Intensive course on
Laplacian eigenvalues and optimality

Lecturers: R. A. Bailey and Peter J. Cameron (QMUL)

Rockefeller Building, Gower Street, London
13–14 June 2012

A request
If you were at the lectures on Wednesday afternoon,
and didn't get your name on the attendance sheet,
could you please email Nisha Jones (address below).
It is helpful to the LTCC to have accurate data!
Thank you for your consideration.

Registration | Synopsis | Course materials | Other materials

Eigenvalues of the Laplacian matrix of a graph have been widely used in studying connectivity and expansion properties of networks.

Independently, statisticians introduced various optimality criteria in experimental design, the goal being to obtain more accurate estimates of quantities of interest in an experiment. It turns out that the most popular of these optimality criteria for block designs are determined by the Laplacian eigenvalues of the concurrence graph, or the Levi graph, of the design.

The most important optimality criteria, called A (average), D (determinant) and E (extreme), are related to the conductance of the graph as an electrical network, the number of spanning trees, and the isoperimetric properties of the graphs.

The number of spanning trees is also an evaluation of the Tutte polynomial of the graph, and is the subject of the Merino--Welsh conjecture relating it to acyclic and totally cyclic orientations, of interest in their own right.

Course details and registration

The course is over now. But here is Nisha Jones' email for correspondence about technical aspects of the course:

For mathematical aspects, email one of the lecturers: R.A.Bailey or P.J.Cameron at

Details of all LTCC intensives are available here.

Synopsis of the course

The course was organised as four 2-hour lectures, commencing at 1pm on Wednesday 13 June 2012, and finishing at around 1pm the next day.

Lecture 1: Block designs (RAB)
Block designs in use; complete- and incomplete-block designs; balanced incomplete-block designs; estimation using block designs.
Lecture 2: Graphs and Laplacians (PJC)
The Laplacian matrix of a graph and its spectrum; connections with spanning trees, isoperimetric properties, electrical networks and random walks.
Lecture 3: Designs, graphs and optimality (RAB)
The concurrence and Levi graphs of a block design. Variance and resistance, spanning trees. Definitions of optimality; optimality of BIBDs and group divisible designs.
Lecture 4: More on graphs (PJC)
Optimality and graph properties; variance-balanced designs. The Tutte polynomial; spanning trees, acyclic and totally cyclic orientations of graphs.

Course materials

The lecture slides have been updated since the lectures were given. Current versions from 15 June.

Other material

Here is some recommended further reading.

Peter J. Cameron
15 June 2012