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

Chapter 5The Number of Spanning Trees ofCertain Families of Planar MapsIn this chapter, we are going to derive several easily computable formulae for certainfamilies of planar maps, and similar formulae for related generating series.5.1 IntroductionThe number of spanning trees of a map C is the total number of distinct spanning subgraphsof C that are trees. Even though the classic result, Matrix Tree Theorem expressesthe number of spanning trees τ(C) of a map C as a function of the determinant of amatrix that can be easily constructed from C’s incidence matrix, in practice, this methodof counting spanning trees by calculating determinants is infeasible for large planar maps(graphs embedded in the plane without edge-crossings). For some special classes of planarmaps, it is possible to give explicit, simple formulae for the number of spanning trees. Inthis chapter, we apply our methods which we have seen in Chapter 4 to give the numberof spanning trees in some special planar maps and derive several simple formulae forthe number of spanning trees of special families of planar maps. They are known as then-Fan chains, the n-Grid chains, the n-tent chains, the n-Hexagonal chains, the n-Eightchains, the n-Barrel chains the n-Light chains, the n-Home chains, the n-Kite chains, then-Envelope chains and the n-Diphenylene chains ... etc.5.2 The case of one cycleEach map having two faces possesses one cycle. Let C be a planar map with two faces(i.e., a cycle) with n vertices, then the complexity of C is equal to the length of its cycle(the length of a cycle is the number of edges that form this cycle); see Figure. 5.1.85