enumeration of the number of spanning trees in some ... - Toubkal

4.3. Main Results 73This recursion shows that the function τ is recursively enumerable, but the resultingalgorithm in general requires on the order of 2 |E(C)| arithmetic operations, and so it isnot suitable for large computations. It is a fortune that we have several computationallysimplified and totally generalized formulae for τ(C) which we are going to derive them inthis chapter and in upcoming chapters.Example 4.3.13 Here is an example of a map C with Recursive calculation of τ(C)Figure 4.8: Recursive calculation of τ(C)We consider finite undirected simple maps and multimaps with no loops; the term multimapis used when multiple edges are allowed in a map. For a map C, we denote by V (C)and E(C) the vertex set and edge set of C, respectively. The multiplicity of a vertex-pair(v, u) of a map C, denoted by l C (vu), is the number of edges joining the vertices v and uin C.Remark 4.3.14 If C is a connected loopless map with no cycle of length at least 3, thenτ(C) is the product of the edge multiplicities. A disconnected map has no spanning trees.Counting trees recursively requires initial conditions for maps (graphs) in which all edgesare loops. Such a map has one spanning tree if it has only one vertex, and it has nospanning trees if it has more than one vertex. If a computer completes the computationby deleting or contracting every edge in a loopless map C, then it may compute as manyas 2 E(C) terms. Even with savings from Remark 4.3.14, the amount of computation growsexponentially with the size of the map; this is impractical, therefore, we shall providenew non-trivial methods for calculating the number of spanning trees in maps in general,then in particular in some special maps and prove new simplified results in this chapter.Now, we consider the case where the edge e is replaced by a simple path p = v 1 ,v 2 , ..., v k , v k+1 of length k which connects v 1 with v k+1 , i.e., If there are k edges betweentwo vertices v 1 , v 2 of a map C.