OFFSET
0,4
COMMENTS
Heuberger and Wagner consider how many different maximum matchings a tree of n vertices may have. They determine the unique tree (free tree) of n vertices with the largest number of maximum matchings, or at n=6 and n=34 the two trees with equal largest number. a(n) is the largest number of maximum matchings. They show that a(n) grows as O(1.391...^n), where the power is ((11 + sqrt(85))/2)^(1/7) = A333346.
They note an algebraic interpretation too, that a(n) is the largest possible absolute value of the product of the nonzero eigenvalues of the adjacency matrix of a tree of n vertices. This is simply that, in the usual way, a term +- m*x^j in the characteristic polynomial of that matrix means there are m matchings which have j vertices unmatched. The smallest j with a nonzero m is the maximum matchings, and that m is also the product of the nonzero roots.
In Heuberger and Wagner's Sage code, optimal_m(n) is a(n) for the general case tree forms. Their general case symbolic calculations are in terms of lambda = (11 + sqrt(85))/2 = A333345 and its quadratic conjugate lambdabar = (11 - sqrt(85))/2 (called alpha and alphabar in the code). The resulting coefficients give constants c_0 through c_6 in their paper for a(n) -> c_{n mod 7} * lambda^(n/7) (theorem 1.2).
The combinations of powers of lambda and lambdabar occurring are linear recurrences. Recurrence coefficients can be found from a symbolic calculation, or from explicit values and an upper bound on recurrence orders from the patterns of branch lengths and powers. Each case n mod 7 is a recurrence of order up to 44. The simplest is G_k = A190872(k) for n == 1 (mod 7) in the formulas below. Other cases are G variants, and possible additional terms growing slower than G.
The full recurrence for all n is order 574 applying at n=31 onwards (after the last initial exception at n=30). See the links for recurrence coefficients and generating function.
LINKS
Kevin Ryde, Table of n, a(n) for n = 0..1500
Clemens Heuberger and Stephan Wagner, The Number of Maximum Matchings in a Tree, Discrete Mathematics, volume 311, issue 21, November 2011, pages 2512-2542; arXiv preprint, arXiv:1011.6554 [math.CO], 2010.
Clemens Heuberger and Stephan Wagner, Number of Maximum Matchings In a Tree - Sage Code Worksheet.
Kevin Ryde, recurrence and generating function, in PARI/GP.
Kevin Ryde, vpar examples/most-maximum-matchings.gp creating, counting, and recurrences, in PARI/GP.
Kevin Ryde, Sample Tree Drawings.
FORMULA
CROSSREFS
KEYWORD
nonn
AUTHOR
Kevin Ryde, Mar 15 2020
STATUS
approved