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

5.8. The number of spanning trees in some particular planar maps 103(EnFigure 5.18: The n-Envelope chains E n)= MQ n( )En= MQ n(En−1), where M =Q n−1( )En−1Q n−1( 114 -5555 -24= ... = M n−1 (E1Q 1),we compute M n−1 :det (M − λI 2 ) = λ 2 −90λ + 289 = 0, λ 1 = 45 − 2 √ 434 and λ 2 = 45 + 2 √ 434, λ 1 ≠ λ 2 thenthere is a matrix 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 =M n−1 = P D n−1 P −1 whereD n−1 =P −1 =69+2 √ 43455−554 √ 434from which we obtain M n−1 , hence the result.(69−2 √ 4345569−2 √ 434-155−69−2 √ 434155( √ )(45 − 2 434)n−100 (45 + 2 √ 434) n−1))□