# The bipartite distance matrix of a nonsingular tree

Research Article

## Publication Title

Linear Algebra and Its Applications

## Abstract

Let G be a labeled, connected, bipartite graph with the bi-partition (L={l1,…,lk},R={r1,…,rp}) of the vertex set V. Let D be the usual distance matrix of G, where rows and columns are indexed by l1,…,lk,r1,…,rp. For X,Y⊆V, let us define DG[X,Y] as the submatrix of D induced by the rows indexed in X and columns indexed in Y. Let us call DG[L,R] the bipartite distance matrix of G. If G has a unique perfect matching, then k=p and we assume that the bi-partition is canonical, that is, [li,ri] are matching edges. For a nonsingular tree T, let us denote the bipartite distance matrix of T by B(T). We observe that det⁡B(T) is always a multiple of 2p−1. This is similar to the well known result of Graham and Pollak (1971) [1] which tells that the determinant of the usual distance matrix D is a multiple of 2n−2. Call the number bd(T):=det⁡B(T)/(−2)p−1 the bipartite distance index of T. We supply a recursive formula to compute this index. We show that this index satisfies an interesting inclusion-exclusion type of principle at any matching edge of the tree. Even more interestingly, we show that the index is completely characterized by the structure of T via what we call the f-alternating sums, that is, the sum f(T):=∑[d(u)−2][d(v)−2]S|Puv|/2, where the sum is taken over all u-v-alternating paths Puv, and S is the sequence (1,1,3,3,5,5,…). A well known result by Graham, Hoffman and Hosoya (1977) [2] is that the determinant of the distance matrix of a graph only depends on the blocks and is independent of how they are assembled. Such a result does not hold true for B(T). However, we identify some basic elements and a merging operation and show that each of the trees that can be constructed from a given set of elements, sequentially using this operation, have the same det⁡B(T), independent of the order in which the sequence is followed. For the class of trees that can be obtained in this way, we give a surprisingly simple way to evaluate the determinant of B(T).

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## DOI

10.1016/j.laa.2021.09.001

12-15-2021

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