scholarly journals Chip-Firing Game and a Partial Tutte Polynomial for Eulerian Digraphs

10.37236/3924 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Kévin Perrot ◽  
Trung Van Pham
Keyword(s):  
Generating Function ◽  
Directed Graphs ◽  
Discrete Dynamical System ◽  
Tutte Polynomial ◽  
Direct Consequence ◽  
Undirected Graphs ◽  
Underlying Graph ◽  
Eulerian Digraph ◽  
Chip Firing ◽  
Strong Relation

The Chip-firing game is a discrete dynamical system played on a graph, in which chips move along edges according to a simple local rule. Properties of the underlying graph are of course useful to the understanding of the game, but since a conjecture of Biggs that was proved by Merino López, we also know that the study of the Chip-firing game can give insights on the graph. In particular, a strong relation between the partial Tutte polynomial $T_G(1,y)$ and the set of recurrent configurations of a Chip-firing game (with a distinguished sink vertex) has been established for undirected graphs. A direct consequence is that the generating function of the set of recurrent configurations is independent of the choice of the sink for the game, as it characterizes the underlying graph itself. In this paper we prove that this property also holds for Eulerian directed graphs (digraphs), a class on the way from undirected graphs to general digraphs. It turns out from this property that the generating function of the set of recurrent configurations of an Eulerian digraph is a natural and convincing candidate for generalizing the partial Tutte polynomial $T_G(1,y)$ to this class. Our work also gives some promising directions of looking for a generalization of the Tutte polynomial to general digraphs.

Chip firing and the tutte polynomial

1997 ◽  
Vol 1 (1) ◽  
pp. 253-259 ◽  
Author(s):  
Criel Merino López
Keyword(s):  
Tutte Polynomial ◽  
Chip Firing

scholarly journals RANDOM WALKS ON DIRECTED NETWORKS: THE CASE OF PAGERANK

2007 ◽  
Vol 17 (07) ◽  
pp. 2343-2353 ◽  
Author(s):  
SANTO FORTUNATO ◽  
ALESSANDRO FLAMMINI
Keyword(s):  
Random Walk ◽  
Power Law ◽  
Degree Distribution ◽  
Directed Graphs ◽  
Damping Factor ◽  
Web Pages ◽  
Stationary Probability ◽  
Exact Results ◽  
Limit Values ◽  
Underlying Graph

PageRank, the prestige measure for Web pages used by Google, is the stationary probability of a peculiar random walk on directed graphs, which interpolates between a pure random walk and a process where all nodes have the same probability of being visited. We give some exact results on the distribution of PageRank in the cases in which the damping factor q approaches the two limit values 0 and 1. When q → 0 and for several classes of graphs the distribution is a power law with exponent 2, regardless of the in-degree distribution. When q → 1 it can always be derived from the in-degree distribution of the underlying graph, if the out-degree is the same for all nodes.

Unsupervised and Semi-Supervised Graph-Spectral Algorithms for Robust Extraction of Arbitrarily Shaped Fuzzy Clusters

2007 ◽  
Vol 11 (6) ◽  
pp. 554-560 ◽  
Author(s):  
Weiwei Du ◽  
◽  
Kiichi Urahama
Keyword(s):  
Image Retrieval ◽  
Directed Graphs ◽  
Noisy Data ◽  
Lagrangian Function ◽  
Color Images ◽  
Web Page ◽  
Undirected Graphs ◽  
Face Images ◽  
Spectral Algorithms ◽  
Similarity Searches

We present unsupervised and semi-supervised algorithms for extracting fuzzy clusters in weighted undirected regular, undirected bipartite, and directed graphs. We derive the semi-supervised algorithms from the Lagrangian function in unsupervised methods for extracting dominant clusters in a graph. These algorithms are robust against noisy data and extract arbitrarily shaped clusters. We demonstrate applications for similarity searches of data such as image retrieval in face images represented by undirected graphs, quantized color images represented by undirected bipartite graphs, and Web page links represented by directed graphs.

A GENERATING FUNCTION APPROACH TO THE SURFACE AREAS OF SOME INTERCONNECTION NETWORKS

2009 ◽  
Vol 10 (03) ◽  
pp. 189-204 ◽  
Author(s):  
EDDIE CHENG ◽  
KE QIU ◽  
ZHIZHANG SHEN
Keyword(s):  
Generating Function ◽  
Interconnection Networks ◽  
Interconnection Network ◽  
Direct Consequence ◽  
Alternating Group ◽  
Function Technique ◽  
Explicit Formulas ◽  
Star Graphs ◽  
Surface Areas ◽  
Whitney Number

An important and interesting parameter of an interconnection network is the number of vertices of a specific distance from a specific vertex. This is known as the surface area or the Whitney number of the second kind. In this paper, we give explicit formulas for the surface areas of the (n, k)-star graphs and the arrangement graphs via the generating function technique. As a direct consequence, these formulas will also provide such explicit formulas for the star graphs, the alternating group graphs and the split-stars since these graphs are related to the (n, k)-star graphs and the arrangement graphs. In addition, we derive the average distances for these graphs.

