Multiplicities of distance Laplacian eigenvalues and forbidden subgraphs

In this work, the graphs of order n having the second distance Laplacian eigenvalue of multiplicity n−2 are determined. Besides that, this result also characterizes the graphs where the multiplicity of some distance Laplacian eigenvalue is equal to n−2. In addition, all connected graphs of order n where the largest eigenvalue of the distance Laplacian matrix has multiplicity n−3 are determined. Finally, we determine some graphs with a distance Laplacian eigenvalue having multiplicity n−3.