On Restricted Unitary Cayley Graphs and Symplectic Transformations Modulo $n$

Abstract

We present some observations on a restricted variant of unitary Cayley graphs modulo $n$, and implications for a decomposition of elements of symplectic operators over the integers modulo $n$. We define quadratic unitary Cayley graphs $G_n$, whose vertex set is the ring ${\Bbb Z}_n$, and where residues $a,b$ modulo $n$ are adjacent if and only if their difference is a quadratic residue. By bounding the diameter of such graphs, we show an upper bound on the number of elementary operations (symplectic scalar multiplications, symplectic row swaps, and row additions or subtractions) required to decompose a symplectic matrix over ${\Bbb Z}_n$. We also characterize the conditions on $n$ for $G_n$ to be a perfect graph.