On restricted unitary Cayley graphs and symplectic transformations modulo n

TL;DR

This paper investigates quadratic unitary Cayley graphs modulo n and explores their implications for decomposing symplectic matrices over the integers modulo n, providing bounds on elementary operations needed.

Contribution

It introduces quadratic unitary Cayley graphs G_n, analyzes their diameter and perfectness, and links these properties to symplectic matrix decomposition complexity.

Findings

Bounded the diameter of G_n graphs, leading to an upper bound on symplectic matrix decomposition steps.

Characterized conditions under which G_n is a perfect graph.

Provided insights into the structure of symplectic transformations modulo n.

Abstract

We present some observations on a restricted variant of unitary Cayley graphs modulo n, and the 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 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 Z_n. We also characterize the conditions on n for G_n to be a perfect graph.