# Graphs that are cospectral for the distance Laplacian

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## Abstract

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 $\mathcal{D}(G)$. Several general methods are established for producing $\mathcal{D}^L$-cospectral graphs that can be used to construct infinite families. Examples are provided to show that various properties are not preserved by $\mathcal{D}^L$-cospectrality, including examples of $\mathcal{D}^L$-cospectral strongly regular and circulant graphs. It is established that the absolute values of coefficients of the distance Laplacian characteristic polynomial are decreasing, i.e., $|\delta^L_{1}|\geq \cdots \geq |\delta^L_{n}|$, where $\delta^L_{k}$ is the coefficient of $x^k$.

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