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

5.8. The number of spanning trees in some particular planar maps 105(dnFigure 5.19: The n-Diphenylene chains D n)= Mq n( )dn= Mq n(dn−1), where M =q n−1( )dn−1q n−1( 23 -46 -1= ... = M n−1 (d1q 1),we compute M n−1 :det (M − λI 2 ) = λ 2 −22λ + 1 = 0, λ 1 = 11 − 2 √ 30 and λ 2 = 11 + 2 √ 30, 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 =, P −1 = √ −1 6− √ )30-12306+ √ 302M n−1 = P D n−1 P −1 where,D n−1 =6− √ 302from which we obtain M n−1 , hence the result.−6− √ 3012( √ )(11 − 2 30)n−100 (11 + 2 √ 30) n−1Numerical results The Table 5.8 illustrates some of the values of the number of spanningtrees in the n-Diphenylene chains D n and Q n by using the formula given in Theorem5.8.5:)□