Exploring a Graph Complement in Quadratic Congruence
In this work, we investigate essential definitions, defining <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi></mrow></semantics></math></inline-formula> as a s...
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| Format: | Article |
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MDPI AG
2024-02-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/16/2/213 |
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| author | Hamza Daoub Osama Shafah Ahmad A. Almutlg |
| author_facet | Hamza Daoub Osama Shafah Ahmad A. Almutlg |
| author_sort | Hamza Daoub |
| collection | DOAJ |
| description | In this work, we investigate essential definitions, defining <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi></mrow></semantics></math></inline-formula> as a simple graph with vertices in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi mathvariant="double-struck">Z</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></semantics></math></inline-formula> and subgraphs <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mo>Γ</mo></mrow><mrow><mi>u</mi></mrow></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mo>Γ</mo></mrow><mrow><mi>q</mi></mrow></msub></mrow></semantics></math></inline-formula> as unit residue and quadratic residue graphs modulo <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow></semantics></math></inline-formula>, respectively. The investigation extends to the degree of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mo>Γ</mo></mrow><mrow><mi>u</mi></mrow></msub></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mo>Γ</mo></mrow><mrow><mi>q</mi></mrow></msub></mrow></semantics></math></inline-formula>, illuminating the properties of these subgraphs in the context of quadratic congruences. |
| first_indexed | 2024-03-07T22:12:28Z |
| format | Article |
| id | doaj.art-d5b7aeaf56cf482cabb2e218b07bf533 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-07T22:12:28Z |
| publishDate | 2024-02-01 |
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| series | Symmetry |
| spelling | doaj.art-d5b7aeaf56cf482cabb2e218b07bf5332024-02-23T15:36:02ZengMDPI AGSymmetry2073-89942024-02-0116221310.3390/sym16020213Exploring a Graph Complement in Quadratic CongruenceHamza Daoub0Osama Shafah1Ahmad A. Almutlg2Department of Mathematics, Faculty of Science, Zawia University, Zawia 16418, LibyaDepartment of Mathematics, Faculty of Science, Sabratha University, Sabratha 00218, LibyaDepartment of Mathematics, College of Science and Arts, Methnab, Qassim University, Buraidah 51931, Saudi ArabiaIn this work, we investigate essential definitions, defining <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi></mrow></semantics></math></inline-formula> as a simple graph with vertices in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi mathvariant="double-struck">Z</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></semantics></math></inline-formula> and subgraphs <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mo>Γ</mo></mrow><mrow><mi>u</mi></mrow></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mo>Γ</mo></mrow><mrow><mi>q</mi></mrow></msub></mrow></semantics></math></inline-formula> as unit residue and quadratic residue graphs modulo <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow></semantics></math></inline-formula>, respectively. The investigation extends to the degree of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mo>Γ</mo></mrow><mrow><mi>u</mi></mrow></msub></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mo>Γ</mo></mrow><mrow><mi>q</mi></mrow></msub></mrow></semantics></math></inline-formula>, illuminating the properties of these subgraphs in the context of quadratic congruences.https://www.mdpi.com/2073-8994/16/2/213quadratic residuesquadratic congruencesimple graphcomposite modulusvertex degreegraph complement |
| spellingShingle | Hamza Daoub Osama Shafah Ahmad A. Almutlg Exploring a Graph Complement in Quadratic Congruence Symmetry quadratic residues quadratic congruence simple graph composite modulus vertex degree graph complement |
| title | Exploring a Graph Complement in Quadratic Congruence |
| title_full | Exploring a Graph Complement in Quadratic Congruence |
| title_fullStr | Exploring a Graph Complement in Quadratic Congruence |
| title_full_unstemmed | Exploring a Graph Complement in Quadratic Congruence |
| title_short | Exploring a Graph Complement in Quadratic Congruence |
| title_sort | exploring a graph complement in quadratic congruence |
| topic | quadratic residues quadratic congruence simple graph composite modulus vertex degree graph complement |
| url | https://www.mdpi.com/2073-8994/16/2/213 |
| work_keys_str_mv | AT hamzadaoub exploringagraphcomplementinquadraticcongruence AT osamashafah exploringagraphcomplementinquadraticcongruence AT ahmadaalmutlg exploringagraphcomplementinquadraticcongruence |