Diameter and Girth of Representation Graphs of Quadratic Forms

Abstract

Let $q$ be a non-degenerate quadratic form defined on an $F$ vector space $V$ and $a \in F$. We consider the Cayley graph on $V$ with generating set $\{x \in V \mid q(x) = a\}$ and study its diameter and girth. In particular, if $F$ is a finite field, we calculate these invariants and the number of cycles of minimal length in these graphs.