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

122 CHAPTER 7. MAXIMAL PLANAR MAPSProof:1. W (v, E n ) = ∑u∈V (E n)= ∑u∈V (En)d(u,v)=1d(u, v) (we use Remark 7.2.14)d(u, v) += deg(v) + 2 ∑u∈V (En)d(u,v)=2∑u∈V (En)d(u,v)=21d(u, v)= deg(v) + 2(n − deg(v) − 1)= 2n − deg(v) − 22. W (E n ) = 1 2= 1 2∑W (v, E n )v∈V (E n )∑v∈V (E n )= (n − 1)(2n − deg(v) − 2) (from 1)∑1 − 1 2v∈V (E n)∑deg(v)v∈V (E n)= n(n − 1) − 1 2 × 2|E(E n)|= (n − 2) 2 + 2□Lemma 7.2.16 Let C n be a simple planar map with n vertices (n ≥ 2) and let v be avertex of C n , thenW (v, C n ) ≥ 2n − deg(v) − 2.Proof:W (v, C n ) = ∑u∈V (C n )= ∑u∈V (Cn)d(u,v)=1d(u, v)d(u, v) +∑u∈V (Cn)d(u,v)≥2d(u, v)≥ deg(v) + 2(n − deg(v) − 1)≥ 2n − deg(v) − 2□Let T n be a tree with n vertices, then the Wiener index W (T n ) = ∑ u,vd(u, v) is minimizedby star tree with n vertices and maximized by path with n vertices, both uniquely, i.e.,