Discriminants, symmetrized graph monomials, and sums of squares

TL;DR

This paper investigates when symmetrized graph monomials, constructed from graphs without loops, are non-negative or sums of squares, motivated by invariant theory and a conjecture related to polynomial discriminants.

Contribution

It provides a detailed analysis of symmetrized graph monomials for graphs with four and six edges, including computational results and examples.

Findings

Identifies graphs with symmetrized monomials that are non-negative but not sums of squares.

Provides computational data on symmetrized graph monomials for specific graph classes.

Highlights the connection to invariant theory and polynomial discriminants.

Abstract

Motivated by the necessities of the invariant theory of binary forms J. J. Sylvester constructed in 1878 for each graph with possible multiple edges but without loops its symmetrized graph monomial which is a polynomial in the vertex labels of the original graph. In the 20-th century this construction was studied by several authors. We pose the question for which graphs this polynomial is a non-negative resp. a sum of squares. This problem is motivated by a recent conjecture of F. Sottile and E. Mukhin on discriminant of the derivative of a univariate polynomial, and an interesting example of P. and A. Lax of a graph with 4 edges whose symmetrized graph monomial is non-negative but not a sum of squares. We present detailed information about symmetrized graph monomials for graphs with four and six edges, obtained by computer calculations.