Abstract: The largest eigenvalue of the distance Laplacian matrix of a connected graph G is called the distance Laplacian spectral radius of the graph $G$. In this paper we obtain a sharp lower bound of distance Laplacian spectral radius, and then using the bound we determine the unique graph which has the minimum distance Laplacian spectral radius among all unicyclic graphs. Finally, by using the bound again as well as the characteristics polynomial of a distance Laplacian matrix, we characterize the unique graph with the minimum distance Laplacian spectral radius among all bicyclic graphs.

Key words: distance Laplacian spectral radius, unicyclic graph, bicyclic graph