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

CHAPTER 5.102THE NUMBER OF SPANNING TREES OF CERTAIN FAMILIES OFPLANAR MAPSNumerical results The Table 5.6 illustrates some of the values of the number of spanningtrees in the n-Kite chains K n and Q n by using the formula given in Theorem 5.8.2:n τ(K n ) τ(Q n )1 45 212 1584 7563 55404 264604 1937520 9253445 67756176 323598246 2369471616 11316412807 82861755840 395741324168 2897722232064 13839296808969 101334977144064 4839679974528010 3543741176832000 1692463322317824Table 5.6: Some values of τ(K n ) and τ(Q n )5.8.3 Formulae for the number of spanning trees in the n-Envelope chainsIn this section, we derive simple formulae for the number of spanning trees of a specialfamily of planar maps which is called the n-Envelope chains.Theorem 5.8.4 (The n-Envelope chains) The complexity of the n-Envelope chains E nand Q n (see Figure. 5.18) is given by the following formulae:1τ(E n ) =((4841+2284 √ √ 434)(45+2 √ 434) n−1 −(4841−228 √ 434)(45−2 √ )434) n−1 ,434n ≥ 1, andτ(Q n ) = 55 ((454 √ + 2 √ 434) n − (45 − 2 √ )434) n , n ≥ 1.434Proof: We put E n = τ(E n ) and Q n = τ(Q n ). E 1 = 114, Q 1 = 55, in the sequenceof maps E n , we cut the last Envelope (see Figure 5.18), and we use Theorem 4.3.31(the same goes for the sequence of maps Q n ), then we obtain: τ(Q n ) = 55τ(E n−1 ) −24τ(Q n−1 ), τ(E n ) = 114τ(E n−1 ) − 55τ(Q n−1 ) therefore, we have the following system:{En = 114E n−1 − 55Q n−1Q n = 55E n−1 − 24Q n−1 with E 1 = 114 and Q 1 = 55