Eigenpolytopes, Spectral Polytopes and Edge-Transitivity

TL;DR

This paper explores the properties of eigenpolytopes derived from graphs, introduces a geometric criterion for spectrality, and classifies distance-transitive polytopes, revealing their spectral nature and symmetry characteristics.

Contribution

It introduces a new geometric condition to identify spectral polytopes and provides a complete classification of distance-transitive polytopes, linking spectral properties with symmetry.

Findings

Every edge-transitive polytope is $ heta_2$-spectral.

Edge-transitive polytopes are uniquely determined by their graphs.

Complete classification of distance-transitive polytopes.

Abstract

Starting from a finite simple graph $G$, for each eigenvalue $\theta$ of its adjacency matrix one can construct a convex polytope $P_G(\theta)$, the so called $\theta$-eigenpolytop of $G$. For some polytopes this technique can be used to reconstruct the polytopes from its edge-graph. Such polytopes (we shall call them spectral) are still badly understood. We give an overview of the literature for eigenpolytopes and spectral polytopes. We introduce a geometric condition by which to prove that a given polytope is spectral (more exactly, $\theta_2$-spectral). We apply this criterion to the edge-transitive polytopes. We show that every edge-transitive polytope is $\theta_2$-spectral, is uniquely determined by this graph, and realizes all its symmetries. We give a complete classification of distance-transitive polytopes.