Work

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

Year: 2025

Type: article

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 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.

Source: Mathematics

DOI: 10.3390/math13071040

Authors: Subha A B, Sreekumar K G, Elsayed M. Elsayed, Manilal K, Turki D. Alharbi

Institutions: University of Kerala, Mansoura University, King Abdulaziz University, Umm al-Qura University

Language: English

Topic: Graph theory and applications

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $3,137

Funder: Umm Al-Qura University