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arXiv · 2603.08895

Degree-based weighted adjacency matrices: spectra, integrality, and edge deletion effects

Bilal Ahmad Rather·Hilal Ahmad Ganie
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Abstract

We study degree-based weighted adjacency matrices associated with symmetric edge-weight functions $\phi(d_u,d_v)$, with emphasis on complete multipartite graphs and spectral changes caused by edge modification. We characterize its families with three distinct eigenvalues and identifies integral matrices. For complete graphs, an exact threshold is obtained that determines whether deleting one edge increases, preserves, or decreases both the spectral radius and the weighted energy; the generalized Randi\'c family is classified completely, thereby correcting and refining earlier published results in [Bilal and Munir, Int. J. Quantum Chem. (2024)]. We further determine the $ISI$ spectrum under edge deletion from regular complete multipartite graphs, derive the complete weighted spectrum, energy, and inertia of crown multipartite graphs, and prove that adding an edge between two leaves of $S_n$ strictly increases its $ISI$ energy.

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Bilal Ahmad Rather, Hilal Ahmad Ganie. 2026-03-09. Degree-based weighted adjacency matrices: spectra, integrality, and edge deletion effects. https://arxiv.org/abs/2603.08895

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