Irreducibility of the Dispersion Polynomial for Periodic Graphs

TL;DR

This paper investigates the mathematical properties of the dispersion polynomial in periodic graphs, demonstrating its irreducibility and applying these findings to various lattice structures including hexagonal and diamond lattices.

Contribution

It introduces new algebraic and geometric methods to analyze the irreducibility of dispersion polynomials in periodic graphs, with broad applications.

Findings

Dispersion polynomial is irreducible after changing the period lattice.

Results apply to dense periodic graphs and specific lattice families.

Provides algebraic tools for analyzing periodic operators on graphs.

Abstract

We use methods from algebra and discrete geometry to study the irreducibility of the dispersion polynomial of a discrete periodic operator associated to a periodic graph after changing the period lattice. We provide numerous applications of these results to discrete periodic operators associated to families of graphs which include dense periodic graphs, and a family containing the hexagonal and diamond lattices.