A generalization of the Whitney rank generating function

1993 ◽  
Vol 113 (2) ◽  
pp. 267-280 ◽  
Author(s):  
G. E. Farr
Keyword(s):  
Generating Function ◽  
Linear Code ◽  
Tutte Polynomial ◽  
Natural Generalization ◽  
Chromatic Polynomial ◽  
Weight Enumerator ◽  
Hadamard Transform ◽  
Percolation Probability ◽  
Special Cases

AbstractThe Whitney quasi-rank generating function, which generalizes the Whitney rank generating function (or Tutte polynomial) of a graph, is introduced. It is found to include as special cases the weight enumerator of a (not necessarily linear) code, the percolation probability of an arbitrary clutter and a natural generalization of the chromatic polynomial. The crucial construction, essentially equivalent to one of Kung, is a means of associating, to any function, a rank-like function with suitable properties. Some of these properties, including connections with the Hadamard transform, are discussed.

scholarly journals Chip-Firing and Riemann-Roch Theory for Directed Graphs

2011 ◽  
Vol 38 ◽  
pp. 63-68 ◽  
Author(s):  
Arash Asadi ◽  
Spencer Backman
Keyword(s):  
Directed Graphs ◽  
Chip Firing

scholarly journals Solving Multi-agent Path Finding on Strongly Biconnected Digraphs

2018 ◽  
Vol 62 ◽  
pp. 273-314
Author(s):  
Adi Botea ◽  
Davide Bonusi ◽  
Pavel Surynek
Keyword(s):  
Empirical Analysis ◽  
Polynomial Time ◽  
Directed Graphs ◽  
Path Finding ◽  
Undirected Graphs ◽  
Suboptimal Solutions ◽  
Polynomial Time Algorithms ◽  
Asymmetric Communication ◽  
Multi Agent

Much of the literature on suboptimal, polynomial-time algorithms for multi-agent path finding focuses on undirected graphs, where motion is permitted in both directions along a graph edge. Despite this, traveling on directed graphs is relevant in navigation domains, such as path finding in games, and asymmetric communication networks.We consider multi-agent path finding on strongly biconnected directed graphs. We show that all instances with at least two unoccupied positions have a solution, except for a particular, degenerate subclass where the graph has a cyclic shape. We present diBOX, an algorithm for multi-agent path finding on strongly biconnected directed graphs. diBOX runs in polynomial time, computes suboptimal solutions and is complete for instances on strongly biconnected digraphs with at least two unoccupied positions. We theoretically analyze properties of the algorithm and properties of strongly biconnected directed graphs that are relevant to our approach. We perform a detailed empirical analysis of diBOX, showing a good scalability. To our knowledge, our work is the first study of multi-agent path finding focused on directed graphs.

scholarly journals Disjoint Cycles of Different Lengths in Graphs and Digraphs

10.37236/6921 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Julien Bensmail ◽  
Ararat Harutyunyan ◽  
Ngoc Khang Le ◽  
Binlong Li ◽  
Nicolas Lichiardopol
Keyword(s):  
Minimum Degree ◽  
Directed Graphs ◽  
Probabilistic Method ◽  
Undirected Graphs ◽  
Disjoint Cycles ◽  
Vertex Disjoint ◽  
Degree Bound ◽  
Directed Cycles

In this paper, we study the question of finding a set of $k$ vertex-disjoint cycles (resp. directed cycles) of distinct lengths in a given graph (resp. digraph). In the context of undirected graphs, we prove that, for every $k \geq 1$, every graph with minimum degree at least $\frac{k^2+5k-2}{2}$ has $k$ vertex-disjoint cycles of different lengths, where the degree bound is best possible. We also consider other cases such as when the graph is triangle-free, or the $k$ cycles are required to have different lengths modulo some value $r$. In the context of directed graphs, we consider a conjecture of Lichiardopol concerning the least minimum out-degree required for a digraph to have $k$ vertex-disjoint directed cycles of different lengths. We verify this conjecture for tournaments, and, by using the probabilistic method, for some regular digraphs and digraphs of small order.

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