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

6.4. An explicit formula for the number of spanning trees in S n,k 113By the sum of all the previous equations, we obtain:τ(S n,1 ) = 2(3) n−1 (n − 1) + 2(3) n −1 then, we take 2(3) n−1 as a factor, hence the result. □Example 6.4.4 In the star flower planar maps S 2,1 and S 3,1 shown in Figure. 6.7; wehave n = 2 in S 2,1 and n = 3 in S 3,1 then, we apply Corollary 6.4.3 to find the number ofspanning trees of these star flower planar maps S 2,1 and S 3,1 , we obtain:τ(S 2,1 ) = 2 ∗ 2(3) 2−1 = 12 and τ(S 3,1 ) = 2 ∗ 3(3) 3−1 = 54.Numerical results The Table 6.1 illustrates some of the values of the number ofspanning trees in the star flower planar map S n,1 by using the formula given in Corollary6.4.3:n 2 3 4 5 6 7 8 9 10τ(S n,1 ) 12 54 216 810 2916 10206 34992 118098 393660Table 6.1: Some values of τ(S n,1 )2) In the previous Theorem 6.4.1, if we take k = 2 (see the star flower planar mapS n,k shown in figure 6.3), then we obtain the star flower planar map S n,2 (see Figure. 6.9).Figure 6.9: The star flower planar map S n,2Corollary 6.4.5 The number of spanning trees of the star flower planar map S n,2 (seeFigure. 6.9) is given by the following formula:τ(S n,2 ) = n(2) 2n , n ≥ 2.