Maximal trees with log-concave independence polynomials

If s k denotes the number of independent sets of cardinality k in graph G, and alpha(G) is the size of a maximum independent set, then I (G; x) = Sigma(alpha(G))(k=0) S(k)x(k) is the independence polynomial of G (I. Gutman and F. Harary, 1983, [ 8]). The Merrifield-Simmons index sigma (G) (known also as the Fibonacci number) of a graph G is defined as the number of all independent sets of G. Y. Alavi, P. J. Malde, A. J. Schwenk and P. Erd " os (1987, [ 2]) conjectured that I (T; x) is unimodal whenever T is a tree, while, in general, they proved that for each permutation pi of f {1; 2;...; alpha} there is a graph G with alpha (G) = alpha such that s(pi) (1) < s(pi)(2) < ... <s(pi) (alpha). By maximal tree on n vertices we mean a tree having a maximum number of maximal independent sets among all the trees of order n. B. Sagan proved that there are three types of maximal trees, which he called batons [24]. In this paper we derive closed formulas for the independence polynomials and MerrifieldSimmons indices of all the batons. In addition, we prove that I (T; x) is log-concave for every maximal tree T having an odd number of vertices. Our findings give support to the above mentioned conjecture.