# Graphs that are cospectral for the distance Laplacian

## TL;DR

This paper develops methods to construct pairs of graphs that are cospectral with respect to the distance Laplacian matrix, revealing that many graph properties are not preserved under this spectral equivalence.

## Contribution

The authors introduce general techniques for generating infinite families of $ ext{D}^L$-cospectral graphs and analyze properties of the distance Laplacian characteristic polynomial.

## Key findings

- Multiple methods for constructing $ ext{D}^L$-cospectral graphs.
- Examples showing property non-preservation under $ ext{D}^L$-cospectrality.
- The coefficients of the distance Laplacian characteristic polynomial decrease in absolute value.

## 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 $\mathcal{D}(G)$. 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 $\delta^L_{k}$ is the coefficient of $x^k$.