Partial geometric designs having circulant concurrence matrices

TL;DR

This paper explores the structure and existence of partial geometric designs with circulant concurrence matrices, classifying their parameters and providing new constructions and feasible parameter sets for small orders.

Contribution

It identifies new partial geometric designs with specific concurrence properties and characterizes which symmetric circulant matrices can serve as their concurrence matrices.

Findings

Existence of partial geometric designs with two or three distinct concurrences.

Characterization of feasible parameters for designs up to order 12.

Construction methods for designs with circulant concurrence matrices.

Abstract

We survey partial geometric designs and investigate their concurrences of points. The concurrence matrix of a design, which encodes the concurrences of pairs of points, can be used in the classification of designs in some extent. An ordinary 2-$(v,k,\lambda)$ design has concurrence $\lambda$ for any pair of distinct points, and its concurrence matrix is circulant. A partial geometry has two concurrences $1$ and $0$ and a transversal design TD$_{\lambda}(k, u)$ has two concurrences $\lambda$ and $0$. It is also known that the concurrence matrix of a partial geometric design can have at most three distinct eigenvalues, all of which are non-negative integers. In this paper, we show the existence of other partial geometric designs having two or three distinct concurrences, and investigate which symmetric circulant matrices are realized as the concurrence matrices of partial geometric designs. We collect known sources of partial geometric designs and study their structural characteristics and construction methods. We then give a list of feasible parameter sets for partial geometric designs of order up to 12 each of which has a circulant concurrence matrix. We also consider the combinatorial properties and constructions of partial geometric designs satisfying these parameter sets.