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An independent set in a graph is a collection of vertices that are notadjacent to each other. The cardinality of the largest independent set in $G$is represented by $\alpha(G)$. The independence polynomial of a graph $G = (V,E)$ was introduced by Gutman and Harary in 1983 and is defined as  \[  I(G;x) =\sum_{k=0}^{\alpha(G)}{s_k}x^{k}={s_0}+{s_1}x+{s_2}x^{2}+...+{s_{\alpha(G)}}x^{\alpha(G)},  \]  where $s_k$ represents the number of independent sets in $G$ of size $k$. Theconjecture made by Alavi, Malde, Schwenk, and Erd\"os in 1987 stated that theindependence polynomials of trees are unimodal, and many researchers believedthat this conjecture could be strengthened up to its corresponding log-concaveversion. However, in our paper, we present evidence that contradicts thisassumption by introducing infinite families of trees whose independencepolynomials are not log-concave.

The independence polynomial of trees is not always log-concave starting from order 26