Degree-Based Weighted Adjacency Matrices: Spectra, Integrality, and Edge Deletion Effects

TL;DR

This paper studies the spectral properties of weighted adjacency matrices of complete multipartite graphs, revealing conditions for integrality, effects of edge deletion on energy and spectral radius, and resolving open problems in spectral graph theory.

Contribution

It characterizes the spectra of weighted adjacency matrices for certain multipartite graphs, identifies integral cases, and clarifies the impact of edge deletion on spectral properties, correcting previous results.

Findings

Weighted adjacency spectrum of complete multipartite graphs characterized.

Edge deletion generally decreases energy and spectral radius, with exceptions.

Resolved open problems on $ISI$ energy change and provided counterexamples.

Abstract

The article presents weighted adjacency spectrum of complete multipartite graphs, characterize its families with three distinct eigenvalues and identifies integral matrices. Also, we observe that for almost all weighted matrices, the energy and the spectral radius of a complete graph decreases upon edge deletion, thereby correcting and refining earlier published results in [Bilal and Munir, Int. J. Quantum Chem. (2024)]. Furthermore, we give counter examples related to $ISI$ energy decrease of regular tripartite graph by edge deletion and give its correct $ISI$ spectrum and $ISI$ energy and settle an open problem related to $ISI$ energy change of the multipartite graph. Also, we calculate the weighted adjacency spectrum of crown multipartite graph and discuss its integral spectral weighted spectrum.