scholarly journals Differentiability of L-spectrum and multifractal decomposition by using infinite graph-directed IFSs

2017 ◽  
Vol 311 ◽  
pp. 190-237 ◽  
Author(s):  
Guotai Deng ◽  
Sze-Man Ngai
Keyword(s):  
Infinite Graph ◽  
Multifractal Decomposition

A multifractal decomposition according to rate of returns

2007 ◽  
Vol 280 (8) ◽  
pp. 888-896 ◽  
Author(s):  
Jihua Ma ◽  
Zhiying Wen
Keyword(s):  
Rate Of Returns ◽  
Multifractal Decomposition

Critical Probabilities of 1-Independent Percolation Models

2012 ◽  
Vol 21 (1-2) ◽  
pp. 11-22 ◽  
Author(s):  
PAUL BALISTER ◽  
BÉLA BOLLOBÁS
Keyword(s):  
Infinite Graph ◽  
Percolation Model ◽  
Bond Percolation ◽  
Locally Finite

Given a locally finite connected infinite graphG, let the interval [pmin(G),pmax(G)] be the smallest interval such that ifp>pmax(G), then every 1-independent bond percolation model onGwith bond probabilityppercolates, and forp<pmin(G) none does. We determine this interval for trees in terms of the branching number of the tree. We also give some general bounds for other graphsG, in particular for lattices.

scholarly journals Bounding the Distinguishing Number of Infinite Graphs and Permutation Groups

10.37236/3046 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Simon M. Smith ◽  
Mark E. Watkins
Keyword(s):  
Connected Graph ◽  
Automorphism Groups ◽  
Cardinal Number ◽  
Infinite Graph ◽  
Infinite Graphs ◽  
Finite Radius ◽  
Locally Finite ◽  
Distinguishing Number ◽  
Identity Permutation ◽  
Imprimitive Group

A group of permutations $G$ of a set $V$ is $k$-distinguishable if there exists a partition of $V$ into $k$ cells such that only the identity permutation in $G$ fixes setwise all of the cells of the partition. The least cardinal number $k$ such that $(G,V)$ is $k$-distinguishable is its distinguishing number $D(G,V)$. In particular, a graph $\Gamma$ is $k$-distinguishable if its automorphism group $\rm{Aut}(\Gamma)$ satisfies $D(\rm{Aut}(\Gamma),V\Gamma)\leq k$.Various results in the literature demonstrate that when an infinite graph fails to have some property, then often some finite subgraph is similarly deficient. In this paper we show that whenever an infinite connected graph $\Gamma$ is not $k$-distinguishable (for a given cardinal $k$), then it contains a ball of finite radius whose distinguishing number is at least $k$. Moreover, this lower bound cannot be sharpened, since for any integer $k \geq 3$ there exists an infinite, locally finite, connected graph $\Gamma$ that is not $k$-distinguishable but in which every ball of finite radius is $k$-distinguishable.In the second half of this paper we show that a large distinguishing number for an imprimitive group $G$ is traceable to a high distinguishing number either of a block of imprimitivity or of the induced action by $G$ on the corresponding system of imprimitivity. An immediate application is to automorphism groups of infinite imprimitive graphs. These results are companion to the study of the distinguishing number of infinite primitive groups and graphs in a previous paper by the authors together with T. W. Tucker.

scholarly journals On the Turán Properties of Infinite Graphs

10.37236/771 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Andrzej Dudek ◽  
Vojtěch Rödl
Keyword(s):  
Infinite Graph ◽  
Infinite Graphs ◽  
The Other ◽  
Other Hand ◽  
Turan's Theorem ◽  
Vertex Set ◽  
Turán’S Theorem ◽  
Positive Integers

Let $G^{(\infty)}$ be an infinite graph with the vertex set corresponding to the set of positive integers ${\Bbb N}$. Denote by $G^{(l)}$ a subgraph of $G^{(\infty)}$ which is spanned by the vertices $\{1,\dots,l\}$. As a possible extension of Turán's theorem to infinite graphs, in this paper we will examine how large $\liminf_{l\rightarrow \infty} {|E(G^{(l)})|\over l^2}$ can be for an infinite graph $G^{(\infty)}$, which does not contain an increasing path $I_k$ with $k+1$ vertices. We will show that for sufficiently large $k$ there are $I_k$–free infinite graphs with ${1\over 4}+{1\over 200} < \liminf_{l\rightarrow \infty} {|E(G^{(l)})|\over l^2}$. This disproves a conjecture of J. Czipszer, P. Erdős and A. Hajnal. On the other hand, we will show that $\liminf_{l\rightarrow \infty} {|E(G^{(l)})|\over l^2}\le{1\over 3}$ for any $k$ and such $G^{(\infty)}$.

scholarly journals Matroid and Tutte-Connectivity in Infinite Graphs

10.37236/2832 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Henning Bruhn
Keyword(s):  
Infinite Graph ◽  
Infinite Graphs ◽  
Dual Graphs ◽  
Connectivity Function ◽  
Cycle Matroid ◽  
Infinite Cycles

We relate matroid connectivity to Tutte-connectivity in an infinite graph. Moreover, we show that the two cycle matroids, the finite-cycle matroid and the cycle matroid, in which also infinite cycles are taken into account, have the same connectivity function. As an application we re-prove that, also for infinite graphs, Tutte-connectivity is invariant under taking dual graphs.

scholarly journals An infinite graph of girth $12$

1984 ◽  
Vol 283 (2) ◽  
pp. 575-575
Author(s):  
Asia Ivi{ć Weiss
Keyword(s):  
Infinite Graph

scholarly journals Approximating graphs with polynomial growth

2000 ◽  
Vol 42 (1) ◽  
pp. 1-8 ◽  
Author(s):  
Norbert Seifter ◽  
Wolfgang Woess
Keyword(s):  
Inverse Limit ◽  
Polynomial Growth ◽  
Infinite Sequence ◽  
Growth Conditions ◽  
Infinite Graph ◽  
Subject Classification ◽  
Transitive Graph ◽  
Locally Finite ◽  
Finite Graphs ◽  
Growth Properties

Let X be an infinite, locally finite, almost transitive graph with polynomial growth. We show that such a graph X is the inverse limit of an infinite sequence of finite graphs satisfying growth conditions which are closely related to growth properties of the infinite graph X.1991 Mathematics Subject Classification. Primary 05C25, Secondary 20F8.

On a class of packet-switching networks on an infinite graph

1990 ◽  
Vol 6 (1) ◽  
pp. 391-403
Author(s):  
M. Ya Kelbert ◽  
Yu. M. Suhov
Keyword(s):  
Packet Switching ◽  
Infinite Graph ◽  
Switching Networks ◽  
Packet Switching Networks
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