#### Title

The bipartite distance matrix of a nonsingular tree

#### Article Type

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 detB(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):=detB(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 detB(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).

#### First Page

254

#### Last Page

281

#### DOI

10.1016/j.laa.2021.09.001

#### Publication Date

12-15-2021

#### Recommended Citation

Bapat, R. B.; Jana, Rakesh; and Pati, S., "The bipartite distance matrix of a nonsingular tree" (2021). *Journal Articles*. 1650.

https://digitalcommons.isical.ac.in/journal-articles/1650