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.