关于三圈图的拉普拉斯谱半径的一些结果

边数等于点数加二的连通图称为三圈图.~设 ~$\Delta(G)$~和~$\mu(G)$~
分别表示图~$G$~的最大度和其拉普拉斯谱半径,设${\mathcal
T}(n)$~表示所有~$n$~阶三圈图的集合,证明了对于~${\mathcal
T}(n)$~的两个图~$H_{1}$~和~$H_{2}$~,~若~$\Delta(H_{1})>
\Delta(H_{2})$ ~且 ~$\Delta(H_{1})\geq \frac{n+7}{2}$,~则~$\mu
(H_{1})> \mu (H_{2}).$ 作为该结论的应用,~确定了~${\mathcal
T}(n)(n\geq9)$~中图的第七大至第十九大的拉普拉斯谱半径及其相应的极图.

Some Results on the Laplacian Spectral Radii of Tricyclic Graphs

A tricyclic graph is a connected graph in which the number of edges equals the number of vertices plus two. Let $\Delta(G)$ and $\mu(G)$ denote the maximum degree and the Laplacian spectral radius of a graph $G$, respectively. Let $\mathcal {T}(n)$ be the set of tricyclic graphs on $n$ vertices.~In this paper,~it is proved that,~for two graphs $H_{1}$ and $H_{2}$ in $\mathcal {T}(n)$,~if $\Delta(H_{1})> \Delta(H_{2})$ and $\Delta(H_{1})\geq \frac{n+7}{2}$,~then $\mu (H_{1})> \mu (H_{2}).$ As an application of this result,~we determine the seventh to the nineteenth largest values of the Laplacian spectral radii among all the graphs in $\mathcal {T}(n)(n\geq9)$ together with the corresponding graphs.