The Effect on the Largest Eigenvalue of Degree-Based Weighted Adjacency Matrix by Perturbations

Let G be a connected graph. Denote by d_i the degree of a vertex v_i in G . Let f(x,y)>0 be a real symmetric function. Consider an edge-weighted graph in such a way that for each edge v_iv_j of G , the weight of v_iv_j is equal to the value f(d_i, d_j) . Therefore, we have a degree-based weighted adjacency matrix A_f(G) of G , in which the ( i , j )-entry is equal to f(d_i,d_j) if v_iv_j is an edge of G and is equal to zero otherwise. Let x be a positive eigenvector corresponding to the largest eigenvalue λ _1(A_f(G)) of the weighted adjacency matrix A_f(G) . In this paper, we first consider the unimodality of the eigenvector x on an induced path of G . Second, if f ( x , y ) is increasing in the variable x , then we investigate how the largest weighted adjacency eigenvalue λ _1(A_f(G)) changes when G is perturbed by vertex contraction or edge subdivision. The aim of this paper is to unify the study of spectral properties for the degree-based weighted adjacency matrices of graphs.