Independence polynomials of well-covered graphs: generic counterexamples for the unimodality conjecture (Q2493111)

Let \(G\) be a graph, let \(s_k\) denote the number of independent sets (stable sets) of cardinality \(k\), and let \(\alpha(G)\) denote the maximum cardinality of an independent set in \(G\). Then \(I(G; x) = \sum_{k=0}^{\alpha(G)} s_kx^k\) is the independence polynomial of \(G\). \(I(G;x)\) is called unimodal if there exists some \(j\in\{0, 1, \dots, \alpha(G)\}\) such that \(s_0\leq \ldots\leq s_{j-1}\leq s_j\geq s_{j+1}\geq\ldots\geq s_{\alpha(G)}\). A graph \(G\) is well-covered if all of its inclusion-maximal independent sets have the same size. Given any integer \(\alpha\geq 4\), the authors construct a well-covered graph \(G\) with \(\alpha(G) = \alpha\), whose independence polynoimal \(I(G; x)\) is not unimodal. This disproves the unimodality conjecture for well-covered graphs, due to \textit{J. I. Brown} et al. [J. Algebr. Comb. 11, 197--210 (2000; Zbl 0994.05109)].