A Unified Spectral Framework for Weighted Adjacency-Degree Matrices of Dendrimer and Bethe Trees

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, $A_{\alpha }$, 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 $U_{j+1}(x)+\delta U_j(x)=0,$ covering $L\mathrm{/}{L}^{+}$ and $A_{\alpha }$; explicit cosine spectra occur only in special cases, such as the adjacency matrix and the $L\mathrm{/}{L}^{+}$ 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.