# The normalized distance Laplacian

## TL;DR

This paper introduces the normalized distance Laplacian matrix for graphs, explores its spectral properties, bounds, and relationships with other matrices, and investigates its cospectrality characteristics.

## Contribution

The paper defines the normalized distance Laplacian, establishes bounds and properties, and studies its spectral and cospectral features compared to other graph matrices.

## Key findings

- Bounds on the spectral radius of the normalized distance Laplacian
- Connections with the normalized Laplacian matrix
- Fewer cospectral pairs for the normalized distance Laplacian among small graphs

## Abstract

The distance matrix $\mathcal{D}(G)$ of a graph $G$ is the matrix containing the pairwise distances between vertices. The transmission of a vertex $v_i$ in $G$ is the sum of the distances from $v_i$ to all other vertices and $T(G)$ is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, $\mathcal{D}^{\mathcal {L}}(G)=I-T(G)^{-1/2}\mathcal{D}(G) T(G)^{-1/2}$, is introduced. This is analogous to the normalized Laplacian matrix, $\mathcal{L}(G)=I-D(G)^{-1/2}A(G)D(G)^{-1/2}$, where $D(G)$ is the diagonal matrix of degrees of the vertices of the graph and $A(G)$ is the adjacency matrix. Bounds on the spectral radius of $\mathcal{D}^{\mathcal {L}}$ and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The generalized distance characteristic polynomial is defined and its properties discussed. Finally, $\mathcal{D}^{\mathcal {L}}$-cospectrality is studied for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.