32,567 research outputs found

    Abelian sandpiles: an overview and results on certain transitive graphs

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    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

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    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

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    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 GG with a linear arrangement of bandwidth bb can be embedded into a distribution T\mathcal T of spanning trees such that the expected stretch of each edge of GG is O(b2)O(b^2). Our proof implies a linear time algorithm for sampling from T\mathcal T. Therefore, we have a linear time algorithm that finds a spanning tree of GG with average stretch O(b2)O(b^2) with high probability. We also describe a deterministic linear-time algorithm for computing a spanning tree of GG with average stretch O(b3)O(b^3). For graphs of cutwidth cc, we construct a spanning tree with stretch O(c2)O(c^2) in linear time. Finally, when GG has treewidth kk we provide a dynamic programming algorithm computing a minimum stretch spanning tree of GG that runs in polynomial time with respect to the number of vertices of GG

    Minimum spanning trees on random networks

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    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 (P(ϵ)P(\epsilon)) found using uniform disorder. We show that the MST energy for other disorder distributions is simply related to P(ϵ)P(\epsilon). 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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