Counterexamples to Unimodality of the Connected Set Polynomial

For a finite graph G, let C(G;x) count connected induced vertex sets by cardinality. We construct four explicit pairwise nonisomorphic connected counterexample families for every order n>=8 to a question of Mol on unimodality when the number of edges is at least the number of vertices. A common rooted-block formula proves the fixed coefficient pattern 10,11,10. A complete scan of all 11,117 connected eight-vertex graph classes proves that eight is the minimum counterexample order and that the four first family members are exactly all minimum-order counterexamples. Status: Public Beta; internally verified candidate proof; external mathematical review pending.