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

3.3. Counting the number of spanning trees in graphs combinatorically 57Another combinatorial method of counting the number of spanning trees in a graph Gis the Prüfer’s code or Prüfer’s sequence approach, which is first described by Prüfer[100] for proving Cayles’s theorem [25]: τ(K n ) = n n−2 , for the complete graph K n . Asan example, the 16 spanning trees of K 4 are shown in Figure 3.3. The key idea of thePrüfer’s code approach is to build a bijection between the set of all trees in a graph anda set of some sequences, then count the spanning trees by counting the sequences in thesequences set.Figure 3.3: The 16 different spanning trees in K 4 .Theorem 3.3.1 (Cayley’s Theorem) The number of spanning trees of the complete graphK n is n (n−2) .The proof of Cayley’s Theorem presented here is the work of Jim Pitman, who used clevercounting in two ways to prove the theorem 3.3.1. Cayley’s theorem can also be provedusing the Matrix Tree Theorem [117] presented below, but here we give an alternate proofto show a different way of proving the theorem. Before presenting the proof, we need tostate a few definitions. A rooted forest on the vertex set {1, ..., n} is a forest togetherwith a choice of a root in each component tree. Additionally, a forest F is said to containanother forest F ′ if F contains F ′ as a directed graph. Then clearly if F properly containsF ′ , then F has fewer components than F ′ . And finally, let F n,k be the set of all rootedforests that consist of k rooted trees. Then we call a sequence F 1 , ..., F k of forests a refiningsequence if F i ∈ F n,i and F i contains F i+1 , for all i. Now we can begin the proof.