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

7.3. Formulae for the Number of Spanning Trees in a Maximal Planar Map 123(n − 1) 2 ≤ W (T n ) ≤ n(n2 −1)6[117]. The goal of this work is to give an inequality similar inthe case of planar maps. Let C n be a planar map with n vertices, then the Wiener indexof W (C n ) is minimized by the maximal planar map W (E n ) with n vertices and maximizedby the path W (P n ) with n vertices. Now we can state the following theorem.Theorem 7.2.17 Let C n be a simple planar map with n vertices, thenW (E n ) ≤ W (C n ) ≤ W (P n )Proof: By Remark 7.2.10, In each deleted edge of C, we expand the Wiener index. Theconnected planar map obtained after deleting all possible edges is a spanning tree of C n ,On the other hand in the trees, the Wiener index is maximized by the path P n with nvertices, hence W (C n ) ≤ W (P n ).The remaining premises is to show that: W (E n ) ≤ W (C n )• For n = 2, 3 or 4 :W (E n ) = 1 2 n(n − 1) and as W (C n) ≥ 1 2 n(n − 1) = W (E n), hence the result.• For n ≥ 5Since W (v, C n ) ≥ 2n − deg(v) − 2, we have :W (C n ) = 1 2≥ 1 2∑v∈V (E n )∑v∈V (E n )≥ n(n − 1) − 1 2W (v, E n )(2n − deg(v) − 2)∑v∈V (E n )deg(v)≥ n(n − 1) − |E(E n )| = W (E n )Remark 7.2.18 For a planar map C with n vertices, W (C n ) ≤ W (P n ) ( P n is the pathwith n vertices). The lower bound of W (C n ) is not yet known [54, 117]. But in case ofplanar maps, we have given in this section the upper bound that is the Wiener index inthe maximal planar map W (E n ).7.3 Formulae for the Number of Spanning Trees in aMaximal Planar MapIn this section, we derive the explicit formula for the number of spanning trees of themaximal planar map and deduce a formula for the number of spanning trees of the crystalplanar map.□