  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 $\mathcal{R}$.
