scholarly journals An Algebraic Analogue of a Formula of Knuth

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Lionel Levine
Keyword(s):  
Directed Graph ◽  
Spanning Trees ◽  
Line Graph ◽  
Torsion Subgroup ◽  
Directed Line ◽  
De Bruijn Graphs ◽  
International Audience ◽  
De Bruijn ◽  
Kautz Graphs ◽  
Sandpile Group

International audience We generalize a theorem of Knuth relating the oriented spanning trees of a directed graph $G$ and its directed line graph $\mathcal{L} G$. The sandpile group is an abelian group associated to a directed graph, whose order is the number of oriented spanning trees rooted at a fixed vertex. In the case when $G$ is regular of degree $k$, we show that the sandpile group of $G$ is isomorphic to the quotient of the sandpile group of $\mathcal{L} G$ by its $k$-torsion subgroup. As a corollary we compute the sandpile groups of two families of graphs widely studied in computer science, the de Bruijn graphs and Kautz graphs. Nous généralisons un théorème de Knuth qui relie les arbres couvrants dirigés d'un graphe orienté $G$ au graphe adjoint orienté $\mathcal{L} G$. On peut associer à tout graphe orienté un groupe abélien appelé groupe du tas de sable, et dont l'ordre est le nombre d'arbres couvrants dirigés enracinés en un sommet fixé. Lorsque $G$ est régulier de degré $k$, nous montrons que le groupe du tas de sable de $G$ est isomorphe au quotient du groupe du tas de sable de $\mathcal{L} G$ par son sous-groupe de $k$-torsion. Comme corollaire, nous déterminons les groupes de tas de sable de deux familles de graphes étudiées en informatique: les graphes de de Bruijn et les graphes de Kautz.

A Bijective Proof of a Theorem of Knuth

2010 ◽  
Vol 20 (1) ◽  
pp. 11-25 ◽  
Author(s):  
HODA BIDKHORI ◽  
SHAUNAK KISHORE
Keyword(s):  
Spanning Trees ◽  
Line Graph ◽  
Binary Sequences ◽  
Critical Groups ◽  
De Bruijn Graphs ◽  
Bijective Proof ◽  
The Matrix ◽  
De Bruijn ◽  
Kautz Graphs ◽  
Number Of Spanning Trees

The line graph G of a directed graph G has a vertex for every edge of G and an edge for every path of length 2 in G. In 1967, Knuth used the Matrix Tree Theorem to prove a formula for the number of spanning trees of G, and he asked for a bijective proof [6]. In this paper, we give a bijective proof of Knuth's formula. As a result of this proof, we find a bijection between binary de Bruijn sequences of degree n and binary sequences of length 2n−1. Finally, we determine the critical groups of all the Kautz graphs and de Bruijn graphs, generalizing a result of Levine [7].

scholarly journals The number of Euler tours of a random $d$-in/$d$-out graph

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Páidí Creed ◽  
Mary Cryan
Keyword(s):  
Asymptotic Distribution ◽  
Directed Graph ◽  
Polynomial Time ◽  
Simple Approach ◽  
Uniform Sampling ◽  
Concentration Result ◽  
International Audience ◽  
De Bruijn ◽  
Euler Tours

International audience In this paper we obtain the expectation and variance of the number of Euler tours of a random $d$-in/$d$-out directed graph, for $d \geq 2$. We use this to obtain the asymptotic distribution and prove a concentration result. We are then able to show that a very simple approach for uniform sampling or approximately counting Euler tours yields algorithms running in expected polynomial time for almost every $d$-in/$d$-out graph. We make use of the BEST theorem of de Bruijn, van Aardenne-Ehrenfest, Smith and Tutte, which shows that the number of Euler tours of a $d$-in/$d$-out graph is the product of the number of arborescences and the term $[(d-1)!]^n/n$. Therefore most of our effort is towards estimating the asymptotic distribution of the number of arborescences of a random $d$-in/$d$-out graph.

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 SIFAT KEPRIMAAN MODUL SEDERHANA CHEN UNTUK GRAF 𝐴∞

2019 ◽  
Vol 12 (2) ◽  
Author(s):  
Risnawita Risnawita ◽  
Irawati Irawati ◽  
Intan Muchtadi Alamsyah
Keyword(s):  
Directed Graph ◽  
Infinite Sequence ◽  
Line Graph ◽  
Simple Module ◽  
Path Algebra ◽  
Leavitt Path Algebra ◽  
Directed Line ◽  
Simple Modules ◽  
Infinite Path ◽  
Prime Module

Let 𝐾𝐾 be a field, 𝐸𝐸 is a directed graph. Let 𝐴𝐴~ is a directed line graph. Suppose that 𝑉𝑉[𝑝𝑝] is a class of Chen simple module for the Leavitt path algebra (𝐿𝐿𝐾𝐾 (𝐸𝐸)), with [p] being equivalent classes containing an infinite path. An infinite path p is an infinite sequence from the sides of a graph. In this paper it will be shown that 𝑉𝑉[𝑝𝑝]is not a prime module of the Leavitt path algebra for graph 𝐴𝐴∞ .Keywords : Leavitt path algebra, Graph 𝐴𝐴~, Chen simple modules, Prime modules

Power domination in generalized undirected de Bruijn graphs and Kautz graphs

2015 ◽  
Vol 07 (01) ◽  
pp. 1550003 ◽  
Author(s):  
Jyhmin Kuo ◽  
Wei-Lun Wu
Keyword(s):  
Power System ◽  
Dominating Set ◽  
Vertex Cover ◽  
Domination Number ◽  
Electric Power System ◽  
Minimum Cardinality ◽  
Power Domination ◽  
De Bruijn Graphs ◽  
De Bruijn ◽  
Kautz Graphs

To monitor an electric power system by placing as few phase measurement units (PMUs) as possible is closely related to the famous vertex cover problem and domination problem in graph theory. A set P is a power dominating set (PDS) of a graph G = (V, E), if every vertex and every edge in the system is observed following the observation rules of power system monitoring. The minimum cardinality of a PDS of a graph G is the power domination number γp(G). In this paper, we determine the upper bounds of power domination number of generalized undirected de Bruijn graphs and generalized undirected Kautz graphs.

scholarly journals A Note on the Critical Group of a Line Graph

10.37236/611 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
David Perkinson ◽  
Nick Salter ◽  
Tianyuan Xu
Keyword(s):  
Directed Graph ◽  
Line Graph ◽  
Critical Group ◽  
Directed Line

This note answers a question posed by Levine. The main result shows that under certain circumstances a critical group of a directed graph is the quotient of a critical group of its directed line graph.

scholarly journals Buffering Updates Enables Efficient Dynamic de Bruijn Graphs

2021 ◽  
Author(s):  
Jarno Alanko ◽  
Bahar Alipanahi ◽  
Jonathen Settle ◽  
Christina Boucher ◽  
Travis Gagie
Keyword(s):  
De Bruijn Graphs ◽  
De Bruijn
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