关于双圈图的拉普拉斯谱半径的注记(英文)

A Note on the Laplacian Spectral Radii of Bicyclic Graphs

边数等于点数加1的连通图称为双圈图.设B(n)表示所有n阶双圈图的集合,μ(G)和Δ(G)分别表示图G的拉普拉斯谱半径和其最大度.本文证明了对于B(n)中的两个图G1和G_2,若Δ(G_1)>Δ(G_2)且△(G_1)≥(n+5)/2,则μ(G_1)>μ(G_2).作为该结论的应用,本文确定了B(n)中图的第五大至第八大的拉普拉斯谱半径以及相应的极图(其中前四大的拉普拉斯谱半径以及相应的极图在文献[C.X.He,J.Y.Shao,J.L.He,On the Laplacian spectral radii of bicyclicgraphs,Discrete Mathematics,2008,308:5981-5995.]中已确定).

Bicyclic graph is a connected graph in which the number of edges equals the number of vertices plus one.Denote byβ（n） the set of bicyclic graphs on n vertices.Letμ（G） and△（G） denote the Laplacian spectral radius and the maximum degree of a graph G, respectively.In this paper,it is proved that,for two graphs G_1 and G_2 inβ（n）,if△（G_1）△（G_2） and A（G_1）≥（n＋5）/2,thenμ（G_1）μ（G_2）.As an application of this result,we determine the fifth to the eighth largest values of the Laplacian spectral radii among all the graphs inβ（n）（the first four values were determined in[C.X.He,J.Y.Shao,J.L.He,On the Laplacian spectral radii of bicyclic graphs,Discrete Mathematics,2008,308：5981-5995.]） together with the corresponding graphs.