Discriminants, symmetrized graph monomials, and sums of squares certificates

TL;DR

This paper provides certificates demonstrating that certain symmetrized graph monomials for specific 6-edged multigraphs can be expressed as sums of squares but cannot be represented as linear combinations of partition square graphs, expanding understanding of their algebraic properties.

Contribution

It introduces certificates for five classes of 6-edged multigraphs showing their symmetrized graph monomials are sums of squares but not linear combinations of partition square graphs, complementing prior results.

Findings

Identified five classes of 6-edged multigraphs with sum of squares certificates.

Proved these symmetrized graph monomials cannot be expressed as linear combinations of partition square graphs.

Extended the understanding of algebraic representations of symmetrized graph monomials.

Abstract

Here we present certificates for 5 classes of 6-edged multigraphs whose symmetrized graph monomials may be represented as sum of squares, but not as linear combinations of partition square graphs. This is a complement to the results presented in Discriminants, symmetrized graph monomials, and sums of squares.