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Distance Sequences In Locally Infinite Vertex-Transitive Digraphs

Wesley Pegden
2006 Combinatorica  
We prove that the out-distance sequence {f + (k)} of a vertex-transitive digraph of finite or infinite degree satisfies f + (k + 1) ≤ f + (k) 2 for k ≥ 1, where f + (k) denotes the number of vertices at  ...  As a corollary, we prove that for a connected vertextransitive undirected graph of infinite degree d, we have f (k) = d for all k, 1 ≤ k < diam(G). This answers a question  ...  vertex-transitive graphs, and has asked if "valleys" can occur in the locally infinite case [B] .  ... 
doi:10.1007/s00493-006-0033-y fatcat:xckh6n2ezzbqjlv67ctmplzism

Distance Sequences of Locally Infinite Primitive Graphs [article]

Katalin Berlow
2020 arXiv   pre-print
In this paper, we prove a classification of the possible distance sequences of locally infinite primitive graphs.  ...  We also prove a constraint on the distance sequences of locally finite infinite graphs.  ...  Pegden in [2] has noted that for locally infinite vertex-transitive graphs, the possible distance sequences are limited.  ... 
arXiv:2009.11276v1 fatcat:s7dzcegvv5exfhqd4bzpeitjry


2013 Journal of Topology and Analysis (JTA)  
In this paper, we consider the Voronoi decompositions of an arbitrary infinite vertex-transitive graph G.  ...  We show that all vertex transitive graphs with polynomial growth have a finite s(G); vertex transitive graphs with infinitely many ends have an infinite s(G); the lamplighter graph LL(Z), which has exponential  ...  Every infinite vertex-transitive graph G has s(G) ≥ 2. Proof. Let v be a vertex in G.  ... 
doi:10.1142/s1793525313500088 fatcat:b6yiu2ettvacro5yhh54a2jzz4

A census of infinite distance-transitive graphs

Peter J. Cameron
1998 Discrete Mathematics  
This paper describes some classes of infinite distance-transitive graphs. It has no pretensions to give a complete list, but concentrates on graphs which have no finite analogues.  ...  A very complete reference on finite distance-transitive graphs is the book of Brouwer et al. [l]. By contrast, the theory of infinite distance-transitive graphs is open.  ...  However, only in two cases do we get distance-transitive graphs:  ... 
doi:10.1016/s0012-365x(98)00063-6 fatcat:tac5nwiyrnh4xboixwqzyk2hga

Infinite vertex-transitive, edge-transitive non-$1$-transitive graphs

Carsten Thomassen, Mark E. Watkins
1989 Proceedings of the American Mathematical Society  
We show that every vertex-transitive, edge-transitive graph of odd valence and subexponential growth is 1-transitive, thus extending to infinite graphs a theorem of W. T. Tutte for finite graphs.  ...  We describe a number of counterexamples in the case of exponential growth.  ...  Acknowledgment The authors are grateful to the Laboratoire de Recherche en Informatique, Université de Paris-Sud, Orsay, France, for having brought us there at the same time in February 1986.  ... 
doi:10.1090/s0002-9939-1989-0973847-6 fatcat:ycdyzvniwjasfata7p4livwny4

Sum rules for effective resistances in infinite graphs

Greg Markowsky, José Luis Palacios
2017 Journal of Statistical Mechanics: Theory and Experiment  
Extending work of Foster, Doyle, and others, we show how the Foster Theorems, a family of results concerning effective resistances on finite graphs, can in certain cases be extended to infinite graphs.  ...  The results are illustrated with some of the most common grids in the plane, including the square, triangular, and hexagonal grids.  ...  As was noted in Section 3, Foster's First and Second easily give the resistances between adjacent points in 1-arc transitive graphs and points of distance 2 in 2-arc transitive graphs, with the corresponding  ... 
doi:10.1088/1742-5468/aa6503 fatcat:f6br43njcffj3jsfc7ueyy44j4

Page 654 of Mathematical Reviews Vol. , Issue 95b [page]

1995 Mathematical Reviews  
In particular, they have infinitely many ends. The author shows that for an infinite, locally finite, connected graph with more than one end, 2-distance transitivity implies distance transitivity.  ...  (ICE-UICE-SI; Reykjavik) Distance-transitivity in infinite graphs. (English summary) J. Combin. Theory Ser. B 60 (1994), no. 1, 36-39.  ... 

