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

Bilal Ahmad Rather, Hilal Ahmad Ganie

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.

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

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 energy decrease of regular tripartite graph by edge deletion and give its correct spectrum and energy and settle an open problem related to energy change of the multipartite graph. Also, we calculate the weighted adjacency spectrum of crown multipartite graph and discuss its integral spectral weighted spectrum.
Paper Structure (4 sections, 13 theorems, 67 equations, 5 tables)

This paper contains 4 sections, 13 theorems, 67 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Weighted spectra of the complete multipartite graphs
  3. Weighted adjacency energy of some special families of graphs
  4. Conclusion

Key Result

Theorem 2.1

Let $G$ be a connected graph with vertex set $V(G)= \{v_{1},v_{2},\dots,v_{n}\}$ and let $S=\{v_{1}, v_{2},\dots, v_{\alpha}\}$ be an independent subset of $G$ such that $N(v_{i})=N(v_{j})$, for all $i,j\in \{1,2,\dots,\alpha \}$. Then $0$ is the eigenvalue of $\dot{A}(G)$ with multiplicity at least

Theorems & Definitions (15)

  • Theorem 2.1: bilalmdpi
  • Theorem 2.2: bilalmdpi
  • Theorem 2.3
  • Corollary 2.4
  • Remark 2.5
  • Corollary 2.6
  • Corollary 2.7
  • Corollary 2.8
  • Theorem 3.1
  • Conjecture 1
  • ...and 5 more