Counting the Number of Spanning Trees in the Star Flower Planar Map

Abstract The number of spanning trees of a graph G is the total number of distinct spanning subgraphs of G that are trees (tree that visiting all the vertices of the graph G). Let Cn be a cycle with n vertices. The Star flower planar map is a simple graph G formed from a cycle Cn by adding a vertex adjacent to every edge of Cn and we connect this vertex with two end vertices of each edge of Cn, i.e., we replace each edge of Cn by a triangulation. If there are k edges between every two vertices of each edge of the cycle Cn, then we obtain the star flower planar map in the general case. In this work, we denote the star flower planar map by Sn,k where n is the number of triangles of the star flower planar map, k is the number of edges between each two vertices of each edge of the cycle Cn; and derive the explicit formula for τ(Sn,k) the number of spanning trees in Sn,k to be τ(Sn,k) = 2kn(k + 2)n−1, n ≥ 2.