A Nonunimodal Acyclic Polynomial with All Roots in the Left Half-Plane
Let $A(G,x)$ count vertex subsets of a graph $G$ that induce forests, by their cardinalities. Barton, Brown, and Pike asked whether failure of coefficient unimodality forces $A(G,x)$ to have a root with positive real part. We give a negative answer. The join of $K_{46}$ with three disjoint copies of $P_3$ has a nonunimodal acyclic polynomial of degree nine, yet all its complex roots have strictly negative real part. We derive the polynomial by classifying induced forests and give its exact positive Hurwitz determinants. A Lean formalization verifies the graph-to-polynomial identity and the…
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