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A chip-firing game and Dirichlet eigenvalues

2002
*
Discrete Mathematics
*

The

doi:10.1016/s0012-365x(02)00434-x
fatcat:cy4yrqp7bzamrblhi25zme6kua
*game*continues until no vertex can be*fired*. We will give an upper bound, in terms of*Dirichlet**eigenvalues*, for the number of*firings*needed before*a**game*terminates. ... We consider*a*variation of the*chip*-*firing**game*in*a*induced subgraph S of*a*graph G. ... After preliminary definitions in Section 2, in Section 3 we obtain*a*bound on the length of the*Dirichlet**game*in terms of the number of*chips**and*the*Dirichlet**eigenvalues*of the graph. ...##
###
Page 6549 of Mathematical Reviews Vol. , Issue 2003i
[page]

2003
*
Mathematical Reviews
*

Lass.”
2003i:05087 05C50 91443
Chung, Fan (1-UCSD; La Jolla, CA);
Ellis, Robert B. (1-UCSD; La Jolla, CA)

*A**chip*-*firing**game**and**Dirichlet**eigenvalues*. ... We give an upper bound, in terms of*Dirichlet**eigenvalues*, for the number of*firings*needed before*a**game*terminates. ...##
###
Page 7564 of Mathematical Reviews Vol. , Issue 98M
[page]

1998
*
Mathematical Reviews
*

One variant of the

*chip**firing**games*, called the dollar*game**and*introduced by the author, gives rise to the so-called critical configurations which may be thought of as elements in*a*group. ...*A*number of short proofs are included. Among the more recent results discussed in the survey are the*chip**firing**games*which relate in*a*natural way to the Laplacian. ...##
###
Author index to volume

2002
*
Discrete Mathematics
*

Ellis,

doi:10.1016/s0012-365x(02)00700-8
fatcat:hkrv7qelo5copoel65lcrt6mzi
*A**chip*-*firing**game**and**Dirichlet**eigenvalues*(2-3) 341-355 Corbas, B.*and*G.D. Williams, Congruence classes in M 3 ðF q Þ ðq even) (1) 15-27 Cowen, L.J., J.R. Griggs*and*D.B. ... errez,*A*., see J.A. Rodr! ıguez (1) 101-109 Gy! arf! as,*A*., see Z. F . uredi (2-3) 385-403 Gy! arf! as,*A*., A.E. K! ezdy*and*J. Lehel,*A*finite basis characterization of*a*-,*A*.*and*K. ...##
###
Multi-commodity Allocation for Dynamic Demands Using PageRank Vectors
[chapter]

2012
*
Lecture Notes in Computer Science
*

In this process, the demands for several types of commodities are initially given at some specified vertices

doi:10.1007/978-3-642-30541-2_11
fatcat:nqrcgryd2fbgbcp3fthci33kcu
*and*then the demands spread interactively on*a*contact graph. ... While not strictly necessary, we assume that there is*a*self loop at each vertex. In this case, d v includes v in the count of neighbors*and*hence the loop counts as 1 towards the degree. ... Unlike in classical*chip*-*firing**games*, the number of*chips*is not static,*and*the process is continuous time. We restate the possible transitions in terms of demand*chips*. ...##
###
Multicommodity Allocation for Dynamic Demands Using PageRank Vectors

2014
*
Internet Mathematics
*

In this process, the demands for several types of commodities are initially given at some specified vertices

doi:10.1080/15427951.2013.833148
fatcat:lunxw4giwbdy7bmmimpdz6vovu
*and*then the demands spread interactively on*a*contact graph. ... While not strictly necessary, we assume that there is*a*self loop at each vertex. In this case, d v includes v in the count of neighbors*and*hence the loop counts as 1 towards the degree. ... Unlike in classical*chip*-*firing**games*, the number of*chips*is not static,*and*the process is continuous time. We restate the possible transitions in terms of demand*chips*. ...##
###
Discrete Green's functions for products of regular graphs
[article]

2003
*
arXiv
*
pre-print

These Green's functions may be used in conjunction with diffusion-like problems on graphs such as electric potential, random walks,

arXiv:math/0309080v2
fatcat:slbnkp557rh4rllcbayoup5jg4
*and**chip*-*firing**games*or other balancing*games*. ... Explicit formulas are derived for the cycle, torus,*and*3-dimensional torus, as is an inductive formula for the t-dimensional torus with n vertices, from which the Green's function can be completely determined ...*games*or other balancing*games*. ...##
###
A majorization bound for the eigenvalues of some graph Laplacians
[article]

