3,246 research outputs found
Completely independent spanning trees in some Cartesian product graphs
Let be spanning trees of a graph . For any two vertices of , if the paths from to in these trees are pairwise openly disjoint, then we say that are completely independent. Hasunuma showed that there are two completely independent spanning trees in any 4-connected maximal planar graph, and that given a graph , the problem of deciding whether there exist two completely independent spanning trees in is NP-complete. In this paper, we consider the number of completely independent spanning trees in some Cartesian product graphs such as
Dimers, Tilings and Trees
Generalizing results of Temperley, Brooks, Smith, Stone and Tutte and others
we describe a natural equivalence between three planar objects: weighted
bipartite planar graphs; planar Markov chains; and tilings with convex
polygons. This equivalence provides a measure-preserving bijection between
dimer coverings of a weighted bipartite planar graph and spanning trees on the
corresponding Markov chain. The tilings correspond to harmonic functions on the
Markov chain and to ``discrete analytic functions'' on the bipartite graph.
The equivalence is extended to infinite periodic graphs, and we classify the
resulting ``almost periodic'' tilings and harmonic functions.Comment: 23 pages, 5 figure
Searching for a Connection Between Matroid Theory and String Theory
We make a number of observations about matter-ghost string phase, which may
eventually lead to a formal connection between matroid theory and string
theory. In particular, in order to take advantage of the already established
connection between matroid theory and Chern-Simons theory, we propose a
generalization of string theory in terms of some kind of Kahler metric. We show
that this generalization is closely related to the Kahler-Chern-Simons action
due to Nair and Schiff. In addition, we discuss matroid/string connection via
matroid bundles and a Schild type action, and we add new information about the
relationship between matroid theory, D=11 supergravity and Chern-Simons
formalism.Comment: 28 pages, LaTex, section 6 and references adde
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