On the transitivity of Gilbert graphs and their complements

TL;DR

This paper classifies when Gilbert graphs and their complements are edge- or distance-transitive, revealing precise conditions and applying spectral methods to analyze their symmetry properties and compute Lovász theta functions.

Contribution

It provides a complete classification of transitivity properties for Gilbert graphs and their complements, including spectral analysis and exact Lovász theta function calculations.

Findings

Gilbert graphs are edge- and distance-transitive only for specific parameters.

Complement graphs' transitivity depends on spectral properties and differs from Gilbert graphs.

Exact Lovász theta functions are computed for cases with transitivity.

Abstract

The Gilbert graph $\text{Gilbert}(q,n,d)$, which arises naturally in graph theory and coding theory, is the regular graph on $\mathbb{F}_q^n$ in which two vertices are adjacent if their Hamming distance is less than $d$, and it is vertex-transitive. We classify all parameters $(q,n,d)$ for which $\text{Gilbert}(q,n,d)$ is edge-transitive or distance-transitive, and separately classify all parameters for which its complement has these properties. We prove that $\text{Gilbert}(q,n,d)$ is edge-transitive if and only if it is distance-transitive, and that this occurs precisely when $d=2$, $(q,d)=(2,3)$, or $(q,d)=(2,n)$. For the complement graphs, we determine all parameters yielding edge- or distance-transitivity using spectral methods based on Krawtchouk polynomials and the structure of the Hamming association scheme. In contrast to the Gilbert graphs, where the parameter sets corresponding to edge- and distance-transitivity coincide, we show that for their complements the set of parameters yielding distance-transitivity is strictly contained in the set yielding edge-transitivity. As an application, we compute the exact values of the Lov\'{a}sz $\vartheta$-function of Gilbert graphs, as well as of their complements, in all cases where either one of them is edge-transitive.