A general bridge theorem for self-avoiding walks☆
Section snippets
Introduction and results
Let be a simple, locally finite, connected, infinite graph consisting of a countable set of vertices and a symmetric neighbourhood relation . We consider a walk on the graph as a sequence of vertices, where consecutive vertices in the walk must be adjacent in the graph and call it self-avoiding (or a SAW), if no vertex of the graph is contained twice in this sequence. The length of a walk is the length of the sequence reduced by one, so the number of “steps” in the walk.
We shall
Terminology and preliminaries
We consider a graph as a countable set of vertices together with a symmetric neighbourhood relation on . We usually write for the graph and omit . Two vertices and are called adjacent if . The degree of a vertex in is the number of vertices of which are adjacent to . We call locally finite, if for every .
A walk on a graph is a sequence of vertices of such that any two consecutive vertices of the sequence are adjacent in . We
The bridge theorem
One of the main results of [6] is the bridge theorem.
Theorem 3.1 Let possess an unimodular graph height function . Then .Thm. 4.3 in [6]
Note that as a consequence, the bridge constant does not depend on the choice of the unimodular graph height function . However, there are simple examples showing that unimodularity is required in this theorem, one of them being the Grandparent graph, which will be discussed in Section 4.
The main result in this paper is the following extension of this bridge
Bridges in the Grandparent graph
In this section we provide an example of a graph which does not possess a unimodular graph height function. We calculate the bridge constants and use the bridge theorem to obtain the connective constant.
An end of a tree is an equivalence class of one-way infinite SAWs, where two walks are equivalent if they share all but finitely many initial vertices. Fix some end of the infinite 3-regular tree and let the graph “hang down” from the end . Then the graph can be seen as the union of
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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This work was partially supported by the Austrian Science Fund : FWF P31889-N35 and FWF W1230.