Graphs that are cospectral for the distance Laplacian

The distance matrix $\mathcal{D}(G)$ of a graph $G$ is the matrix containing the pairwise distances between vertices, and the distance Laplacian matrix is ${\mathcal{D}}^L(G)=T(G)-\mathcal{D}(G)$, where $T(G)$ is the diagonal matrix of row sums of $\mathcal{D}(G)$. We establish several general methods for producing ${\mathcal{D}}^L$-cospectral graphs that can be used to construct infinite families. We provide examples showing that various properties are not preserved by ${\mathcal{D}}^L$-cospectrality, including examples of ${\mathcal{D}}^L$-cospectral strongly regular and circulant graphs. We establish that the absolute values of coefficients of the distance Laplacian characteristic polynomial are decreasing, i.e., $|\delta^L_{1}|\geq \dots \geq |\delta^L_{n}|$where$\delta^L_{k}$isthecoefficientof$x^k$.