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.
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
Let K be the quasi-Laplacian matrix of a graph G and B be the adjacency matrix of the line graph of G, respectively. In this paper, we first present two sharp upper bounds for the largest Laplacian eigenvalue of G by ...Let K be the quasi-Laplacian matrix of a graph G and B be the adjacency matrix of the line graph of G, respectively. In this paper, we first present two sharp upper bounds for the largest Laplacian eigenvalue of G by applying the non-negative matrix theory to the similar matrix D-1/2 KD 1/2 and U-1/2 BU 1/2, respectively, where D is the degree diagonal matrix of G and U=diag(dudv: uv ∈ E(G)).And then we give another type of the upper bound in terms of the degree of the vertex and the edge number of G. Moreover, we determine all extremal graphs which achieve these upper bounds. Finally,some examples are given to illustrate that our results are better than the earlier and recent ones in some sense.展开更多
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
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.
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.
基金This work was supported by the Natural Science Foundation of Sichuan Province (Grant No.2006C040)
文摘Let K be the quasi-Laplacian matrix of a graph G and B be the adjacency matrix of the line graph of G, respectively. In this paper, we first present two sharp upper bounds for the largest Laplacian eigenvalue of G by applying the non-negative matrix theory to the similar matrix D-1/2 KD 1/2 and U-1/2 BU 1/2, respectively, where D is the degree diagonal matrix of G and U=diag(dudv: uv ∈ E(G)).And then we give another type of the upper bound in terms of the degree of the vertex and the edge number of G. Moreover, we determine all extremal graphs which achieve these upper bounds. Finally,some examples are given to illustrate that our results are better than the earlier and recent ones in some sense.
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.
