Partial geometric designs having circulant concurrence matrices

Abstract

We classify small partial geometric designs (PGDs) by spectral characteristics of their concurrence matrices. It is well known that the concurrence matrix of a PGD can have at most three distinct eigenvalues, all of which are nonnegative integers. The matrix contains useful information on the incidence structure of the design. An ordinary 2-$\left(v,k,\lambda \right)$ design has a single concurrence $\lambda$, and its concurrence matrix is circulant, a partial geometry has two concurrences 1 and 0, and a transversal design ${\text{TD}}_{\lambda }\left(k,u\right)$ has two concurrences $\lambda$ and 0, and its concurrence matrix is circulant. In this paper, we survey the known PGDs by highlighting their concurrences and constructions. Then we investigate which symmetric circulant matrices are realized as the concurrence matrices of PGDs. In particular, we try to give a list of all PGDs of order up to 12 each of which has a circulant concurrence matrix. We then describe these designs along with their combinatorial properties and constructions. This work is part of the second author's Ph.D. dissertation [46].