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

6.3. Main Results 109Figure 6.2: A cycle C n and the star flower planar mapIf there are k edges (simple path in the general case) between every two vertices of eachedge of the cycle C n , then we obtain the star flower planar map shown in Figure. 6.3.In this work, we denote the star flower planar map by S n,k where n is the number oftriangles of the star flower planar map, k is the number of edges (the length of simplepath) between each two vertices of each edge of the cycle C n ; the star flower planar maphas n(k + 1) vertices (n vertices of degree 4 and n + (k − 1)n of degree 2), n(k + 2) edgesand n + 2 faces (n faces of degree k + 2, one face of degree nk and the other face of degree2n)(see Figure. 6.3).Figure 6.3: The star flower planar map S n,kExample 6.3.1 Here is an example of a star flower planar map S 2,k with n = 2 (2triangles) and k edges, and a star flower planar map S 3,k with n = 3 (3 triangles) and kedges (see Figure. 6.4).