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