dot_clear dot_clear
Four different views of Glasgow University Tower

My research interests at the present time are in the area of classification of combinatorial designs of various different sorts, including Hadamard mareices, 2-designs, Two-graphs, strongly regular graphs and cospectral graphs. In some cases I have been able to classify the designs completely and where this has been possible I have stored the designs on disc. They can be accessed via the links in the side-bar on the left . These files will be updated at intervals, as I find the time.

Recently, along with Andries Brouwer [1], I have extended some of the results that Willem Haemers and I obtained concerning cospectral graphs [2]. In that paper Haemers and I considered graphs with n <= 11 vertices and determined the number that were cospectral with respect to the usual graph spectrum, together with the number cospectral with respect the Laplace and unsigned Laplace spectrum. Moreover, we also found the number of graphs G, H for which both G and H are cospectral and also their complements.

All the results of [2] have now been extended to the case n =12. I have completed the investigation for n = 12 and the Laplace spectrum as well as the signless Laplace spectrum. One further aspect considered is the question of the number of graphs G for which G and its complement G' have the same pair of characteristic polynomials, ( P, Q ) say. For graphs on less than seven vertices the answer is none.

The details of the data for all these different spectra may be found by accessing the appropriate link on the sidebar on the left-hand side of the page.

[1] A. E. Brouwer and E Spence, Cospectral graphs on 12 vertices, Elec. Journ. Combin., Vol. 16 (20) (2009).

[2] W. H. Haemers, E. Spence, Enumeration of cospectral graphs, European J. Combin, 25 (2004), 199-211.