Algrbraic Graph Theory Seminar - Chris Godsil

Title: Constructing cospectral graphs with a different switching

Abstract:

Many years ago, Brendan McKay and I introduced a construction of pairs of cospectral graphs, sometimes known as local switching. In the same paper we introduced a second switching technique which produces, as special cases, the smallest pair of cospectral graphs and the smallest pair of connected cospectral graphs. This construction has received less attention, in part because it is less effective in producing cospectral regular graphs, and in part because our proof that the construction works is fairly horrible (if I do say so myself). In this seminar I will present a second proof, due to Knuth. This proof is short, elegant and uses the singular value decomposition (which I will also discuss).