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math.CO — Combinatorics

A Nonunimodal Acyclic Polynomial with All Roots in the Left Half-Plane

Contributed by YURAN XIU

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 full counterexample, using an exact rational Lyapunov certificate for strict stability. HAL deposit: hal-05800220v1.

Primarily AI-generated textHuman understanding: no partsHurwitz stabilityLean formalizationacyclic polynomialcounterexamplegraph polynomial

Oct 18, 2026PDF ↗View work →

hexagon:2610.00326v1

math.CO — Combinatorics

A Bipartite Graph with a Nonunimodal Acyclic Polynomial

Contributed by YURAN XIU

We construct a bipartite graph on 230 vertices whose acyclic polynomial, which enumerates vertex subsets inducing forests, is not unimodal. This answers Question 3.3 of Barton, Brown, and Pike negatively. The graph has a common set of 50 vertices and 15 branches, each with two sets of six vertices. A partition according to the number of selected common vertices gives a three-term formula for the complete induced-forest polynomial. Its coefficients have a strict descent from degree 62 to degree 63 and a later strict ascent from degree 66 to degree 67. The actual graph, the unrestricted subset-counting formula, and the exact inequalities have been checked in Lean. HAL deposit: hal-05800225v1.

Primarily AI-generated textHuman understanding: no partsLean formalizationacyclic polynomialbipartite graphscounterexampleunimodality

Oct 18, 2026PDF ↗View work →

hexagon:2610.00320v1

math.CO — Combinatorics

The $\mathfrak{sl}(2)$ weight system at $c=-3/32$ extends to a graph 4-invariant

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya

We prove unconditionally the conjecture of Fomichev, Karev, Pavutnitskiy and Usanov that the specialization of the $\mathfrak{sl}(2)$ weight system at $c=-3/32$ extends to a $4$ -invariant of graphs. The extension is $\xi(G)=(-32)^{-n}S_G(-2)$ , where $S_G(N)$ counts with signs the assignments of $2$ -element subsets of $\{1,\dots,N\}$ to the $n$ vertices of $G$ .

A mix of human-written and AI-generated textHuman understanding: some partsClifford algebrafour-term relationintersection graphoscillator representationweight system

Oct 17, 2026PDF ↗View work →

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