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

5.8. The number of spanning trees in some particular planar maps 99(Cn)= MQ n(Cn−1Q n−1)= ... = M n−1 (C1Q 1), we compute M n−1 :det (M − λI 2 ) = λ 2 −7λ + 1 = 0, λ 1 = 7−3√ 5and λ2 2 = 7+3√ 5, λ2 1 ≠ λ 2 then there is amatrix P invertible such that M = P DP −1 where( )λ1 0D =0 λ 2and, P is the invertible transformation matrix formed by eigenvectors( 1 1P =3+ √ 523− √ 52M n−1 = P D n−1 P −1 where D n−1 =(M n−1 = √ −15hence the result.)(; P −1 = √ 15(( 7−3√ 53− √ 52-1−3− √ 512)) n−1 02, from which:0 ( 7+3√ 5) n−1 2)( 7−3√ 5) n−1 ( 3−√ 5) + ( 7+3√ 5) n−1 ( −3−√ 5) −( 7−3√ 5) n−1 + ( 7+3√ 5) n−12 2 2 2 2 2( 7−3√ 5) n−1 − ( 7+3√ 5) n−1 ( 3−√ 5)( 7+3√ 5) n−1 − ( 3+√ 5)( 7−3√ 5) n−1 2 2 2 2 2 2)□Numerical results The Table 5.5 illustrates some of the values of the number of spanningtrees in the sequence of planar maps τ(C n ) and τ(Q n ) by using the formula given inTheorem 5.8.1:n 1 2 3 4 5 6 7 8 9 10τ(C n ) 8 55 377 2584 17711 121393 832040 5702887 39088169 267914296τ(Q n ) 3 21 144 987 6765 46368 317811 2178309 14930352 102334155Table 5.5: Some values of τ(C n ) and τ(Q n )5.8.2 Formulae for the number of spanning trees in the n-KitechainsWe derive simple formulae for the number of spanning trees of a special family of planarmaps which is called the n-Kite chains.