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 $(n{q}^3–q^3+n{q}^2,n{q}^2–q^2+nq–q,2qn–3q,qn–q)$ 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.