Super Stable Tensegrities and the Colin de Verdi\`{e}re Number $\nu$

TL;DR

This paper establishes a precise link between super stable tensegrities' realizability in certain dimensions and the Colin de Verdière number, providing a combinatorial characterization for 3D cases.

Contribution

It reveals an exact relationship between super stable tensegrities' maximum realization dimension and the Colin de Verdière number, advancing understanding in spectral graph theory and tensegrity structures.

Findings

Exact relation between tensegrity dimension and Colin de Verdière number

Characterization of multigraphs realizable as 3D super stable tensegrities

Bridges spectral graph theory with geometric rigidity concepts

Abstract

A super stable tensegrity introduced by Connelly in 1982 is a globally rigid discrete structure made from stiff bars or struts connected by cables with tension. In this paper we show an exact relation between the maximum dimension that a multigraph can be realized as a super stable tensegrity and Colin de Verdi\`{e}re number~$\nu$ from spectral graph theory. As a corollary we obtain a combinatorial characterization of multigraphs that can be realized as 3-dimensional super stable tensegrities.