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.

翻译：我们发展了一种基于族的方法，用于构造其独立多项式为单峰但不满足对数凹性的单圈图。本文围绕一个核心结论展开：对于明确的KL闭包族$U_{k,r}$，其中$r\in\{0,1,2\}$且$k$为可允许参数，其独立多项式是单峰但不满足对数凹性的。证明过程将闭包多项式分解为占优的卷积项与实根修正项。在非对数凹性方面，我们通过符号计算证明，对所有可允许参数，倒数第二个对数凹性不等式均不成立。在单峰性方面，我们证明了主卷积项$H_{k,r}=G_kF_{k+r}$具有受控众数的单峰性，这是通过组合使用精确系数公式、Ibragimov强单峰性原理以及剩余类增长论证得到的。Darroch定位化方法与邻接众数桥引理进一步将该众数结论迁移至完整的KL闭包多项式。这构造了一个显式的无穷单圈图族，其独立多项式单峰但不满足对数凹性。在精确范围$k\le 400$内，我们进一步验证了倒数第二个断点具有唯一性，并确定了$H_{k,r}$、二项式修正项以及$I(U_{k,r};x)$本身的精确众数公式。本文还将KL族置于涉及Galvin树、Ramos-Sun树与Bautista-Ramos树的更广泛库程序框架中，由此获得了有限范围内的若干普适精确定理。