Intriguing Sets of Partial Quadrangles

Finite geometry is the study of incidence structures, a system of points and lines with some points on some lines according to an incidence relation, such that the number of points and lines is finite. Generalised quadrangles arose from an effort by Tits [5] to give geometric constructions of the finite simple groups of Lie type. Tits' key observation was that the incidence graph of an n-gon always has diameter n and girth 2n. So when considering generalised quadrangles, we were concerned with finite incident structures with incidence graph of diameter 4 and girth 8. In particular, we focused on partial quadrangles, defined by Cameron [2]. Triangle-free strongly regular graphs, of which only seven are known [3], were our source of partial quadrangles, as they satisfy all the axioms of a partial quadrangle. A generalised quadrangle is also a partial quadrangle. The focus of this research was intriguing sets, found as tight sets and ovoids in finite geometry, upper and lower bounds of average degrees in graph theory and correspond to perfect error correcting codes of radius 1 in coding theory. We take a graph with adjacency matrix A and a map f that sends vertices to real numbers. We are looking for real constants α and β such that