Spectral Properties and Energy of Weighted Adjacency Matrices for Graphs with Degree-based Edge-weight Functions

Abstract

Let G be a graph and d_i denote the degree of a vertex v_i 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 v_iv_j of G, the weight of v_iv_j is assigned by the value f(d_i,d_j). Hence, we have a weighted adjacency matrix \(\mathcal{A}_{f}(G)\) of G, in which the ij-entry is equal to f(d_i,d_j) if v_iv_j ∈ 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.