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 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.

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 Computing the number of h-edge spanning forests in complete bipartite graphs

2014 ◽  
Vol Vol. 16 no. 1 (Analysis of Algorithms) ◽  
Author(s):  
Rebecca Stones
Keyword(s):  
Bipartite Graph ◽  
Spanning Trees ◽  
Analysis Of Algorithms ◽  
Complete Bipartite Graph ◽  
Bipartite Graphs ◽  
Spanning Forests ◽  
International Audience ◽  
Complete Bipartite Graphs ◽  
Number Of Spanning Trees

Analysis of Algorithms International audience Let fm,n,h be the number of spanning forests with h edges in the complete bipartite graph Km,n. Kirchhoff\textquoterights Matrix Tree Theorem implies fm,n,m+n-1=mn-1 nm-1 when m ≥1 and n ≥1, since fm,n,m+n-1 is the number of spanning trees in Km,n. In this paper, we give an algorithm for computing fm,n,h for general m,n,h. We implement this algorithm and use it to compute all non-zero fm,n,h when m ≤50 and n ≤50 in under 2 days.

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.

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 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

scholarly journals On the number of spanning trees of K_n^m #x00B1 G graphs

2006 ◽  
Vol Vol. 8 ◽  
Author(s):  
Stavros D. Nikolopoulos ◽  
Charis Papadopoulos
Keyword(s):  
Complete Graph ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Star Graph ◽  
Complete Multipartite Graph ◽  
The Family ◽  
International Audience ◽  
Multipartite Graph ◽  
Complete Multipartite ◽  
Number Of Spanning Trees

International audience The K_n-complement of a graph G, denoted by K_n-G, is defined as the graph obtained from the complete graph K_n by removing a set of edges that span G; if G has n vertices, then K_n-G coincides with the complement øverlineG of the graph G. In this paper we extend the previous notion and derive determinant based formulas for the number of spanning trees of graphs of the form K_n^m #x00b1 G, where K_n^m is the complete multigraph on n vertices with exactly m edges joining every pair of vertices and G is a multigraph spanned by a set of edges of K_n^m; the graph K_n^m + G (resp. K_n^m - G) is obtained from K_n^m by adding (resp. removing) the edges of G. Moreover, we derive determinant based formulas for graphs that result from K_n^m by adding and removing edges of multigraphs spanned by sets of edges of the graph K_n^m. We also prove closed formulas for the number of spanning tree of graphs of the form K_n^m #x00b1 G, where G is (i) a complete multipartite graph, and (ii) a multi-star graph. Our results generalize previous results and extend the family of graphs admitting formulas for the number of their spanning trees.

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