A q-analogue of the bipartite distance matrix of a nonsingular tree

Abstract

All graphs considered here are simple and finite. Let G be a labeled, connected, bipartite graph and let (L,R) be a vertex bipartition of G. The bipartite adjacency matrix A of G is the submatrix of its adjacency matrix determined by the rows corresponding to L and the columns corresponding to R, see [6]. In a similar way, one can consider the bipartite distance matrix of G. It is the submatrix of the usual distance matrix D of G determined by the rows corresponding to L and the columns corresponding to R. Let G=T be a tree with a unique perfect matching. Very recently, in [2], it was shown that this matrix possesses many interesting properties and has a very deep combinatorial relation with the tree structure. One of those results, is the elegant description of the determinant of the bipartite distance matrix of T by a combinatorial sum over the collection of all alternating paths. Another result is reminiscent of the well known result of Graham and Pollack, which tells that the determinant of the usual distance matrix a tree of order 2p is a multiple of 22p−2. It was shown that the determinant of the bipartite distance matrix of a tree of order 2p with a unique perfect matching is a multiple of 2p−1.There are many studies of distance-like matrices available in the literature. In this article, we consider two of them. The first one is the q-distance matrix which is a generalization of the usual distance matrix. We show that the determinant of the q-bipartite distance matrix of a tree with a unique perfect matching can still be expressed as a combinatorial sum over the set of alternating paths with respect to a suitable sequence. The result of [2] follows as a corollary when q=1.The second distance-like matrix we study is the exponential distance matrix, which has a very simple expression for the determinant. We show that an extremely similar result hold here too. This is unexpected, as our matrix is of half the order.Recall that, the determinant of the usual distance matrix of a tree is just dependent on the number of vertices and is independent of the tree structure. In [2], it was shown that the determinant of the bipartite distance matrix of a tree with a unique perfect matching was dependent on the tree structure and on some smaller classes it was independent of the tree structure. We prove similar results for the q-bipartite distance matrix.However, as another surprise, we show that the determinant of the exponential bipartite distance matrix of a tree with a unique perfect matching is independent of the tree structure.