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 assigne...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(d_(i),d_(j)).Hence,we have a weighted adjacency matrix Af(G)of G,in which the ij-entry is equal to f(d_(i),d_(j))if v_(i)v_(j)∈E(G)and 0 otherwise.This paper attempts to unify the study of spectral properties for the weighted adjacency matrix Af(G)of graphs with a degree-based edge-weight function f(x,y).Some lower and upper bounds of the largest weighted adjacency eigenvalueλ1 are given,and the corresponding extremal graphs are characterized.Bounds of the energy for the ε_(f)(G)weighted adjacency matrix A_(f)(G)are also obtained.By virtue of the unified method,this makes many earlier results become special cases of our results.展开更多
基金Supported by NSFC(Grant Nos.12131013 and 12161141006)。
文摘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(d_(i),d_(j)).Hence,we have a weighted adjacency matrix Af(G)of G,in which the ij-entry is equal to f(d_(i),d_(j))if v_(i)v_(j)∈E(G)and 0 otherwise.This paper attempts to unify the study of spectral properties for the weighted adjacency matrix Af(G)of graphs with a degree-based edge-weight function f(x,y).Some lower and upper bounds of the largest weighted adjacency eigenvalueλ1 are given,and the corresponding extremal graphs are characterized.Bounds of the energy for the ε_(f)(G)weighted adjacency matrix A_(f)(G)are also obtained.By virtue of the unified method,this makes many earlier results become special cases of our results.