2004
*
arXiv
*
pre-print

We also show that

arXiv:math/0411153v1
fatcat:m52hmi7fsjbtnpee6glr3p6cym
*a*generalization of this conjecture to graphs with*Dirichlet*boundary conditions is equivalent to the original conjecture. ... It is conjectured that the Laplacian spectrum of*a*graph is majorized by its conjugate degree sequence. In this paper, we prove that this majorization holds for*a*class of graphs including trees. ... Recently [2] used the spectrum of the*Dirichlet*Laplacian in the analysis of "*chip*-*firing**games*", which are processes on graphs that have an absorbing (*Dirichlet*) boundary at some vertices. ...##
###
A Majorization Bound for the Eigenvalues of Some Graph Laplacians

2007
*
SIAM Journal on Discrete Mathematics
*

We also show that this conjecture

doi:10.1137/040619594
fatcat:loygeh7rqbfurl6apqcjil7mpu
*and*its generalization to graphs with*Dirichlet*boundary conditions are equivalent. ... Grone*and*Merris [5] conjectured that the Laplacian spectrum of*a*graph is majorized by its conjugate vertex degree sequence. ... Recently [1] used the spectrum of the*Dirichlet*Laplacian in the analysis of "*chip*-*firing**games*", which are processes on graphs that have an absorbing (*Dirichlet*) boundary at some vertices. ...##
###
Green functions on product networks

2019
*
Discrete Applied Mathematics
*

*and*the

*eigenvalues*

*and*eigenfunctions of some Schrödinger operator on the other factor network. ... We apply our results to compute the Green function of P m × S h , where P m is

*a*Path with m vertices

*and*S h is

*a*Star network with h + 1 vertices. (A.M. Encinas), margarida.mitjana@upc.edu (M. ... Acknowledgment This work has been supported by the Spanish Research Council, Spain, under projects MTM2014-60450-R

*and*MTM2017-85996-R. ...

##
###
A Schur-Weyl Duality Approach to Walking on Cubes
[article]

2014
*
arXiv
*
pre-print

Walks on the representation graph R_V(G) determined by

arXiv:1409.8154v2
fatcat:lss6vb4vfvfxhjoypjtodpyghy
*a*group G*and**a*G-module V are related to the centralizer algebras of the action of G on the tensor powers V^⊗ k via Schur-Weyl duality. ... We describe*a*basis for the centralizer algebras in terms of labeled partition diagrams. ... Walks on graphs are also related to*chip*-*firing**games*, or to what is often referred to in physics as the (abelian) sandpile model. In*a**chip*-*firing**game*, each node starts with*a*pile of*chips*. ...##
###
Topic Discovery through Data Dependent and Random Projections
[article]

2013
*
arXiv
*
pre-print

We present

arXiv:1303.3664v2
fatcat:bvpqfxjxcza4rnf6ban7t3wzua
*a*suite of highly efficient algorithms based on data-dependent*and*random projections of word-frequency patterns to identify novel words*and*associated topics. ... This is*a*condition on existence of novel-words that are unique to each topic. ... RP*chip*circuit noise analog current voltage gates DDP*chip*circuit analog voltage pulse vlsi device Gibbs analog circuit*chip*output figure current vlsi RecL2 N/*A*RP visual cells spatial ocular ...##
###
Diffuse Scattering on Graphs
[article]

2016
*
arXiv
*
pre-print

In addition, we outline

arXiv:1401.4428v2
fatcat:6dhgvhsosvdk7m7cfpnf4jahpe
*a*method for finding Green's functions for Cayley graphs for both abelian*and*non-abelian groups. ... Finally, we conclude with*a*discussion of the effects of sparsity on our method*and*results, outlining the simplifications that can be made provided that the scatterers are weak*and*well-separated. ... Additionally, Green's function methods have yielded interesting results in many areas including the properties of random walks [17, 30] ,*chip*-*firing**games*[25] , analysis of online communities [33] ...##
###
Diffuse scattering on graphs

2016
*
Linear Algebra and its Applications
*

In addition, we outline

doi:10.1016/j.laa.2016.01.012
fatcat:rk6mx57z2nhytdjfpjnvpvrziq
*a*method for finding Green's functions for Cayley graphs for both abelian*and*non-abelian groups. ... Finally, we conclude with*a*discussion of the effects of sparsity on our method*and*results, outlining the simplifications that can be made provided that the scatterers are weak*and*wellseparated. ... Additionally, Green's function methods have yielded interesting results in many areas including the properties of random walks [17, 30] ,*chip*-*firing**games*[25] , analysis of online communities [33] ...##
###
Page 2758 of Mathematical Reviews Vol. , Issue Index
[page]

*
Mathematical Reviews
*

*A*

*chip*-

*firing*

*game*

*and*

*Dirichlet*

*eigenvalues*. ... of lattices induced by (extended)

*chip*

*firing*

*games*. ...

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