Sombor characteristic values of cographs

Area of Science:

• Graph theory
• Spectral graph theory
• Combinatorics

Background:

• Cographs form a unique graph class with specific recursive definitions.
• Spectral graph theory utilizes eigenvalues of matrices associated with graphs to understand their properties.
• The Sombor index and Sombor energy are graph invariants with applications in chemistry and network analysis.

Purpose of the Study:

• To analyze a specific recursive class of cographs.
• To compute the Sombor spectrum, spectral radius, and bounds for the Sombor index and energy of these cographs.
• To characterize cographs based on the simplicity or uniqueness of their Sombor characteristic values.

Main Methods:

• Recursive definition of the cograph class.
• Calculation of the Sombor adjacency matrix.
• Eigenvalue analysis to determine the Sombor spectrum and related invariants.
• Derivation of sharp bounds for graph indices and energy.
• Characterization based on spectral properties.

Main Results:

• The Sombor spectrum, spectral radius, and bounds for the Sombor index and energy are determined for the studied cograph class.
• Graphs attaining the established inequalities for Sombor energy are identified.
• Cographs with simple and unique Sombor characteristic values are fully characterized.

Conclusions:

• The spectral properties of this unique cograph class are comprehensively analyzed.
• The study provides a deeper understanding of Sombor spectral invariants in graph theory.
• Characterizations offer criteria for identifying specific types of cographs based on their spectral features.