Unicyclic Graphs Whose Completely Regular Endomorphisms form a Monoid

oleh: Rui Gu, Hailong Hou

Format: Article
Diterbitkan: MDPI AG 2020-02-01

Deskripsi

In this paper, completely regular endomorphisms of unicyclic graphs are explored. Let <i>G</i> be a unicyclic graph and let <inline-formula> <math display="inline"> <semantics> <mrow> <mi>c</mi> <mi>E</mi> <mi>n</mi> <mi>d</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> be the set of all completely regular endomorphisms of <i>G</i>. The necessary and sufficient conditions under which <inline-formula> <math display="inline"> <semantics> <mrow> <mi>c</mi> <mi>E</mi> <mi>n</mi> <mi>d</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> forms a monoid are given. It is shown that <inline-formula> <math display="inline"> <semantics> <mrow> <mi>c</mi> <mi>E</mi> <mi>n</mi> <mi>d</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> forms a submonoid of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>E</mi> <mi>n</mi> <mi>d</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> if and only if <i>G</i> is an odd cycle or <inline-formula> <math display="inline"> <semantics> <mrow> <mi>G</mi> <mo>=</mo> <mi>G</mi> <mo>(</mo> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> for some odd <inline-formula> <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>&#8805;</mo> <mn>3</mn> </mrow> </semantics> </math> </inline-formula> and integer <inline-formula> <math display="inline"> <semantics> <mrow> <mi>m</mi> <mo>&#8805;</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>.