q-Analogs of distance matrices of 3-hypertrees

Abstract

We consider the distance matrix of trees in 3-uniform hypergraphs (which we call 3-hypertrees). We give a formula for the inverse of a few q-analogs of distance matrices of 3-hypertrees T. Some results are analogs of results by Bapat et al. for graphs. We give an alternate proof of the result that the determinant of the distance matrix of a 3-hypertree T depends only on n, the number of vertices of T. Further, we give a Pfaffian identity for a principal submatrix of some (skew-symmetrized) distance matrices of 3-hypertrees when we fix an ordering of the vertices and assign signs appropriately. A result of Graham, Hoffman and Hosoya relates the determinant of the distance matrix of a graph and the determinants of its 2-connected blocks. When the graph has as blocks a fixed connected graph H which satisfy some conditions, we give a formula for the inverse of its distance matrix. This result generalises a result of Graham and Lovasz. When each block of G is a fixed graph G, we also give some corollaries about the sum of the entries of the inverse of the distance matrix of G and some of its analogs. (C) 2009