Realizability of a constructed matrix as a distance Laplacian matrix

Ascertain whether the matrix M defined in Proposition \ref{Qmat contains distinct eigenvalues of M} is realizable as the distance Laplacian matrix D^{L}(H) of some graph H.

Background

After presenting a specific matrix M whose smallest equitable quotient captures all distinct eigenvalues, the authors remark that it resembles a Laplacian-type structure but do not know whether it is a distance Laplacian matrix of any graph.

Establishing such a realization would connect the constructed algebraic example with a concrete graph-theoretic object and clarify the scope of the method for distance-based graph matrices.