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

CHAPTER 5.98THE NUMBER OF SPANNING TREES OF CERTAIN FAMILIES OFPLANAR MAPS5.8.1 Formulae for the number of spanning trees in particularplanar mapLet C n and Q n be the sequences of planar maps as following (see Figure 5.15):Figure 5.15: Particular case of maps C n and Q nTheorem 5.8.1 For the sequence maps C n and Q n in Figure 5.15, we have:τ(C n ) = √ 1 ((9 + 4 √ 5)( 7 + 3√ 5) n−1 − (9 − 4 √ 5)( 7 − 3√ 5)), n−1 n ≥ 15 22τ(Q n ) = 1 √5(( 7 + 3√ 52) n − ( 7 − 3√ 5)), n n ≥ 1.2Proof: We put C n = τ(C n ) and Q n = τ(Q n ). C 1 = 8, Q 1 = 3, in the sequence of mapsC n ; we then cut the last triangle (see Figure 5.15), and use the Theorem 4.3.31 (the samegoes for the sequence of maps Q n ), then we obtain:τ(Q n ) = 3τ(C n−1 ) − τ(Q n−1 ), τ(C n ) = 3τ(Q n ) − τ(C n−1 )therefore, we have the following system:{Qn = 3C n−1 − Q n−1C n = 3Q n − C n−1By replacing Q n by its value in the second equation we get:{Cn = 8C n−1 − 3Q n−1Q n = 3C n−1 − Q n−1 , with C 1 = 8 and Q 1 = 3(Cn)= MQ n(Cn−1Q n−1), where M =( 8 -33 -1)