Identical Neighbor Structure: Effects on Spectrum and Independence in CN s 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 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.