On stress matrices of chordal bar frameworks in general position

TL;DR

This paper characterizes the rigidity of chordal bar frameworks in general position using stress matrices, establishing conditions for universal and global rigidity, and linking stress matrix rank properties to positive semidefinite matrices.

Contribution

It introduces a stress matrix-based characterization of rigidity for chordal frameworks in general position, including conditions for positive semidefinite stress matrices.

Findings

Stress matrices characterize rigidity in chordal frameworks.

Rank conditions on stress matrices imply positive semidefinite solutions.

Chordal frameworks in general position have specific rigidity properties.

Abstract

A bar framework in R^r, denoted by G(p), is a simple connected graph G whose vertices are points p^1,...,p^n in R^r that affinely span R^r, and whose edges are line segments between pairs of these points. In this paper, we use stress matrices to characterize the universal and global rigidities of chordal bar frameworks in general position in R^r, i.e., bar frameworks where graph G is chordal and the points p^1,...,p^n are in general position in R^r. We also prove that if a chordal bar framework in R^r admits a stress matrix of rank n-r-1 with generic rank profile, then it admits a positive semidefinite stress matrix of rank n-r-1.