We first apply non-negative matrix theory to the matrix K=D+A,where D and A are the degree-diagonal and adjacency matrices of a graph G,respectively,to establish a relation on the largest Laplacian eigenvalue λ_1(G)o...We first apply non-negative matrix theory to the matrix K=D+A,where D and A are the degree-diagonal and adjacency matrices of a graph G,respectively,to establish a relation on the largest Laplacian eigenvalue λ_1(G)of G and the spectral radius ρ(K)of K.And then by using this relation we present two upper bounds for λ_1(G)and determine the extremal graphs which achieve the upper bounds.展开更多
A signed graph is a graph with a sign attached to each edge. This paper extends some fundamental concepts of the Laplacian matrices from graphs to signed graphs. In particular, the relationships between the least Lapl...A signed graph is a graph with a sign attached to each edge. This paper extends some fundamental concepts of the Laplacian matrices from graphs to signed graphs. In particular, the relationships between the least Laplacian eigenvalue and the unbalancedness of a signed graph are investigated.展开更多
The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second smallest eigenvalue of the Laplacian matrix of the graph. In our recent work, we have determined the graphs...The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second smallest eigenvalue of the Laplacian matrix of the graph. In our recent work, we have determined the graphs with maximal Laplacian spreads among all trees of fixed order and among all unicyclic graphs of fixed order, respectively. In this paper, we continue the work on Laplacian spread of graphs, and prove that there exist exactly two bicyclic graphs with maximal Laplacian spread among all bicyclic graphs of fixed order, which are obtained from a star by adding two incident edges and by adding two nonincident edges between the pendant vertices of the star, respectively.展开更多
In this paper,we determine graphs with the largest Laplacian spectral radius among the unicyclic and the bicyclic graphs on n vertices with k pendant vertices,respectively.
We characterize all connected graphs with exactly three distinct normalized Laplacian eigenvalues among which one is equal to 1, and determine all connected bipartite graphs with at least one vertex of degree 1 having...We characterize all connected graphs with exactly three distinct normalized Laplacian eigenvalues among which one is equal to 1, and determine all connected bipartite graphs with at least one vertex of degree 1 having exactly four distinct normalized Laplacian eigenvalues. In addition, we find all unicyclic graphs with three or four distinct normalized Laplacian eigenvalues.展开更多
Suppose that the vertex set of a graph G is V(G) ={v1,v2,...,vn}.The transmission Tr(vi) (or Di) of vertex vi is defined to be the sum of distances from vi to all other vertices.Let Tr(G) be the n × n diagonal ma...Suppose that the vertex set of a graph G is V(G) ={v1,v2,...,vn}.The transmission Tr(vi) (or Di) of vertex vi is defined to be the sum of distances from vi to all other vertices.Let Tr(G) be the n × n diagonal matrix with its (i,i)-entry equal to TrG(vi).The distance signless Laplacian spectral radius of a connected graph G is the spectral radius of the distance signless Laplacian matrix of G,defined as L(G) =Tr(G) + D(G),where D(G) is the distance matrix of G.In this paper,we give a lower bound on the distance signless Laplacian spectral radius of graphs and characterize graphs for which these bounds are best possible.We obtain a lower bound on the second largest distance signless Laplacian eigenvalue of graphs.Moreover,we present lower bounds on the spread of distance signless Laplacian matrix of graphs and trees,and characterize extremal graphs.展开更多
Let G be a simple connected graph with n vertices and m edges,L G be the line graph of G and λ 1(L G)≥λ 2(L G)≥...≥λ m(L G) be the eigenvalues of the graph L G.In this paper,the range of eigenvalues of a...Let G be a simple connected graph with n vertices and m edges,L G be the line graph of G and λ 1(L G)≥λ 2(L G)≥...≥λ m(L G) be the eigenvalues of the graph L G.In this paper,the range of eigenvalues of a line graph is considered.Some sharp upper bounds and sharp lower bounds of the eigenvalues of L G are obtained.In particular,it is proved that-2cos(πn)≤λ n-1 (L G)≤n-4 and λ n(L G)=-2 if and only if G is bipartite.展开更多
基金Supported by National Natural Science Foundation of China(Grant No.19971086)
文摘We first apply non-negative matrix theory to the matrix K=D+A,where D and A are the degree-diagonal and adjacency matrices of a graph G,respectively,to establish a relation on the largest Laplacian eigenvalue λ_1(G)of G and the spectral radius ρ(K)of K.And then by using this relation we present two upper bounds for λ_1(G)and determine the extremal graphs which achieve the upper bounds.
