32,567 research outputs found
Abelian sandpiles: an overview and results on certain transitive graphs
We review the Majumdar-Dhar bijection between recurrent states of the Abelian
sandpile model and spanning trees. We generalize earlier results of Athreya and
Jarai on the infinite volume limit of the stationary distribution of the
sandpile model on Z^d, d >= 2, to a large class of graphs. This includes: (i)
graphs on which the wired spanning forest is connected and has one end; (ii)
transitive graphs with volume growth at least c n^5 on which all bounded
harmonic functions are constant. We also extend a result of Maes, Redig and
Saada on the stationary distribution of sandpiles on infinite regular trees, to
arbitrary exhaustions.Comment: 44 pages. Version 2 incorporates some smaller changes. To appear in
Markov Processes and Related Fields in the proceedings of the meeting:
Inhomogeneous Random Systems, Stochastic Geometry and Statistical Mechanics,
Institut Henri Poincare, Paris, 27 January 201
The number and degree distribution of spanning trees in the Tower of Hanoi graph
The number of spanning trees of a graph is an important invariant related to topological and dynamic properties of the graph, such as its reliability, communication aspects, synchronization, and so on. However, the practical enumeration of spanning trees and the study of their properties remain a challenge, particularly for large networks. In this paper, we study the number and degree distribution of the spanning trees in the Hanoi graph. We first establish recursion relations between the number of spanning trees and other spanning subgraphs of the Hanoi graph, from which we find an exact analytical expression for the number of spanning trees of the n-disc Hanoi graph. This result allows the calculation of the spanning tree entropy which is then compared with those for other graphs with the same average degree. Then, we introduce a vertex labeling which allows to find, for each vertex of the graph, its degree distribution among all possible spanning trees.Postprint (author's final draft
Low-Stretch Spanning Trees of Graphs with Bounded Width
We study the problem of low-stretch spanning trees in graphs of bounded
width: bandwidth, cutwidth, and treewidth. We show that any simple connected
graph with a linear arrangement of bandwidth can be embedded into a
distribution of spanning trees such that the expected stretch of
each edge of is . Our proof implies a linear time algorithm for
sampling from . Therefore, we have a linear time algorithm that
finds a spanning tree of with average stretch with high
probability. We also describe a deterministic linear-time algorithm for
computing a spanning tree of with average stretch . For graphs of
cutwidth , we construct a spanning tree with stretch in linear
time. Finally, when has treewidth we provide a dynamic programming
algorithm computing a minimum stretch spanning tree of that runs in
polynomial time with respect to the number of vertices of
Minimum spanning trees on random networks
We show that the geometry of minimum spanning trees (MST) on random graphs is
universal. Due to this geometric universality, we are able to characterise the
energy of MST using a scaling distribution () found using uniform
disorder. We show that the MST energy for other disorder distributions is
simply related to . We discuss the relationship to invasion
percolation (IP), to the directed polymer in a random media (DPRM) and the
implications for the broader issue of universality in disordered systems.Comment: 4 pages, 3 figure
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