3 research outputs found
Distance matrices of a tree: two more invariants, and in a unified framework
Graham-Pollak showed that for the distance matrix of a tree ,
det depends only on its number of edges. Several other variants of ,
including directed/multiplicative/- versions were studied, and always,
det depends only on the edge-data.
We introduce a general framework for bi-directed weighted trees, with
threefold significance. First, we improve on state-of-the-art for all known
variants, even in the classical Graham-Pollak case: we delete arbitrary pendant
nodes (and more general subsets) from the rows/columns of , and show these
minors do not depend on the tree-structure.
Second, our setting unifies all known variants (with entries in a commutative
ring). We further compute in closed form the inverse of , extending a result
of Graham-Lovasz [Adv. Math. 1978] and answering a question of Bapat-Lal-Pati
[Lin. Alg. Appl. 2006].
Third, we compute a second function of the matrix : the sum of all its
cofactors, cof. This was worked out in the simplest setting by
Graham-Hoffman-Hosoya (1978), but is relatively unexplored for other variants.
We prove a stronger result, in our general setting, by computing cof for
minors as above, and showing these too depend only on the edge-data.
Finally, we show our setting is the 'most general possible', in that with
more freedom in the edgeweights, det and cof depend on the tree
structure. In a sense, this completes the study of the invariant det
(and cof) for trees with edge-data in a commutative ring.
Moreover: for a bi-directed graph we prove multiplicative
Graham-Hoffman-Hosoya type formulas for det, cof, . We
then show how this subsumes their 1978 result. The final section introduces and
computes a third, novel invariant for trees and a Graham-Hoffman-Hosoya type
result for our "most general" distance matrix .Comment: 42 pages, 2 figures; minor edits in the proof of Theorems A and 1.1
On The Determinant of q-Distance Matrix of a Graph
In this note, we show how the determinant of the q-distance matrix Dq(T) of a weighted directed graph G can be expressed in terms of the corresponding determinants for the blocks of G, and thus generalize the results obtained by Graham et al. [R.L. Graham, A.J. Hoffman and H. Hosoya, On the distance matrix of a directed graph, J. Graph Theory 1 (1977) 85-88]. Further, by means of the result, we determine the determinant of the q-distance matrix of the graph obtained from a connected weighted graph G by adding the weighted branches to G, and so generalize in part the results obtained by Bapat et al. [R.B. Bapat, S. Kirkland and M. Neumann, On distance matrices and Laplacians, Linear Algebra Appl. 401 (2005) 193- 209]. In particular, as a consequence, determinantal formulae of q-distance matrices for unicyclic graphs and one class of bicyclic graphs are presented