoid are given. It is shown that c E n d ( G ) forms a submonoid of E n d ( G ) if and only if G is an odd cycle or G = G ( n , m ) for some odd n ≥ 3 and integer m ≥ 1 .
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.3390/math8020240">doi:10.3390/math8020240</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/llvummkxyzedbm46mmmywqdlxu">fatcat:llvummkxyzedbm46mmmywqdlxu</a> </span>
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