Let G be a graph and di denote the degree of a vertex vi in G, and let f(x,y) be a real symmetric function. Then one can get an edge-weighted graph in such a way that for each edge vivj of G, the weight of vivj is assigned by the value f(di,dj). Hence, we have a weighted adjacency matrix \(\mathcal{A}_{f}(G)\) of G, in which the ij-entry is equal to f(di,dj) if vivj ∈ E(G) and 0 otherwise. This paper attempts to unify the study of spectral properties for the weighted adjacency matrix \(\mathcal{A}_{f}(G)\) of graphs with a degree-based edge-weight function f(x,y). Some lower and upper bounds of the largest weighted adjacency eigenvalue λ1 are given, and the corresponding extremal graphs are characterized. Bounds of the energy \(\mathcal{E}_{f}(G)\) for the weighted adjacency matrix \(\mathcal{A}_{f}(G)\) are also obtained. By virtue of the unified method, this makes many earlier results become special cases of our results.
