Tutte polynomial, complete invariant, and theta series

TL;DR

This paper introduces a new complete polynomial invariant for graphs with a single variable and proposes a method to identify theta series equivalent lattices using Tutte polynomials and matroids.

Contribution

It provides a novel polynomial invariant for graphs and a method to determine theta series equivalence of lattices via Tutte polynomials and matroids.

Findings

New polynomial invariants for graphs with one variable

Method to find theta series equivalent lattices

Connection between Tutte polynomials and lattice equivalence

Abstract

In this study, we present two results that relate Tutte polynomials. First, we provide new and complete polynomial invariants for graphs. We note that the number of variables of our polynomials is one. Second, let L_1 and L_2 be two non-isomorphic lattices. We state that L_1 and L_2 are theta series equivalent if those theta series are the same. The problem of identifying theta series equivalent lattices is discussed in Prof.~Conway's book The Sensual (Quadratic) Form with the title "Can You Hear the Shape of a Lattice?" In this study, we present a method to find theta series equivalent lattices using matroids and their Tutte polynomials.