Symmetric and Spectral Realizations of Highly Symmetric Graphs

Abstract

A realization of a graph $G=(V,E)$ is a map $v:V\to {\mathbb{R}}^d$ that assigns to each vertex a point in $d$-dimensional Euclidean space. We study graph realizations from the perspective of representation theory (expressing certain symmetries), spectral graph theory (satisfying certain self-stress conditions) and rigidity theory (admitting deformations that do not alter the symmetry properties). We explore the connections between these perspectives, with a focus on realizations of highly symmetric graphs (arc-transitive/distance-transitive) and the question of how much symmetry is necessary to ensure that a realization is balanced, spectral, rigid etc. We include many examples to give a broad overview of the possibilities and restrictions of symmetric and spectral graph realizations.

Cited by in corpus (1)

• Eigenpolytopes, Spectral Polytopes and Edge-Transitivity