Local Convergence of Graphs and Enumeration of Spanning Trees

A spanning tree in a connected graph G is a subgraph that contains every vertex of G and is itself a tree. Clearly, if G is a tree then it has only one spanning tree. Every connected graph contains at least one spanning tree: iteratively remove an edge from any cycle that is present until the graph contains no cycles. Counting spanning trees is a very natural problem. Following Lyons [5] we will see how the theory of graph limits does this in an asymptotic sense. There are many other interesting questions that involve understanding spanning trees in large graphs, for example, what is a ‘random spanning tree’ of Z? We will not discuss these questions in this note, however, the interested reader should see chapters 4, 10 and 11 of Lyons and Peres [7]. Let us begin with some motivating examples. Let Pn denote the path on n vertices. Each Pn naturally embeds into the bi-infinite path whose vertices are the set of integers Z with edges between consecutive integers. By an abuse of notation we denote the bi-infiite path as Z. It is intuitive to say that Pn converges to Z as these paths can be embedded into Z in a nested manner such that they exhaust Z. Clearly, both Pn and Z contain only one spanning tree. So the number of spanning trees in Pn converges to the number of spanning trees in Z. Is this part of some general phenomenon?