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### Algebraic Approaches to Combinatorial Design

**Bailey, R; Warren, H; Piera Rogantin, M; Fontana, R; Maruri-Aguilar, H **

Thursday 27 October 2011, 09:00-12:30

Seminar Room 1, Newton Institute

#### Abstract

Hugo Maruri-Aguilar*(Queen Mary, University of London)***Some computational results for block designs**

Rosemary Bailey *(Queen Mary, University of London)*** Connectivity in block design, and Laplacian matrices**

Helen Warren *(London School of Hygiene and Tropical Medicine)*** Robustness of block designs**

A new robustness criteria, Vulnerability, measures the likelihood of an incomplete block design resulting in a disconnected eventual design due to the loss of random observations during the course of the experiment. Formulae have been derived for calculating the vulnerability measure, which aids in design selection and comparison, by producing a full vulnerability ranking of a set of competing designs. For example, this provides a new method for distinguishing between non-isomorphic BIBDs, since despite them all having identical optimality properties, their vulnerabilities can vary. Theory has been developed relating design concurrences to block intersection counts. These combinatorial results have provided further insight into the properties and characteristics of robust designs. Furthermore these have led to interesting closure properties for vulnerability between BIBDs and their complements, between BIBDs and non-balanced designs constructed from them by the removal of or addition of blocks (e.g. Regular Graph Designs, Nearly Balanced Designs), and between BIBDs and replicated BIBDs. It would be interesting to investigate the combinatorial properties of replicated designs in more detail, from connectedness and optimality perspectives, especially since other work on crossover designs has similarly found that replication leads to less robustness, and in order to extend the concept of vulnerability to other blocked designs, e.g. row-column designs, crossover designs and factorial designs. Finally it would be interesting to incorporate prior knowledge of varying probabilities for each observation being lost, rather than assuming observation loss to be random.

Maria Piera Rogantin *(Università degli Studi di Genova)*** Use of indicator functions in design**

Roberto Fontana *(Politecnico di Torino)*** Algebra and factorial designs**