On the distance Laplacian spectra of graphs

The distance Laplacian matrix of a connected graph G is defined in [2] and [3] and it is proved that for a graph G on n vertices, if the complement of G is connected, then the second smallest distance Laplacian eigenvalue is strictly greater than n  . In this article, we consider the graphs whose complement is a tree or a unicyclic graph, and characterize the graphs among them having n+1n+1 as the second smallest distance Laplacian eigenvalue. We prove that the largest distance Laplacian eigenvalue of a path is simple and the corresponding eigenvector has the similar property like that of a Fiedler vector.