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.

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 $\mathcal{R}$.