A note on the growth of transitive graphs

W. Imrich, N. Seifter
1988 Discrete Mathematics  
Let X be a locally finite, connected, growth if and only if X is a strip. infinite, transitive graph. We show that X has linear 0012~365X/88/$3.50 @ 1988, Elsevier !&Science Publishers B.V. (North-  ...  In Proposition 2.1 we prove that a graph X cannot have linear growth if n(E) is infinite for some E E E(X). 'Ihen we show that there exist no transitive graphs X of linear growth with only one end.  ...  By T,(A), m 2 0, we denote the set of all vertices y E V(X) of distance m from x. Further H,(X) = UO F(X). Let v be a fixed vertex in a connected graph X.  ... 
doi:10.1016/0012-365x(88)90138-0 fatcat:elz23bssbzbyhcxearetifkhhy

Page 3540 of Mathematical Reviews Vol. , Issue 93g [page]

1993 Mathematical Reviews  
A graph T is said to be affine distance transitive if some group G of automorphisms acting distance transitively on IT has a regular normal elementary abelian subgroup N.  ...  This paper gives a classification of a certain family of affine distance transitive graphs.  ... 

Percolation on Transitive Graphs as a Coalescent Process: Relentless Merging Followed by Simultaneous Uniqueness [chapter]

Olle Häggström, Yuval Peres, Roberto H. Schonmann
1999 Perplexing Problems in Probability  
Percolation on transitive graphs  ...  In fact, this holds in the larger class of semi-transitive graphs.  ...  Theorem 1.3 Let G be an infinite, locally finite, connected, quasi-transitive graph, and let p c and p u be as in Theorem 1.2.  ... 
doi:10.1007/978-1-4612-2168-5_4 fatcat:vzh6zo4j65e6vcuytml3oq3vci

Page 2460 of Mathematical Reviews Vol. , Issue 2000d [page]

2000 Mathematical Reviews  
The graph I is said to be distance-transitive if it admits a distance-transitive group. This paper outlines a program for classifying the finite distance-transitive graphs.  ...  A subgroup G of automorphisms of a graph T is said to be COMBINATORICS 2460 distance-transitive on I if, for all i between 0 and the diameter of I, G is transitive on the set of ordered pairs of vertices  ... 

Critical percolation on any quasi-transitive graph of exponential growth has no infinite clusters [article]

Tom Hutchcroft
2016 arXiv   pre-print
We prove that critical percolation on any quasi-transitive graph of exponential volume growth does not have a unique infinite cluster.  ...  This allows us to deduce from earlier results that critical percolation on any graph in this class does not have any infinite clusters.  ...  Given a graph G, we write B(x, r) to denote the graph distance ball of radius r about a vertex x of G.  ... 
arXiv:1605.05301v1 fatcat:7vvwfzbvdvebxflczm2y7f2zgi

Fairness, distances and degrees

Philippe Darondeau, Doris Nolte, Lutz Priese, Serge Yoccoz
1992 Theoretical Computer Science  
We show the identity between sets of fair computations in recursive transition graphs, sets of cluster points of finite computations for II: ultra-metrics refining the Baire metrics, and II!  ...  Yoccoz, Fairness, distances and degrees, Theoretical Computer Science 97 (1992) 131-142.  ...  Infinite paths in a recursive tree may be seen as infinite computations in a recursive transition graph with a recursive set of sink states.  ... 
doi:10.1016/0304-3975(92)90390-2 fatcat:gg2jnqhxy5b2zirqi3tauag2ty

Exponential decay of connectivity and uniqueness in percolation on finite and infinite graphs [article]

Kathleen E. Hamilton, Leonid P. Pryadko
2016 arXiv   pre-print
We give an upper bound for the uniqueness transition on an arbitrary locally finite graph G in terms of the limit of the spectral radii ρ[ H( G_t)] of the non-backtracking (Hashimoto) matrices for an increasing  ...  With the added assumption of strong local connectivity for the oriented line graph (OLG) of G, connectivity on any finite subgraph G'⊂ G decays exponentially for p<(ρ[ H( G^')])^-1.  ...  We give lower bounds for all three transitions usually associated with site percolation on infinite graphs. We also identify a region of p where connectivity decays exponentially with the distance.  ... 
arXiv:1610.04897v1 fatcat:tnfdvilumbdvhdhazbbek3y5ca

On distance-balanced graphs

Aleksandar Ilić, Sandi Klavžar, Marjan Milanović
2010 European journal of combinatorics (Print)  
It is shown that the graphs for which the Szeged index equals G ·|G | 2 4 are precisely connected, bipartite, distance-balanced graphs. This enables us to disprove a conjecture proposed in [M.H.  ...  Infinite families of distance-balanced, non-regular graphs that are prime with respect to the Cartesian product are also constructed.  ...  [10] constructed an infinite family of semisymmetric graphs which are not distance-balanced. (A regular graph is called semisymmetric if it is edge-transitive but not vertex-transitive [11] .)  ... 
doi:10.1016/j.ejc.2009.10.006 fatcat:nxlayk3w4veobo7tnkjmtjvwji
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