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

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