The independence polynomial of a graph counts independent sets by size — coefficient of x^k is the number of k-element sets with no two vertices adjacent. For many graph families, these coefficients form a log-concave sequence: each coefficient squared exceeds the product of its neighbors. Log-concavity is the combinatorial norm. It holds for matchings, for matroids, for characteristic polynomials of graphs. When it fails, the failure demands explanation.
Bautista-Ramos, Guillén-Galván, and Gómez-Salgado study infinite families of trees whose independence polynomials are not log-concave. They find that these polynomials satisfy linear recurrence relations — each polynomial in the family is a fixed linear combination of its predecessors. The recurrences arise not from the graphs' structural recursion (trees are already recursive) but from ideal-theoretic properties in the polynomial ring: the independence polynomials, viewed as elements of a polynomial ring, generate ideals with predictable algebraic structure.
The sharpest result concerns zeros. The limit points of zeros of these non-log-concave independence polynomials lie on a specific circle in the complex plane: |z + 1/3| = 1/3. This is a circle centered at -1/3 with radius 1/3, passing through the origin. The circle is determined by the recurrence structure, not by the specific graph family — different non-log-concave tree families converge to the same locus.
Log-concavity failure is usually treated as pathology — a deviation from the expected behavior. But when the deviations organize onto a circle in the complex plane, the “pathology” has more structure than the norm. The zeros know where to go. The non-log-concave families are not random exceptions to log-concavity; they are a coherent phenomenon with its own geometry.