On the transitivity of Gilbert graphs and their complements

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.