The Li-Feng transformation of weighted adjacency matrices for graphs with degree-based edge-weights

Abstract

For a graph G, let $d_v$ be the degree of a vertex v. Given a symmetric real function $f(x,y)$, the weight of edge uv in graph G is equal to the value $f(d_u,d_v)$. The degree-based weighted adjacency matrix is defined as $A_f\left(G\right)$, in which the $(u,v)$-entry is equal to $f(d_u,d_v)$ if uv is an edge of G and 0 otherwise. In this paper, we consider the Li-Feng transformation and show that if a graph G contains two pendant paths on a common vertex, the uniform distribution of pendant paths increases the largest eigenvalue of $A_f\left(G\right)$, when $f(x,y)$ is increasing in x and the length of two pendant paths should be at least 2. We also consider the cycle version of Li-Feng transformation and show that if a graph G contains two pendant cycles on a common vertex, the uniform distribution of pendant cycles decreases the largest eigenvalue of $A_f\left(G\right)$, when ${\lambda }_1\left(A_f\right(G\left)\right)>2f(2,2)$. The purpose of this paper is to unify the study of the graph operation on the largest eigenvalue for the degree-based weighted adjacency matrix.