Identical Neighbor Structure: Effects on Spectrum and Independence in CN s Cartesian Product of Graphs

Subha A B, Sreekumar K G, Elsayed M. Elsayed, Manilal K and Turki D. Alharbi ()
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Subha A B: Department of Mathematics, University College, University of Kerala, Thiruvananthapuram 695034, India
Sreekumar K G: Department of Mathematics, University of Kerala, Thiruvananthapuram 695581, India
Elsayed M. Elsayed: Department of Mathematics, Faculty of Science, King AbdulAziz University, Jeddah 21589, Saudi Arabia
Manilal K: Department of Mathematics, University College, University of Kerala, Thiruvananthapuram 695034, India
Turki D. Alharbi: Department of Mathematics, Al-Leith University College, Umm Al-Qura University, Mecca 24382, Saudi Arabia

Mathematics, 2025, vol. 13, issue 7, 1-17

Abstract: 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 C N s vertices as the vertices having an identical set of neighbors and classified graphs containing such vertices as C N s graphs. Furthermore, we introduced the C N s Cartesian product for these graphs. To formally characterize the relationships between C N s vertices, we constructed an n × n C N s matrix, where an entry is 1 if the corresponding pair of vertices are C N s vertices and 0 otherwise. Utilizing this matrix, we established that the spectrum of the C N s Cartesian product consists exclusively of integral eigenvalues. This finding enhances our understanding of graph spectra and their relation to structural properties.

Keywords: CN S graph; DN S graph; CN S matrix; independence number; CN S spectrum; CN S energy; CN S Cartesian product of graphs (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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