A limit characterization for the number of spanning trees of graphs

In this paper we propose a limit characterization of the behaviour of classes of graphs with respect to their number of spanning trees. Let {Gn} be a sequence of graphs G0,G1,G2, . . . that belong to a particular class. We consider graphs of the form Kn −Gn that result from the complete graph Kn after removing a set of edges that span Gn. We study the spanning tree behaviour of the sequence {Kn −Gn} when n→∞ and the number of edges of Gn scales according to n. More specifically, we define the spanning tree indicator α({Gn}), a quantity that characterizes the spanning tree behaviour of {Kn −Gn}. We derive closed formulas for the spanning tree indicators for certain well-known classes of graphs. Finally, we demonstrate that the indicator can be used to compare the spanning tree behaviour of different classes of graphs (even when their members never happen to have the same number of edges).  2004 Elsevier B.V. All rights reserved.