The article explores the spectral properties of weighted adjacency matrices associated with vertex-degree-based (VDB) and neighborhood-degree-based (NDB) topological indices. Let G be a simple graph on vertices \(1,\dots ,n\) . Let \(d_i\) be the degree of the vertex i in G and \(\delta _i= \sum \nolimits _{\{i,j\} \in E(G)}d_j\) . Let \(\psi \) be either d or \(\delta \) . Then, adjacency matrix associated with \(\psi \) is defined as the \(n\times n\) matrix \(A_\psi (G)\) , whose (i, j)-th entry equals \(F(\psi _i,\psi _j)\) (symmetric function) if \(\{i,j\} \in E(G)\) , and 0 otherwise. In this article, we establish fundamental inequalities connecting the energy \(\mathcal {E}_\psi (G)\) , the spectral radius \(\rho _{\psi }(G)\) , and the smallest positive eigenvalue \(\tau _{\psi }(G)\) of this matrix. For any connected graph G, we prove the lower bound \(\mathcal {E}_\psi (G)\rho _\psi (G)\ge 2\sum \nolimits _{\{i,j\}\in E(G)}F(\psi _i,\psi _j)^2\) . For connected bipartite graphs, we prove the upper bound \(\mathcal {E}_\psi (G)\tau _\psi (G)\le 2\sum \nolimits _{\{i,j\}\in E(G)}F(\psi _i,\psi _j)^2\) . In both these cases, equality is characterized and occurs precisely when G is a complete bipartite graph. Furthermore, we solve the extremal problem for trees, proving that the star graph \(S_n\) uniquely maximizes both \(\rho _{\psi }(T)\) and \(\tau _\psi (T)\) among all trees T on n vertices. Finally, under the natural condition that the underlying topological index is minimized by the star, we show that \(S_n\) also achieves the minimum energy \(\mathcal {E}_\psi (T)\) .
