Sombor spectra of chain graphs

Area of Science:

• Graph Theory
• Mathematical Chemistry
• Spectral Graph Theory

Background:

• The Sombor index is a graph invariant used in mathematical chemistry.
• Understanding spectral properties of graphs is crucial for analyzing their structure and behavior.
• Chain graphs are a fundamental class of graphs with diverse applications.

Purpose of the Study:

• To derive an explicit formula for the Sombor index of chain graphs.
• To investigate the Sombor spectral properties, including eigenvalues and their bounds.
• To characterize chain graphs based on their Sombor eigenvalues and related parameters.

Main Methods:

• Derivation of analytical formulas for graph invariants.
• Spectral analysis of graph matrices.
• Characterization of graph classes based on spectral properties.

Main Results:

• An explicit formula for the Sombor index of chain graphs is established.
• The Sombor eigenvalues of chain graphs are analyzed, with bounds provided for the largest and smallest.
• Chain graphs with a simple Sombor eigenvalue are characterized.
• Formulas for the Frobenius norm and determinant of the Sombor quotient matrix are derived.
• Bounds for the Sombor spread and Sombor energy are presented.

Conclusions:

• The study provides a comprehensive analysis of the Sombor index and spectral properties of chain graphs.
• The findings contribute to the understanding of graph invariants and their spectral characteristics.
• The characterizations offer insights into specific structures within the class of chain graphs.