Closing Trees into Unicyclic Counterexamples

Abstract

We develop a family-based route to unicyclic graphs whose independence polynomials are unimodal but not log-concave. The paper is organized around one flagship statement: for the explicit KL-closure family $U_{k,r}$, with $r\in\{0,1,2\}$ and admissible $k$, the independence polynomial is unimodal but not log-concave. The proof separates the closure polynomial into a dominant convolution term and a real-rooted correction term. On the non-log-concavity side, we prove symbolically that the penultimate log-concavity inequality fails for every admissible parameter. On the unimodality side, we prove that the main convolution term $H_{k,r}=G_kF_{k+r}$ is unimodal with a controlled mode, using a combination of exact coefficient formulas, Ibragimov's strong-unimodality principle, and a residue-class growth argument. Darroch localization and an adjacent-mode bridge lemma then transfer that mode statement to the full KL closure polynomial. This yields an explicit infinite family of unicyclic graphs with unimodal but non-log-concave independence polynomials. In the exact range $k\le 400$, we further verify that the penultimate break is unique and determine exact mode formulas for $H_{k,r}$, the binomial correction term, and $I(U_{k,r};x)$ itself. The paper also places the KL family inside a broader reservoir program involving Galvin, Ramos-Sun, and Bautista-Ramos trees, from which we obtain substantial universal exact theorems for finite ranges.