Algebraic Graph Theory - Sung Song

Monday, June 19, 2023 11:30 am - 11:30 am EDT (GMT -04:00)

Title: Partial geometric designs, directed strongly regular graphs, and association scheme

Speaker: Sung Song
Affiliation: Iowa State University
Location: Please contactSabrina Latofor Zoom link

Abstract: A partial geometric design with parameters $(v, b, k, r; \alpha, \beta)$ is a tactical configuration $(P, \mathcal{B})$ (with $|P|=v$, $|\mathcal{B}|=b$, every point $p\in P$ belonging to $r$ blocks, and every block $B\in\mathcal{B}$ consisting of $k$ points) satisfying the property:

{for any pair $(p, B)\in P\times \mathcal{B}$, the number of flags $(q, C)$ with $q\in B$ and $C\ni p$ equals to $\alpha  \mbox{ if } p\notin B$ and to $\beta  \mbox{ if } p\in B$.}

Neumaier studied partial geometric designs in detail in his article, ``$t\frac12$-designs," [JCT A {\bf 28}, 226-248 (1980)]. He investigated their connection with strongly-regular graphs and gave various characterizations of partial geometries, bipartite graphs, symmetric 2-designs, and transversal designs in terms of partial geometric designs.

In this talk we review a few recent works on partial geometric designs and the related topics, and discuss the concurrence profiles of pair of points together with the incidence relations between points and blocks of the designs. Through the analysis of their concurrence matrices, we give further characterization of partial geometric designs. We also investigate their connection with directed strongly-regular graphs and association schemes (and other finite incidence structures if time permits). This talk is a survey expository on the topics.