Simplices and spectra of graphs, continued

TL;DR

This paper explores the algebraic independence of certain face volumes of simplices and investigates how eigenvalues of Kneser graphs relate to simplex geometry, revealing new geometric and spectral properties.

Contribution

It demonstrates that codimension-2 face volumes of simplices are algebraically independent of edge lengths and constructs non-congruent simplices sharing similar face areas.

Findings

Codimension-2 face volumes are algebraically independent of edge lengths.

Eigenvalues of Kneser graphs relate to simplex face volumes.

Existence of non-congruent simplices with identical codimension-2 face areas.

Abstract

In this note we show that the n-2-dimensional volumes of codimension 2 faces of an n-dimensional simplex are algebraically independent quantities of the volumes of its edge-lengths. The proof involves computation of the eigenvalues of Kneser graphs. We also construct families of non-congurent simplices not determined by their codimension-2 areas.