Graph Complexity and Link Colorings

TL;DR

This paper explores the relationship between graph complexity, link colorings, and algebraic properties of Laplacian polynomials, connecting graph theory, knot theory, and number theory to address open questions like Lehmer's problem.

Contribution

It establishes a link between the torsion complexity of periodic graphs, Mahler measures, and link invariants, providing new insights into algebraic and topological graph properties.

Findings

Complexity growth relates to Mahler measure of Laplacian polynomial.

1-periodic plane graphs correspond to links with unknotted components.

Lehmer's question connects to complexity growth in signed periodic graphs.

Abstract

The (torsion) complexity of a finite signed graph is defined to be the order of the torsion subgroup of the abelian group presented by its Laplacian matrix. When $G$ is $d$-periodic (i.e., $G$ has a free ${\mathbb Z}^d$-action by graph automorphisms with finite quotient) the Mahler measure of its Laplacian polynomial is the growth rate of the complexity of finite quotients of $G$. Any 1-periodic plane graph $G$ determines a link $\ell \cup C$ with unknotted component $C$. In this case the Laplacian polynomial of $G$ is related to the Alexander polynomial of the link. Lehmer's question, an open question about the roots of monic integral polynomials, is equivalent to a question about the complexity growth of signed 1-periodic graphs that are not necessarily embedded.