Real roots of the flow polynomial

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Choose an orientation of a finite graph $G$. A nowhere-zero $q$-flow assigns to every edge a nonzero element of $\mathbb Z/q\mathbb Z$, with the outgoing sum equal to the incoming sum at each vertex. The flow polynomial $Q_G(x)$ is the polynomial whose value at each positive integer $q$ counts these flows, independently of the orientation. Graphs with bridges have the zero flow polynomial.

Statement recorded in the supplied source.

The main recorded conjecture asks whether every real zero $x$ of every nonzero flow polynomial satisfies $x\le4$. The source's later discussion separately proposes the bounds $x\le5$ and $x\le6$ in place of $4$. [S2]

Short English statement

How large can a real root of a nonzero graph flow polynomial be? The source records the historical bounds four, five and six.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3221 (OPG-135). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Real roots of the flow polynomial, original node 135, statement and mathematical discussion. https://www.openproblemgarden.org/op/real_roots_of_the_flow_polynomial.