Identical Neighbor Structure: Effects on Spectrum and Independence in CNs Cartesian Product of Graphs - Subha A B; Sreekumar K G; Elsayed M. Elsayed; Manilal K; Turki D. Alharbi (Mathematics)
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 <semantics>CNs</semantics> vertices as the vertices having an identical set of neighbors and classified graphs containing such vertices as <semantics>CNs</semantics> graphs. Furthermore, we introduced the <semantics>CNs</semantics> Cartesian product for these graphs. To formally characterize the relationships between <semantics>CNs</semantics> vertices, we constructed an <semantics>n×n</semantics><semantics>CNs</semantics> matrix, where an entry is 1 if the corresponding pair of vertices are <semantics>CNs</semantics> vertices and 0 otherwise. Utilizing this matrix, we established that the spectrum of the <semantics>CNs</semantics> Cartesian product consists exclusively of integral eigenvalues. This finding enhances our understanding of graph spectra and their relation to structural properties.