A Unified Spectral Framework for Weighted Adjacency-Degree Matrices of Dendrimer and Bethe Trees
Abstract
We introduce a weighted adjacency-degree matrix Afg(G) with edge weights f(di,dj) and diagonal entries g(di), unifying adjacency, Laplacian, signless Laplacian, Aα, ABC, Randić, Sombor and related matrices. For dendrimer trees Dn,k and Bethe trees Bn,k, the characteristic polynomial is factorized through one recursive sequence Pfg,n. When f2(1,k)=f2(k,k), this sequence admits a Chebyshev reduction to Uj+1(x)+δUj(x)=0, covering L/L+ and Aα; explicit cosine spectra occur only in special cases, such as the adjacency matrix and the L/L+ end factor of Bn,k. We derive positive-semidefinite energy formulas, Gershgorin criteria, spectral-gap estimates, interlacing and non-interlacing results, partial eigenvalue-coincidence information, and McClelland- and Koolen--Moulton-type bounds. Several known results are recovered as special cases of this unified framework.
Related
- Extremal Graphs with Prescribed Parameters for the Spectral Radius of Weighted Adjacency Matrices with Property P∗
- Graph transformations and eigenvectors of the Laplacian, signless Laplacian and Adjacency matrices
- Connected graphs with minimum adjacency spectral gap
- The Distance Between the Adjacency Spectral Center and the Characteristic Set of a Tree
- Adjacency-degree algebras and spectral determination of graphs
BibTeX
@misc{metsidik2026unified,
title = {{A Unified Spectral Framework for Weighted Adjacency-Degree Matrices of Dendrimer and Bethe Trees}},
author = {Metrose Metsidik},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10107},
eprint = {2610.10107},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10107},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Unified Spectral Framework for Weighted Adjacency-Degree Matrices of Dendrimer and Bethe Trees
AU - Metrose Metsidik
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10107
UR - https://arxiv.org/abs/2610.10107
AB - We introduce a weighted adjacency-degree matrix A_fg(G) with edge weights f(d_i,d_j) and diagonal entries g(d_i), unifying adjacency, Laplacian, signless Laplacian, A_α, ABC, Randić, Sombor and related matrices. For dendrimer trees D_n,k and Bethe trees B_n,k, the characteristic polynomial is factorized through one recursive sequence P_fg,n. When f^2(1,k)=f^2(k,k), this sequence admits a Chebyshev reduction to U_j+1(x)+δU_j(x)=0, covering L/L^+ and A_α; explicit cosine spectra occur only in special cases, such as the adjacency matrix and the L/L^+ end factor of B_n,k. We derive positive-semidefinite energy formulas, Gershgorin criteria, spectral-gap estimates, interlacing and non-interlacing results, partial eigenvalue-coincidence information, and McClelland- and Koolen--Moulton-type bounds. Several known results are recovered as special cases of this unified framework.
ER -