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 BBt for some (not necessarily square) nonnegative matrix B. A simple graph G is called a completely positive graph if every matrix realization of G that is both nonnegative and positive semidefinite 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. We compute a matrix 𝒭 such that the inverse of the distance matrix of a class of completely positive graphs is expressed a linear combination of the Laplacian matrix, a rank one matrix of all ones and 𝒭. This expression is similar to the existing result for trees. We also bring out interesting spectral properties of some of the principal submatrices of 𝒭.