Identical Neighbor Structure: Effects on Spectrum and Independence in <i>CN</i><sub>s</sub> Cartesian Product of Graphs
In this study, we introduced a novel graph product derived from the standard Cartesian product and investigated its structural properties, with a particular emphasis on its independence number and spectral characteristics in relation to identical neighbor structures. A key finding is that the spectr...
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2025-03-01
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| author | Subha A B Sreekumar K G Elsayed M. Elsayed Manilal K Turki D. Alharbi |
| author_facet | Subha A B Sreekumar K G Elsayed M. Elsayed Manilal K Turki D. Alharbi |
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| description | In this study, we introduced a novel graph product derived from the standard Cartesian product and investigated its structural properties, with a particular emphasis on its independence number and spectral characteristics in relation to identical neighbor structures. A key finding is that the spectrum of this newly defined product graph consists entirely of integral eigenvalues, a significant property with applications in chemistry, network theory, and combinatorial optimization. We defined <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msub><mi>N</mi><mi>s</mi></msub></mrow></semantics></math></inline-formula> vertices as the vertices having an identical set of neighbors and classified graphs containing such vertices as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msub><mi>N</mi><mi>s</mi></msub></mrow></semantics></math></inline-formula> graphs. Furthermore, we introduced the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msub><mi>N</mi><mi>s</mi></msub></mrow></semantics></math></inline-formula> Cartesian product for these graphs. To formally characterize the relationships between <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msub><mi>N</mi><mi>s</mi></msub></mrow></semantics></math></inline-formula> vertices, we constructed an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msub><mi>N</mi><mi>s</mi></msub></mrow></semantics></math></inline-formula> matrix, where an entry is 1 if the corresponding pair of vertices are <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msub><mi>N</mi><mi>s</mi></msub></mrow></semantics></math></inline-formula> vertices and 0 otherwise. Utilizing this matrix, we established that the spectrum of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msub><mi>N</mi><mi>s</mi></msub></mrow></semantics></math></inline-formula> Cartesian product consists exclusively of integral eigenvalues. This finding enhances our understanding of graph spectra and their relation to structural properties. |
| format | Article |
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| publishDate | 2025-03-01 |
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| spelling | doaj-art-4082ca5bacd443409743e547fd7db1752025-08-20T02:17:00ZengMDPI AGMathematics2227-73902025-03-01137104010.3390/math13071040Identical Neighbor Structure: Effects on Spectrum and Independence in <i>CN</i><sub>s</sub> Cartesian Product of GraphsSubha A B0Sreekumar K G1Elsayed M. Elsayed2Manilal K3Turki D. Alharbi4Department of Mathematics, University College, University of Kerala, Thiruvananthapuram 695034, IndiaDepartment of Mathematics, University of Kerala, Thiruvananthapuram 695581, IndiaDepartment of Mathematics, Faculty of Science, King AbdulAziz University, Jeddah 21589, Saudi ArabiaDepartment of Mathematics, University College, University of Kerala, Thiruvananthapuram 695034, IndiaDepartment of Mathematics, Al-Leith University College, Umm Al-Qura University, Mecca 24382, Saudi ArabiaIn this study, we introduced a novel graph product derived from the standard Cartesian product and investigated its structural properties, with a particular emphasis on its independence number and spectral characteristics in relation to identical neighbor structures. A key finding is that the spectrum of this newly defined product graph consists entirely of integral eigenvalues, a significant property with applications in chemistry, network theory, and combinatorial optimization. We defined <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msub><mi>N</mi><mi>s</mi></msub></mrow></semantics></math></inline-formula> vertices as the vertices having an identical set of neighbors and classified graphs containing such vertices as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msub><mi>N</mi><mi>s</mi></msub></mrow></semantics></math></inline-formula> graphs. Furthermore, we introduced the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msub><mi>N</mi><mi>s</mi></msub></mrow></semantics></math></inline-formula> Cartesian product for these graphs. To formally characterize the relationships between <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msub><mi>N</mi><mi>s</mi></msub></mrow></semantics></math></inline-formula> vertices, we constructed an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msub><mi>N</mi><mi>s</mi></msub></mrow></semantics></math></inline-formula> matrix, where an entry is 1 if the corresponding pair of vertices are <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msub><mi>N</mi><mi>s</mi></msub></mrow></semantics></math></inline-formula> vertices and 0 otherwise. Utilizing this matrix, we established that the spectrum of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msub><mi>N</mi><mi>s</mi></msub></mrow></semantics></math></inline-formula> Cartesian product consists exclusively of integral eigenvalues. This finding enhances our understanding of graph spectra and their relation to structural properties.https://www.mdpi.com/2227-7390/13/7/1040<i>CN<sub>S</sub></i> graph<i>DN<sub>S</sub></i> graph<i>CN<sub>S</sub></i> matrixindependence number<i>CN<sub>S</sub></i> spectrum<i>CN<sub>S</sub></i> energy |
| spellingShingle | Subha A B Sreekumar K G Elsayed M. Elsayed Manilal K Turki D. Alharbi Identical Neighbor Structure: Effects on Spectrum and Independence in <i>CN</i><sub>s</sub> Cartesian Product of Graphs Mathematics <i>CN<sub>S</sub></i> graph <i>DN<sub>S</sub></i> graph <i>CN<sub>S</sub></i> matrix independence number <i>CN<sub>S</sub></i> spectrum <i>CN<sub>S</sub></i> energy |
| title | Identical Neighbor Structure: Effects on Spectrum and Independence in <i>CN</i><sub>s</sub> Cartesian Product of Graphs |
| title_full | Identical Neighbor Structure: Effects on Spectrum and Independence in <i>CN</i><sub>s</sub> Cartesian Product of Graphs |
| title_fullStr | Identical Neighbor Structure: Effects on Spectrum and Independence in <i>CN</i><sub>s</sub> Cartesian Product of Graphs |
| title_full_unstemmed | Identical Neighbor Structure: Effects on Spectrum and Independence in <i>CN</i><sub>s</sub> Cartesian Product of Graphs |
| title_short | Identical Neighbor Structure: Effects on Spectrum and Independence in <i>CN</i><sub>s</sub> Cartesian Product of Graphs |
| title_sort | identical neighbor structure effects on spectrum and independence in i cn i sub s sub cartesian product of graphs |
| topic | <i>CN<sub>S</sub></i> graph <i>DN<sub>S</sub></i> graph <i>CN<sub>S</sub></i> matrix independence number <i>CN<sub>S</sub></i> spectrum <i>CN<sub>S</sub></i> energy |
| url | https://www.mdpi.com/2227-7390/13/7/1040 |
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