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

Chapter 7Maximal Planar MapsIn this chapter, we shall focus on the maximal planar maps. This chapter will be dividedinto two sections; we devote the first section for calculating the Weiner index in the caseof planar maps in general then in particular in the maximal planar maps and in the othersection will study how to count the number of spanning trees in this type of this maps,as well as enumeration of spanning trees in the maximal planar maps, i.e., we derive theexplicit formulae for the number of spanning trees of the maximal planar map.7.1 IntrodutionA maximal planar map is a planar map to which no new edge can be added withoutviolating the planarity of C. The number of the edges in any maximal planar map with nvertices is 3n − 6. Furthermore the (facial) faces (including the infinite outerface) are alltriangles, i.e., cycles of length three. A simple planar map is called maximal planar if itis planar but adding any edge (on the given vertex set) would destroy that property. Allfaces (even the outer ones) are then bounded by three edges, explaining the alternativeterms triangular and triangulated for these graphs. If a triangular graph has v verticeswith v > 2, then it has precisely 3v − 6 edges and 2v − 4 faces. In this chapter, we areinterested in the maximal planar map with two complete vertices to isomorphism, thereis only one maximal planar map which has n vertices, two complete vertices of degreen − 1, two vertices of degree 3 and n − 4 vertices of degree 4, 2(n − 2) faces of degree 3(all faces having degree 3) and 3(n − 2) edges, (see Figure 7.5).Isomorphism It follows from the definition that a graph is completely determinedwhen we know its vertices and edges, and that two graphs are the same if they have thesame vertices and edges. Once we know the vertices and edges, we can draw the graphand, in principle, any picture we draw is as good as any other; the actual way in whichthe vertices and edges are drawn is irrelevant, although, some pictures are easier to usethan others!115