Applied Mathematical Sciences, Vol. 5, 2011, no. 18, 883 - 898 The Number of Spanning Trees of Certain Families of Planar Maps A. Modabish and M. El Marraki Department of Computer Sciences, Faculty of Sciences University of Mohamed V, P.O. Box 1014, Rabat, Morocco hafizmod@yahoo.fr, marraki@fsr.ac.ma Abstract Calculating the number of spanning trees of a planar map by the determinant of Laplacian matrix is tedious and impractical. In this paper, we propose some methods to facilitate the calculation of the number of spanning trees for planar maps. We apply these methods to give the number of spanning trees of some special maps (n-Fan chains, n-Grid chains, n-Hexagonal chains,...etc). Mathematics Subject Classification: 05C85, 05C30 Keywords: Laplacian matrix, maps, complexity, spanning trees 1 Introduction In this paper we will present some useful definitions related to our work as follows: an undirected graph G is a triplet (V G ,E G ,δ ) where V G is the set of vertices of the graph G, E G is the set of edges of the graph G and δ is the application δ : E G →P (v ), e i → δ (e i )= {v j ,v k } with v j and v k are end vertices of the edge e i . We notice that the set {v j ,v k } as a multiset (if v j = v k , the same vertex appears twice in δ (e i )). A loop is an edge e i ∈ E G with v j = v k , if δ (e i )= δ (e j ) with i = j then the edges e i and e j are called multiple. We denote by p(v i ,v j ) (weight) the number of edges that connects v i with v j . A graph which contains neither multiple edges nor loops is called a simple graph. The degree of a vertex v noted deg(v) is the number of edges incident to it. The sum of the degrees of all vertices of a graph is equal to twice the number of its edges i.e. ∑ v∈V G deg (v )=2|E G |. In a graph G,a path is a sequence of vertices and edges p = v 0 ,e 1 ,v 1 ,e 2 , ..., v n-1 ,e n ,v n such that δ (e i )= {v i-1 ,v i }. We say that this path attached both ends v 0 and v n .