Some interlacing results on weighted adjacency matrices of graphs with degree-based edge-weights

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 Af(G) of G, in which the ij-entry is equal to f (di, dj) if vivj is an element of E(G) and 0 otherwise. In this paper, we obtain uniform interlacing inequalities for the weighted adjacency eigenvalues under some kinds of graph operations including edge subdivision, vertex deletion and vertex contraction. In addition, if f(x, y) is increasing in the variable x, then some examples are given to show that the interlacing inequalities are the best possible for each type of the operations. This paper attempts to unify the study of spectral properties for the weighted adjacency matrices of graphs with degree-based edge-weights.(c) 2023 Elsevier B.V. All rights reserved.