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

120 CHAPTER 7. MAXIMAL PLANAR MAPSRemark 7.2.7 We notice that deleting one edge of multiple edges does not affect theWiener index; through this section, we consider only the simple planar maps, i.e., mapswithout loops and multiple edges.Lemma 7.2.8 Let C n be a simple planar map with n vertices and v be a complete vertexin the map C, thendeg(v) = n − 1 and W (v, C n ) = n − 1.Lemma 7.2.9 Let C n be a simple planar map with n vertices (n ≥ 2) and let v be avertex not complete of C n , thenW (v, C n ) ≥ n.Remark 7.2.101. Let C n be a simple planar map and v be a vertex of C n , then W (v, C n ) ≥ n − 1.2. Let C n be a simple planar map with n vertices, e be an edge of C n and let C n − e bethe map obtained by deleting the edge e such that the map C n −e remains connected,then W (C n − e) ≥ W (C n ).Theorem 7.2.11 Let C n be a simple planar map with n vertices, thenW (C n ) ≥n(n − 1).2Proof: Let C n be a planar map with n vertices and let v be a vertex of C n . By Remark7.2.10, we have: W (v, C n ) ≥ n − 1;W (C n ) = 1 2≥ 1 2≥∑v∈V (C n )∑v∈V (C n )n(n − 1).2W (v, C n )(n − 1) = 1 2 (n − 1) ∑v∈V (C n )Definition 7.2.12 (A maximal planar map [39]) Let E n be a family of planar maps thatcontains:• n vertices, two complete vertices of degree n − 1, two vertices of degree 3 and n − 4vertices of degree 4,• 2(n − 2) faces of degree 3 (all faces having degree 3),• 3(n − 2) edges,1□