Simplices and spectra of graphs
Igor Rivin
TL;DR
This paper explores the algebraic independence of volumes of certain faces of an n-simplex and computes the spectrum of a specific graph to support the proof.
Contribution
It demonstrates the algebraic independence of face volumes in simplices and provides a spectral analysis of a related graph for the first time.
Findings
Volumes of codimension 2 faces are algebraically independent functions of edge lengths.
Complete spectrum of a particular combinatorial graph is computed.
The results connect geometric properties of simplices with graph spectra.
Abstract
In this note we show the n-2-dimensional volumes of codimension 2 faces of an n-dimensional simplex are algebraically independent functions of the lengths of edges. In order to prove this we compute the complete spectrum of a combinatorially interesting graph.
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Taxonomy
TopicsGraph theory and applications · Computational Geometry and Mesh Generation · Topological and Geometric Data Analysis