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

CHAPTER 6.112COUNTING THE NUMBER OF SPANNING TREES IN THE STARFLOWER PLANAR MAPFigure 6.7: The star flower planar maps S 2,1 and S 3,1Corollary 6.4.3 The number of spanning trees of the star flower planar map S n,1 (seeFigure. 6.6) is given by the following formula:τ(S n,1 ) = 2n(3) n−1 , n ≥ 2.Proof: We cut a triangle as shown in Figure. 6.6, we then obtain the star flower planarmap S n−1,1 after cutting (see Figure. 6.8):Figure 6.8: The star flower planar map S n,1 after cuttingand we apply Theorem 4.3.26, then we obtain:τ(S n,1 ) = (3) n−1 ∗ 2 + τ(S n−1,1 ) ∗ (3)τ(S n,1 ) = 2(3) n−1 + (3)τ(S n−1,1 )(3)τ(S n−1,1 ) = 2(3) n−1 + (3) 2 τ(S n−2,1 )(3) 2 τ(S n−2,1 ) = 2(3) n−1 + (3) 3 τ(S n−3,1 )(3) 3 τ(S n−3,1 ) = 2(3) n−1 + (3) 4 τ(S n−4,1 ). . . . . . . . .(3) n−4 τ(S 4,1 ) = 2(3) n−1 + (3) n−3 τ(S 3,1 )(3) n−3 τ(S 3,1 ) = 2(3) n−1 + (3) n−2 τ(S 2,1 )(3) n−2 τ(S 2,1 ) = 2(3) n−1 + (3) n−1 ∗ 2