Unicyclic Graphs Whose Completely Regular Endomorphisms form a Monoid

Unicyclic Graphs Whose Completely Regular Endomorphisms form a Monoid

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> &...

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Bibliographic Details
Main Authors: Rui Gu, Hailong Hou
Format: Article
Language:English
Published: MDPI AG 2020-02-01
Series:Mathematics
Subjects:
endomorphism
monoid
completely regular
unicyclic graph
Online Access:https://www.mdpi.com/2227-7390/8/2/240
Description
Summary: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>.
ISSN:2227-7390

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