Extremal results on distance Laplacian spectral radius of graphs

The distance Laplacian spectral radius of a connected graph $G$ is the largest eigenvalue of its distance Laplacian matrix $\mathcal{L}(G)$ defined as $\mathcal{L}(G)=Tr(G)-D(G)$, where $Tr(G)$ is the diagonal matrix of vertex transmissions and $D(G)$ is the distance matrix of $G$. We determine the unique trees with maximum distance Laplacian spectral radius among trees of perfect matching with given maximum degree, the unique trees with second (third, respectively) maximum distance Laplacian spectral radius, and the unique bipartite unicyclic graphs with maximum distance Laplacian spectral radius. We also determine the unique graphs with minimum distance Laplacian spectral radius among bicyclic graphs.