Let G be a graph of order n and μ be an adjacency eigenvalue of G with multiplicity k ≥ 1. A star complement H for μ in G is an induced subgraph of G with n-k vertices and no eigenvalue μ, and the vertex subset X ...Let G be a graph of order n and μ be an adjacency eigenvalue of G with multiplicity k ≥ 1. A star complement H for μ in G is an induced subgraph of G with n-k vertices and no eigenvalue μ, and the vertex subset X = V(G-H) is called a star set for μ in G. The star complement technique provides a spectral tool for reconstructing a certain part of a graph from the remaining part. In this paper, we study the regular graphs with K_(t,s)(s ≥ t ≥ 2) as a star complement for an eigenvalue μ, especially, characterize the case of t = 3 completely, obtain some properties when t = s, and propose some problems for further study.展开更多
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
In this paper, we characterize the trees with the largest Laplacian and adjacency spectral radii among all trees with fixed number of vertices and fixed maximal degree, respectively.
Let G be a graph of order n andμbe an adjacency eigenvalue of G with multiplicity k≥1.A star complement H forμin G is an induced subgraph of G of order n-k with no eigenvalueμ,and the subset X=V(G-H)is called a st...Let G be a graph of order n andμbe an adjacency eigenvalue of G with multiplicity k≥1.A star complement H forμin G is an induced subgraph of G of order n-k with no eigenvalueμ,and the subset X=V(G-H)is called a star set forμin G.The star complement provides a strong link between graph structure and linear algebra.In this paper,the authors characterize the regular graphs with K2,2,s(s≥2)as a star complement for all possible eigenvalues,the maximal graphs with K2,2,s as a star complement for the eigenvalueμ=1,and propose some questions for further research.展开更多
基金Supported by the National Natural Science Foundation of China (Grant Nos. 1197118012271337)+1 种基金the Characteristic Innovation Project of General Colleges and Universities in Guangdong Province (Grant No. 2022KTSCX225)the Guangdong Education and Scientific Research Project (Grant No. 2021GXJK159)。
文摘Let G be a graph of order n and μ be an adjacency eigenvalue of G with multiplicity k ≥ 1. A star complement H for μ in G is an induced subgraph of G with n-k vertices and no eigenvalue μ, and the vertex subset X = V(G-H) is called a star set for μ in G. The star complement technique provides a spectral tool for reconstructing a certain part of a graph from the remaining part. In this paper, we study the regular graphs with K_(t,s)(s ≥ t ≥ 2) as a star complement for an eigenvalue μ, especially, characterize the case of t = 3 completely, obtain some properties when t = s, and propose some problems for further study.
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.
基金Foundation item： the National Natural Science Foundation of China （No. 10601001） the Natural Science Foundation of Anhui Province （Nos. 050460102+3 种基金 070412065） Natural Science Foundation of Department of Education of Anhui Province （No. 2005kj005zd） Project of Anhui University on leading Researchers Construction Foundation of Innovation Team on Basic Mathematics of Anhui University.
文摘In this paper, we characterize the trees with the largest Laplacian and adjacency spectral radii among all trees with fixed number of vertices and fixed maximal degree, respectively.
基金supported by the National Natural Science Foundation of China(No.11971180,12271337)the Guangdong Provincial Natural Science Foundation(No.2019A1515012052)。
文摘Let G be a graph of order n andμbe an adjacency eigenvalue of G with multiplicity k≥1.A star complement H forμin G is an induced subgraph of G of order n-k with no eigenvalueμ,and the subset X=V(G-H)is called a star set forμin G.The star complement provides a strong link between graph structure and linear algebra.In this paper,the authors characterize the regular graphs with K2,2,s(s≥2)as a star complement for all possible eigenvalues,the maximal graphs with K2,2,s as a star complement for the eigenvalueμ=1,and propose some questions for further research.
