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

72CHAPTER 4.NEW METHODS TO COMPUTE THE NUMBER OF SPANNINGTREES OF PLANAR MAPSProof: Let T be the set of spanning trees of C,τ(C) = |T |, T = T 1 ∪ T 2 , T 1 ∩ T 2 = ∅, withT 1 = the set of spanning trees of C which contains the edge eT 2 = the set of spanning trees of C which does not contain the edge e|T | = |T 1 | + |T 2 |,|T 1 | = τ(C.e) and |T 2 | = τ(C − e) then τ(C) = τ(C.e) + τ(C − e).We cannot apply the recurrence of Theorem 4.3.11 when e is a loop. For example, a mapconsisting of one vertex and one loop has one spanning tree, but deleting and contractingthe loop would count it twice. Since loops do not affect the number of spanning trees, wecan delete loops as they arise.Remark 4.3.12 The quantity τ(C) can be computed recursively, using the formulaτ(C) = τ(C − e) + τ(C.e), in which C − e is the map obtained from C by deleting theedge e and C.e is the map obtained from C by contracting the edge e (and removing anyloops produced). [Proof of this formula is a simple case analysis: τ(C − e) counts thespanning trees of C that do not contain the edge e, and τ(C.e) counts the spanning treesof C that do contain the edge e.] Furthermore, if C 1 and C 2 are connected maps whichintersect in exactly one vertex (and no edges) then τ(C 1 • C 2 ) = τ(C 1 ) × τ(C 1 ) (Theorem4.3.3), as is also easily seen. See Figure 4.7 for an example of computation using thismethod. (For each map in the figure, an edge which is deleted-contracted is marked withan asterisk.)□Figure 4.7: Computing of τ(C) by deletion-contraction.