Strongly regular graphs from integral point sets in even dimensional affine spaces over finite fields

Abstract

In the $m$-dimensional affine space $AG(m,q)$ over the finite field ${\mathbb{F}}_q$ of odd order $q$, the analogous of the Euclidean distance gives rise to a graph ${\mathfrak{G}}_{m,q}$ where vertices are the points of $AG(m,q)$ and two vertices are adjacent if their (formal) squared Euclidean distance is a square in ${\mathbb{F}}_q$ (including the zero). In 2009, Kurz and Meyer made the conjecture that if $m$ is even then ${\mathfrak{G}}_{m,q}$ is a strongly regular graph. In this paper we prove their conjecture.

14 pages