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

Abstract

<jats:title>Abstract</jats:title><jats:p>A real symmetric matrix <jats:italic>A</jats:italic> is said to be completely positive if it can be written as <jats:italic>BBt</jats:italic> for some (not necessarily square) nonnegative matrix <jats:italic>B</jats:italic>. A simple graph <jats:italic>G</jats:italic> is called a completely positive graph if every matrix realization of <jats:italic>G</jats:italic> 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 𝒭.</jats:p>

Distance Matrix of Weighted Cactoid-type Digraphs

“…Observe that, for n = 2 with m j = 1, 1 ≤ j ≤ r, the graph is a complete bipartite graph K 2,r+1 , with the following vertex partition {u 0 , u 2 } and {u 1 , v j 1 ; 1 ≤ j ≤ r}. Thus, using [6,16], the result follows. Next to prove (ii), we assume m s = 2l + 1, where l ≥ 1.…”

“…Proof. In literature, the graph n = 1 with m j = 1, 1 ≤ j ≤ r, is denoted by T r and in [6], it was shown that det D(T r ) = (−1) r−1 2 r−2 . To prove (ii), we assume m s = 2l, where l ≥ 1.…”

Distance Matrix of a Multi-block Graph: Determinant and Inverse

“…We also provide the inverse of the distance matrix for a class of multi-block graphs with cofactor zero. Consequently, as a special case to multi-block graphs, we compute the determinant and inverse of the distance matrix for a class completely positive graphs, which improves the class studied in [4]. .…”