On Another Class of Strongly Perfect Graphs
For a commutative ring <i>R</i> with unity, the associate ring graph, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>G</mi><mo stretchy=&quo...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-06-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/12/2014 |
| _version_ | 1797484739062398976 |
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| author | Neha Kansal Bableen Kaur Pravin Garg Deepa Sinha |
| author_facet | Neha Kansal Bableen Kaur Pravin Garg Deepa Sinha |
| author_sort | Neha Kansal |
| collection | DOAJ |
| description | For a commutative ring <i>R</i> with unity, the associate ring graph, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>G</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>, is a simple graph with vertices as nonzero elements of <i>R</i> and two distinct vertices are adjacent if they are associates. The graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>G</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> contains components equal in number to the number of distinct orbits, except for the orbit of an element 0. Moreover, each component is a complete graph. An important finding is that this is a class of strongly perfect graphs. In this article we describe the structure of the associate ring graph of the ring of integers modulo <i>n</i>, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>G</mi><mo stretchy="false">(</mo><msub><mi mathvariant="double-struck">Z</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>. We carried out computer experiments and provide a program for the same. We further characterize cases in which <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>G</mi><mo stretchy="false">(</mo><msub><mi mathvariant="double-struck">Z</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>, its complement <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover><mrow><mi>A</mi><mi>G</mi><mo stretchy="false">(</mo><msub><mi mathvariant="double-struck">Z</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><mo>¯</mo></mover></semantics></math></inline-formula>, and their line graphs are planar, ring graphs, and outerplanar. We also discuss the properties of the associate ring graph of a commutative ring <i>R</i> with unity. |
| first_indexed | 2024-03-09T23:08:45Z |
| format | Article |
| id | doaj.art-583adfd3924c4408a326652caf3fbe3c |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T23:08:45Z |
| publishDate | 2022-06-01 |
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| series | Mathematics |
| spelling | doaj.art-583adfd3924c4408a326652caf3fbe3c2023-11-23T17:48:19ZengMDPI AGMathematics2227-73902022-06-011012201410.3390/math10122014On Another Class of Strongly Perfect GraphsNeha Kansal0Bableen Kaur1Pravin Garg2Deepa Sinha3Department of Mathematics, University of Rajasthan, Jaipur 302004, IndiaDepartment of Mathematics, South Asian University, New Delhi 110021, IndiaDepartment of Mathematics, University of Rajasthan, Jaipur 302004, IndiaDepartment of Mathematics, South Asian University, New Delhi 110021, IndiaFor a commutative ring <i>R</i> with unity, the associate ring graph, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>G</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>, is a simple graph with vertices as nonzero elements of <i>R</i> and two distinct vertices are adjacent if they are associates. The graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>G</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> contains components equal in number to the number of distinct orbits, except for the orbit of an element 0. Moreover, each component is a complete graph. An important finding is that this is a class of strongly perfect graphs. In this article we describe the structure of the associate ring graph of the ring of integers modulo <i>n</i>, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>G</mi><mo stretchy="false">(</mo><msub><mi mathvariant="double-struck">Z</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>. We carried out computer experiments and provide a program for the same. We further characterize cases in which <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>G</mi><mo stretchy="false">(</mo><msub><mi mathvariant="double-struck">Z</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>, its complement <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover><mrow><mi>A</mi><mi>G</mi><mo stretchy="false">(</mo><msub><mi mathvariant="double-struck">Z</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><mo>¯</mo></mover></semantics></math></inline-formula>, and their line graphs are planar, ring graphs, and outerplanar. We also discuss the properties of the associate ring graph of a commutative ring <i>R</i> with unity.https://www.mdpi.com/2227-7390/10/12/2014associate ring graphring graphline graphplanarouterplanar |
| spellingShingle | Neha Kansal Bableen Kaur Pravin Garg Deepa Sinha On Another Class of Strongly Perfect Graphs Mathematics associate ring graph ring graph line graph planar outerplanar |
| title | On Another Class of Strongly Perfect Graphs |
| title_full | On Another Class of Strongly Perfect Graphs |
| title_fullStr | On Another Class of Strongly Perfect Graphs |
| title_full_unstemmed | On Another Class of Strongly Perfect Graphs |
| title_short | On Another Class of Strongly Perfect Graphs |
| title_sort | on another class of strongly perfect graphs |
| topic | associate ring graph ring graph line graph planar outerplanar |
| url | https://www.mdpi.com/2227-7390/10/12/2014 |
| work_keys_str_mv | AT nehakansal onanotherclassofstronglyperfectgraphs AT bableenkaur onanotherclassofstronglyperfectgraphs AT pravingarg onanotherclassofstronglyperfectgraphs AT deepasinha onanotherclassofstronglyperfectgraphs |