Flow polynomials as Feynman amplitudes and their $\alpha$-representation

TL;DR

This paper expresses the flow polynomial of a graph using Feynman amplitude techniques and Legendre symbols, aiming to contribute to the proof of the Tutte 5-flow conjecture.

Contribution

It introduces a new formula for the flow polynomial based on Fourier transforms and Legendre symbols, connecting graph theory with quantum field theory methods.

Findings

Flow polynomial expressed as a linear combination of Legendre symbols.

Coefficients are ±1/q^{(V(H)-1)/2} depending on contracted graphs.

Potential application to prove the Tutte 5-flow conjecture.

Abstract

Let $G$ be a connected graph; denote by $\tau(G)$ the set of its spanning trees. Let $\mathbb F_q$ be a finite field, $s(\alpha,G)=\sum_{T\in\tau(G)} \prod_{e \in E(T)} \alpha_e$, where ${\alpha_e\in \mathbb F_q}$. Kontsevich conjectured in 1997 that the number of nonzero values of $s(\alpha, G)$ is a polynomial in $q$ for all graphs. This conjecture was disproved by Brosnan and Belkale. In this paper, using the standard technique of the Fourier transformation of Feynman amplitudes, we express the flow polynomial $F_G(q)$ in terms of the "correct" Kontsevich formula. Our formula represents $F_G(q)$ as a linear combination of Legendre symbols of $s(\alpha, H)$ with coefficients $\pm 1/q^{(|V(H)|-1)/2}$, where $H$ is a contracted graph of $G$ depending on $\alpha\in \left(\mathbb F^*_q\right)^{E(G)}$, and $|V(H)|$ is odd. The case $q=5$ corresponds to the least number with which all coefficients in the linear combination are positive. This allows us to hope that the obtained result can be applied to prove the Tutte 5-flow conjecture.