Strongly Regular Graphs from Large Arcs in Affine Planes

Abstract

Strongly regular graphs with parameters \((q^3 + 2q^2, q^2 + q, q, q)\), \((q^3 + q^2 + q + 1, q^2 + q, q – 1, q + 1)\), and \((q^3, q^2 + q – 2, q – 2, q + 2)\) are constructed from \(k\)-arcs in affine planes of order \(q\) with \(k = q + 2, q + 1, q\). In addition, strongly regular graphs with parameters \((nq^3 – q^3 + nq^2, nq^2 – q^2 + nq – q, 2qn – 3q, qn – q)\) are constructed from maximal arcs of degree \(n\) in affine planes of order \(q\). Each of these examples generalizes previously known examples when the affine planes were assumed to be Desarguesian.