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Bounds on the connective constants of regular graphs
, 2012
"... Abstract. Bounds are proved for the connective constant µ of an infinite, connected, ∆regular graph G. The main result is that µ ≥ √ ∆ − 1 if G is vertextransitive and simple. This inequality is proved subject to weaker conditions under which it is sharp. 1. ..."
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Abstract. Bounds are proved for the connective constant µ of an infinite, connected, ∆regular graph G. The main result is that µ ≥ √ ∆ − 1 if G is vertextransitive and simple. This inequality is proved subject to weaker conditions under which it is sharp. 1.
Counting selfavoiding walks
, 2013
"... The connective constant µ(G) of a graph G is the asymptotic growth rate of the number of selfavoiding walks on G from a given starting vertex. We survey three aspects of the dependence of the connective constant on the underlying graph G. Firstly, when G is cubic, we study the effect on µ(G) of th ..."
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The connective constant µ(G) of a graph G is the asymptotic growth rate of the number of selfavoiding walks on G from a given starting vertex. We survey three aspects of the dependence of the connective constant on the underlying graph G. Firstly, when G is cubic, we study the effect on µ(G) of the Fisher transformation (that is, the replacement of vertices by triangles). Secondly, we discuss upper and lower bounds for µ(G) when G is regular. Thirdly, we present strict inequalities for the connective constants µ(G) of vertextransitive graphs G, as G varies. As a consequence of the last, the connective constant of a Cayley graph of a finitely generated group decreases strictly when a new relator is added, and increases strictly when a nontrivial group element is declared to be a generator. Special prominence is given to open problems.
STRICT INEQUALITIES FOR CONNECTIVE CONSTANTS OF TRANSITIVE GRAPHS
"... Abstract. The connective constant of a graph is the exponential growth rate of the number of selfavoiding walks starting at a given vertex. Strict inequalities are proved for connective constants of vertextransitive graphs. Firstly, the connective constant decreases strictly when the graph is repl ..."
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Abstract. The connective constant of a graph is the exponential growth rate of the number of selfavoiding walks starting at a given vertex. Strict inequalities are proved for connective constants of vertextransitive graphs. Firstly, the connective constant decreases strictly when the graph is replaced by a nontrivial quotient graph. Secondly, the connective constant increases strictly when a quasitransitive family of new edges is added. These results have the following implications for Cayley graphs. The connective constant of a Cayley graph decreases strictly when a new relator is added to the group, and increases strictly when a nontrivial group element is declared to be a generator. 1.
LOCALITY OF CONNECTIVE CONSTANTS, II. CAYLEY GRAPHS
, 2015
"... Abstract. The connective constant µ(G) of an infinite transitive graph G is the exponential growth rate of the number of selfavoiding walks from a given origin. In earlier work of Grimmett and Li, a locality theorem was proved for connective constants, namely, that the connective constants of two g ..."
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Abstract. The connective constant µ(G) of an infinite transitive graph G is the exponential growth rate of the number of selfavoiding walks from a given origin. In earlier work of Grimmett and Li, a locality theorem was proved for connective constants, namely, that the connective constants of two graphs are close in value whenever the graphs agree on a large ball around the origin. A condition of the theorem was that the graphs support socalled ‘graph height functions’. When the graphs are Cayley graphs of infinite, finitely generated groups, there is a special type of graph height function termed here a ‘group height function’. A necessary and sufficient condition for the existence of a group height function is presented, and may be applied in the context of the bridge constant, and of the locality of connective constants for Cayley graphs. Locality may thereby be established for a variety of infinite groups including those with strictly positive deficiency. It is proved that a large class of transitive graphs (and hence Cayley graphs) support graph height functions that are in addition harmonic on the graph. This extends an earlier constructive proof of Grimmett and Li, but subject to an additional condition of unimodularity which is benign in the context of Cayley graphs. It implies the existence of graph height functions for finitely generated solvable groups. The case of nonunimodular graphs may be handled similarly, but the resulting graph height functions need not be harmonic. Group height functions, as well as the graph height functions of the previous paragraph, are nonconstant harmonic functions with linear growth and an additional property of having periodic differences. The existence of such functions on Cayley graphs is a topic of interest beyond their applications in the theory of selfavoiding walks.