发表机构
College of Mathematical Sciences; Xinjiang Normal University（数学科学学院; 新疆师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出统一谱框架，通过加权邻接-度矩阵统一多种图矩阵，并利用递归序列与切比雪夫约化解析树枝状大分子树与Bethe树的谱，推导能量公式与谱界。
AI 中文摘要
我们引入一个加权邻接-度矩阵 \\(A_{fg}(G)\\)，其边权重为 \\(f(d_i,d_j)\\)，对角元为 \\(g(d_i)\\)，统一了邻接矩阵、拉普拉斯矩阵、无符号拉普拉斯矩阵、\\(A_\alpha\\)、ABC、Randić、Sombor及相关矩阵。对于树枝状大分子树 \\(D_{n,k}\\) 和 Bethe 树 \\(B_{n,k}\\)，其特征多项式通过一个递归序列 \\(P_{fg,n}\\) 进行因式分解。当 \\(f^2(1,k)=f^2(k,k)\\) 时，该序列允许切比雪夫约化至 \\( U_{j+1}(x)+\delta U_j(x)=0, \\) 覆盖 \\(L/L^+\\) 和 \\(A_\alpha\\)；显式余弦谱仅在特殊情况下出现，例如 \\(B_{n,k}\\) 的邻接矩阵和 \\(L/L^+\\) 的末端因子。我们推导了正半定能量公式、Gershgorin准则、谱间隙估计、交错与非交错结果、部分特征值重合信息，以及McClelland型和Koolen--Moulton型界。若干已知结果作为该统一框架的特例被重新获得。
英文摘要
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.
