On Classification of Two Class Partially Balanced Designs

Mohan Shrikhande
Department of Mathematics, Central Michigan University, Mt. Pleasant, Michigan, USA
Currently visiting: Department of Mathematics, University of Wales, Aberystwyth

Let Gamma = (V,E) be a connected strongly regular graph on v vertices and B be a family of k-subsets(blocks) of V. The pair D = (V,B) is a partially balanced (v,k,lambda,mu)-design on Gamma, if for any two distinct vertices x and y, the number of blocks containing {x,y} is lambda if {x,y} is an edge and mu otherwise. In such a design, the replication number r satisfies r>= 2lambda-mu. In joint work with Y.J. Ionin (to appear in the forthcoming issue of Journal of Statistical Planning and Inference), a complete classification of designs is obtained with r=2lambda-mu, in case Gamma has an eigenvalue -2. It is shown that there are three infinite families (and their complements) and several sporadic designs. The sporadic designs correspond to the Petersen, Clebsch, and Shrikhande graphs and the triangular graph T(4). This talk will discuss the main combinatorial ideas and tools needed and a very brief sketch of the proof.