基金supported by the NSF of China(No.19971056)SRP(No.03B019) from the Education Committee of Hunan Province
文摘A signed graph is a graph with a sign attached to each edge. This paper extends some fundamental concepts of the Laplacian matrices from graphs to signed graphs. In particular, the relationships between the least Laplacian eigenvalue and the unbalancedness of a signed graph are investigated.
基金Supported by the National Natural Science Foundation of China (Grant No.10601001)the Natural Science Foundation of Anhui Province (Grant No.070412065)+3 种基金Project of Anhui Province for Young Teachers Research Supportin Universities (Grant No.2008jql083)Natural Science Foundation of Department of Education of Anhui Province(Grant No.2005kj005zd)Project of Anhui University on Leading Researchers ConstructionFoundation of Innovation Team on Basic Mathematics of Anhui University
文摘The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second smallest eigenvalue of the Laplacian matrix of the graph. In our recent work, we have determined the graphs with maximal Laplacian spreads among all trees of fixed order and among all unicyclic graphs of fixed order, respectively. In this paper, we continue the work on Laplacian spread of graphs, and prove that there exist exactly two bicyclic graphs with maximal Laplacian spread among all bicyclic graphs of fixed order, which are obtained from a star by adding two incident edges and by adding two nonincident edges between the pendant vertices of the star, respectively.
基金supported by National Natural Science Foundation of China (Grant No.10871204)the Fundamental Research Funds for the Central Universities (Grant No.09CX04003A)
文摘In this paper,we determine graphs with the largest Laplacian spectral radius among the unicyclic and the bicyclic graphs on n vertices with k pendant vertices,respectively.
基金This work is supported by the National Natural Science Foundation of China (grants No. 11671344, 11531011 and 11701492).
文摘We characterize all connected graphs with exactly three distinct normalized Laplacian eigenvalues among which one is equal to 1, and determine all connected bipartite graphs with at least one vertex of degree 1 having exactly four distinct normalized Laplacian eigenvalues. In addition, we find all unicyclic graphs with three or four distinct normalized Laplacian eigenvalues.
基金The authors are grateful to the two anonymous referees for their careful reading of this paper and strict criticisms, constructive corrections, and valuable comments on this paper, which have considerably improved the presentation of this paperThe first author was supported by the National Research Foundation of the Korean government with grant No. 2017R1D1A1B03028642+2 种基金The second author was supported by the National Natural Science Foundation of China (Grant No. 11771141)the Fundamental Research Fund for the Central Universities (No. 222201714049)The third author was supported by the National Natural Science Foundation of China (Grant No. 11371372).
文摘Suppose that the vertex set of a graph G is V(G) ={v1,v2,...,vn}.The transmission Tr(vi) (or Di) of vertex vi is defined to be the sum of distances from vi to all other vertices.Let Tr(G) be the n × n diagonal matrix with its (i,i)-entry equal to TrG(vi).The distance signless Laplacian spectral radius of a connected graph G is the spectral radius of the distance signless Laplacian matrix of G,defined as L(G) =Tr(G) + D(G),where D(G) is the distance matrix of G.In this paper,we give a lower bound on the distance signless Laplacian spectral radius of graphs and characterize graphs for which these bounds are best possible.We obtain a lower bound on the second largest distance signless Laplacian eigenvalue of graphs.Moreover,we present lower bounds on the spread of distance signless Laplacian matrix of graphs and trees,and characterize extremal graphs.
文摘Let G be a simple connected graph with n vertices and m edges,L G be the line graph of G and λ 1(L G)≥λ 2(L G)≥...≥λ m(L G) be the eigenvalues of the graph L G.In this paper,the range of eigenvalues of a line graph is considered.Some sharp upper bounds and sharp lower bounds of the eigenvalues of L G are obtained.In particular,it is proved that-2cos(πn)≤λ n-1 (L G)≤n-4 and λ n(L G)=-2 if and only if G is bipartite.
