scholarly journals Spanning trees of finite Sierpiński graphs

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Elmar Teufl ◽  
Stephan Wagner
Keyword(s):  
Spanning Trees ◽  
Dynamical Behavior ◽  
The Self ◽  
3 Dimensional ◽  
Sierpiński Graphs ◽  
Polynomial Map ◽  
International Audience ◽  
Self Similar ◽  
Number Of Spanning Trees

International audience We show that the number of spanning trees in the finite Sierpiński graph of level $n$ is given by $\sqrt[4]{\frac{3}{20}} (\frac{5}{3})^{-n/2} (\sqrt[4]{540})^{3^n}$. The proof proceeds in two steps: First, we show that the number of spanning trees and two further quantities satisfy a $3$-dimensional polynomial recursion using the self-similar structure. Secondly, it turns out, that the dynamical behavior of the recursion is given by a $2$-dimensional polynomial map, whose iterates can be computed explicitly.

scholarly journals Asymptotic enumeration on self-similar graphs with two boundary vertices

2009 ◽  
Vol Vol. 11 no. 1 (Combinatorics) ◽  
Author(s):  
Elmar Teufl ◽  
Stephan Wagner
Keyword(s):  
Spanning Trees ◽  
General Setting ◽  
Scaling Factor ◽  
Asymptotic Enumeration ◽  
Spanning Forests ◽  
International Audience ◽  
Self Similar ◽  
Connected Subgraphs ◽  
Graph Parameters ◽  
Number Of Spanning Trees

Combinatorics International audience We study two graph parameters, namely the number of spanning forests and the number of connected subgraphs, for self-similar graphs with exactly two boundary vertices. In both cases, we determine the general behavior for these and related auxiliary quantities by means of polynomial recurrences and a careful asymptotic analysis. It turns out that the so-called resistance scaling factor of a graph plays an essential role in both instances, a phenomenon that was previously observed for the number of spanning trees. Several explicit examples show that our findings are likely to hold in an even more general setting.

scholarly journals Chip-Firing and Rotor-Routing on $\mathbb{Z}^d$ and on Trees

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Itamar Landau ◽  
Lionel Levine ◽  
Yuval Peres
Keyword(s):  
Spanning Trees ◽  
Initial Condition ◽  
Simple Process ◽  
Regular Tree ◽  
Single Vertex ◽  
Cyclic Subgroups ◽  
Chip Firing ◽  
International Audience ◽  
Number Of Spanning Trees ◽  
Sandpile Group

International audience The sandpile group of a graph $G$ is an abelian group whose order is the number of spanning trees of $G$. We find the decomposition of the sandpile group into cyclic subgroups when $G$ is a regular tree with the leaves are collapsed to a single vertex. This result can be used to understand the behavior of the rotor-router model, a deterministic analogue of random walk studied first by physicists and more recently rediscovered by combinatorialists. Several years ago, Jim Propp simulated a simple process called rotor-router aggregation and found that it produces a near perfect disk in the integer lattice $\mathbb{Z}^2$. We prove that this shape is close to circular, although it remains a challenge to explain the near perfect circularity produced by simulations. In the regular tree, we use the sandpile group to prove that rotor-router aggregation started from an acyclic initial condition yields a perfect ball.

scholarly journals Spanning forests, electrical networks, and a determinant identity

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Elmar Teufl ◽  
Stephan Wagner
Keyword(s):  
Spanning Trees ◽  
Symmetric Graph ◽  
Electrical Networks ◽  
Laplace Matrix ◽  
Nous Montrons ◽  
Spanning Forests ◽  
International Audience ◽  
Self Similar ◽  
Algebraic Proofs

International audience We aim to generalize a theorem on the number of rooted spanning forests of a highly symmetric graph to the case of asymmetric graphs. We show that this can be achieved by means of an identity between the minor determinants of a Laplace matrix, for which we provide two different (combinatorial as well as algebraic) proofs in the simplest case. Furthermore, we discuss the connections to electrical networks and the enumeration of spanning trees in sequences of self-similar graphs. Nous visons à généraliser un théorème sur le nombre de forêts couvrantes d'un graphe fortement symétrique au cas des graphes asymétriques. Nous montrons que cela peut être obtenu au moyen d'une identité sur les déterminants mineurs d'une matrice Laplacienne, pour laquelle nous donnons deux preuves différentes (combinatoire ou bien algébrique) dans le cas le plus simple. De plus, nous discutons les relations avec des réseaux électriques et l'énumération d'arbres couvrants dans de suites de graphes autosimilaires.

The Number of Spanning Trees in Self-Similar Graphs

2011 ◽  
Vol 15 (2) ◽  
pp. 355-380 ◽  
Author(s):  
Elmar Teufl ◽  
Stephan Wagner
Keyword(s):  
Spanning Trees ◽  
Self Similar ◽  
Number Of Spanning Trees

scholarly journals On the self-similarity index of 𝑝-adic analytic pro-𝑝 groups

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Francesco Noseda ◽  
Ilir Snopce
Keyword(s):  
Rooted Tree ◽  
Similarity Index ◽  
The Self ◽  
Self Similarity ◽  
Similar Action ◽  
3 Dimensional ◽  
Self Similar

Abstract Let 𝑝 be a prime. We say that a pro-𝑝 group is self-similar of index p k p^{k} if it admits a faithful self-similar action on a p k p^{k} -ary regular rooted tree such that the action is transitive on the first level. The self-similarity index of a self-similar pro-𝑝 group 𝐺 is defined to be the least power of 𝑝, say p k p^{k} , such that 𝐺 is self-similar of index p k p^{k} . We show that, for every prime p ⩾ 3 p\geqslant 3 and all integers 𝑑, there exist infinitely many pairwise non-isomorphic self-similar 3-dimensional hereditarily just-infinite uniform pro-𝑝 groups of self-similarity index greater than 𝑑. This implies that, in general, for self-similar 𝑝-adic analytic pro-𝑝 groups, one cannot bound the self-similarity index by a function that depends only on the dimension of the group.

scholarly journals Dynamical behavior of the strong dispersive nonlinear wave equation

2021 ◽  
Vol XXII C ◽  
pp. 3-11
Author(s):  
Mehmed Kodzha
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Dynamical Behavior ◽  
The Self ◽  
Self Similar ◽  
Similar Solutions

In this paper we consider the dynamical behavior of solutions near explicit self-similar solutions for a strong dispersive nonlinear wave equation. First we construct explicit self-similar solutions, then we investigate dynamical behavior of the solutions near to the self-similar solutions.

scholarly journals The number of spanning trees of an infinite family of outerplanar, small-world and self-similar graphs

2013 ◽  
Vol 392 (12) ◽  
pp. 2803-2806 ◽  
Author(s):  
Francesc Comellas ◽  
Alícia Miralles ◽  
Hongxiao Liu ◽  
Zhongzhi Zhang
Keyword(s):  
Spanning Trees ◽  
Infinite Family ◽  
Small World ◽  
Self Similar ◽  
Number Of Spanning Trees
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