On 5- and 6-Leaved Trees with the Largest Number of Matchings

A matching of a graph is a set of its edges that pairwise do not have common vertices. An important parameter of graphs, which is used in mathematical chemistry, is the Hosoya index, defined as the number of their matchings. Previously, the problems of maximizing this index were considered and completely solved for $n$-vertex trees with two, three and four leaves, for any sufficiently large $n$. In the present paper, a similar problem is completely solved for 5-leaved trees with $n\geqslant 20$ and for 6-leaved trees with $n\geqslant 26$.