Sums of squares and negative correlation for spanning forests of series parallel graphs

TL;DR

This paper demonstrates that spanning forests of series parallel graphs exhibit negative correlation properties similar to spanning trees, using electrical network principles and polynomial sum-of-squares representations.

Contribution

It establishes a new sum-of-squares expression for the Rayleigh difference in spanning forest polynomials of series parallel graphs and links matroid properties to Rayleigh conditions.

Findings

Spanning forests satisfy negative correlation properties.

Rayleigh difference expressed as sum of squares of polynomials.

Binary matroids' Rayleigh properties are characterized.

Abstract

We provide new evidence that spanning forests of graphs satisfy the same negative correlation properties as spanning trees, derived from Lord Rayleigh's monotonicity property for electrical networks. The main result of this paper is that the Rayleigh difference for the spanning forest generating polynomial of a series parallel graph can be expressed as a certain positive sum of monomials times squares of polynomials. We also show that every regular matroid is independent-set-Rayleigh if and only if every basis-Rayleigh binary matroid is also independent-set-Rayleigh.