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

1.2. Maps 31Definition 1.2.1 (Map) A map C is a graph G drawn on a surface X or embedded intoit (that is, a compact 2-dimensional orientable variety) in such a way that:• the vertices of graph are represented as distinct points of the surface.• the edges are represented as curves on the surface that intersect only at the vertices.• if we cut the surface along the graph thus drawn, what remains (that is, the set X\G)is a disjoint union of connected components, called faces, each homeomorphic to anopen disk (for more information on the faces of a map see [43] and [69]).A planar map is a map drawn on the plane. Through this thesis, all maps are planar andconnected.Figure 1.16: One graph gives two planar mapsA planar map (hereafter shortly called a map) is an isotopy class of planar embeddingsof a connected planar graph. Notice that the graphs embedded are unlabelled. To stateit simply, a planar map is a connected unlabelled graph drawn in the plane without edgecrossingsand up to continuous deformation. Planar maps are often called plane graphs inthe literature [43, 69]. As illustrated in Figure 1.17 (a)-(b), a planar graph can have nonisotopicplanar embeddings, so that it gives rise to several maps. Due to the topologicalembedding, a map has more structure than a graph. In particular, a map has faces, eachface corresponding to a connected component of the plane splits by the embedding.Figure 1.17: (a) A representation of a graph; its set of vertices is {1, 2, 3, 4}, and (multi)setof edges is {{1, 2}, {2, 3}, {2, 4}, {2, 4}, {3, 3}, {3, 4}}. (b) Two embeddings of this graphin the sphere, which are not homeomorphic since the second has a triangular face, unlikethe first.