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 Phase transitions in a two-component site-bond percolation model

2000 ◽  
Vol 62 (13) ◽  
pp. 8719-8724 ◽  
Author(s):  
H. M. Harreis ◽  
W. Bauer
Keyword(s):  
Phase Transitions ◽  
Percolation Model ◽  
Bond Percolation ◽  
Two Component

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 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.

scholarly journals Self-organized critical dynamics of a directed bond percolation model

1998 ◽  
Vol 243 (1-2) ◽  
pp. 20-24
Author(s):  
Subhankar Ray ◽  
Tapati Dutta ◽  
Jaya Shamanna
Keyword(s):  
Percolation Model ◽  
Bond Percolation ◽  
Critical Dynamics ◽  
Self Organized

scholarly journals Relating Different Cycle Spaces of the Same Infinite Graph

10.37236/622 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
R. Bruce Richter ◽  
Brendan Rooney
Keyword(s):  
Finite Graph ◽  
Infinite Graph ◽  
Cycle Space ◽  
Locally Finite ◽  
Locally Finite Graph ◽  
Continuous Surjection

Casteels and Richter have shown that if $X$ and $Y$ are distinct compactifications of a locally finite graph $G$ and $f:X\to Y$ is a continuous surjection such that $f$ restricts to a homeomorphism on $G$, then the cycle space $Z_X$ of $X$ is contained in the cycle space $Z_Y$ of $Y$. In this work, we show how to extend a basis for $Z_X$ to a basis of $Z_Y$.

scholarly journals Contractible Edges in 2-Connected Locally Finite Graphs

10.37236/4414 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Tsz Lung Chan
Keyword(s):  
Infinite Graph ◽  
Finite Degree ◽  
Outerplanar Graphs ◽  
Locally Finite ◽  
Finite Graphs ◽  
Locally Finite Graphs

In this paper, we prove that every contraction-critical 2-connected infinite graph has no vertex of finite degree and contains uncountably many ends. Then, by investigating the distribution of contractible edges in a 2-connected locally finite infinite graph $G$, we show that the closure of the subgraph induced by all the contractible edges in the Freudenthal compactification of $G$ is 2-arc-connected. Finally, we characterize all 2-connected locally finite outerplanar graphs nonisomorphic to $K_3$ as precisely those graphs such that every vertex is incident to exactly two contractible edges as well as those graphs such that every finite bond contains exactly two contractible edges.

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