Identical Neighbor Structure: Effects on Spectrum and Independence in <i>CN</i><sub>s</sub> Cartesian Product of Graphs

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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Bibliographic Details
Main Authors: Subha A B, Sreekumar K G, Elsayed M. Elsayed, Manilal K, Turki D. Alharbi
Format: Article
Language:English
Published: MDPI AG 2025-03-01
Series:Mathematics
Subjects:
<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
Online Access:https://www.mdpi.com/2227-7390/13/7/1040
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Summary: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.
ISSN:2227-7390

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