Abstract: 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.

Key words:  Laplacian spectral radius, tricyclic graphs, maximum degree