# Distance Matrix of a Class of Completely Positive Graphs: Determinant and Inverse

## TL;DR

This paper investigates the determinant and inverse of the distance matrix for a specific class of completely positive graphs, revealing relations involving the Laplacian and eigenvalues of submatrices.

## Contribution

It provides explicit formulas for the inverse and determinant of the distance matrix for a class of completely positive graphs, extending known results from trees.

## Key findings

- Derived formulas for the inverse of the distance matrix.
- Identified eigenvalues of principal submatrices of matrix R.
- Established relations involving the Laplacian matrix.

## Abstract

A real symmetric matrix $A$ is said to be completely positive if it can be written as $BB^t$ for some (not necessarily square) nonnegative matrix $B$. A simple graph $G$ is called a completely positive graph if every doubly nonnegative matrix realization of $G$ is a completely positive matrix. Our aim in this manuscript is to compute the determinant and inverse (when it exists) of the distance matrix of a class of completely positive graphs. Similar to trees, we obtain a relation for the inverse of the distance matrix of a class of completely positive graphs involving the Laplacian matrix, a rank one matrix and a matrix $\mathcal{R}$. We also determine the eigenvalues of some principal submatrices of matrix $\mathcal{R}$.

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Source: https://tomesphere.com/paper/1906.04636