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

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 $\mathcal{R}$.