Certain structural properties for Cayley regularity graphs of semigroups and their theoretical applications

Nuttawoot Nupo; Sayan Panma; Nuttawoot Nupo; Sayan Panma

doi:10.3934/math.2023830

AIMS Mathematics

2023, Volume 8, Issue 7: 16228-16239. doi: 10.3934/math.2023830

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Certain structural properties for Cayley regularity graphs of semigroups and their theoretical applications

Nuttawoot Nupo ¹,
Sayan Panma ^{2
,
,}

1.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
2.
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

Received: 28 October 2022 Revised: 28 March 2023 Accepted: 30 March 2023 Published: 06 May 2023
MSC : 05C25, 20B15, 20B30

An element $ x $ in a semigroup is said to be regular if there exists an element $ y $ in the semigroup such that $ x = xyx $. The element $ y $ is said to be a regular part of $ x $. Define the Cayley regularity graph of a semigroup $ S $ to be a digraph with vertex set $ S $ and arc set containing all ordered pairs $ (x, y) $ such that $ y $ is a regular part of $ x $. In this paper, certain classes of Cayley regularity graphs such as complete digraphs, connected digraphs and equivalence digraphs are investigated. Furthermore, structural properties of the Cayley regularity graphs are theoretically applied to study perfect matchings of other algebraic graphs.
- Cayley regularity graphs,
- regular part,
- connectedness,
- completeness,
- perfect matching
Citation: Nuttawoot Nupo, Sayan Panma. Certain structural properties for Cayley regularity graphs of semigroups and their theoretical applications[J]. AIMS Mathematics, 2023, 8(7): 16228-16239. doi: 10.3934/math.2023830

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Abstract

An element $ x $ in a semigroup is said to be regular if there exists an element $ y $ in the semigroup such that $ x = xyx $. The element $ y $ is said to be a regular part of $ x $. Define the Cayley regularity graph of a semigroup $ S $ to be a digraph with vertex set $ S $ and arc set containing all ordered pairs $ (x, y) $ such that $ y $ is a regular part of $ x $. In this paper, certain classes of Cayley regularity graphs such as complete digraphs, connected digraphs and equivalence digraphs are investigated. Furthermore, structural properties of the Cayley regularity graphs are theoretically applied to study perfect matchings of other algebraic graphs.

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