Algebraic Graph Theory Seminar - Chris Godsil

Title: Cospectral and strongly cospectral vertices

Abstract:

If $a$ is a vertex in a graph with adjacency matrix $A$, the \textsl{walk module} generated by $a$ is the $A$-invariant subspace spanned by the vectors $A^re_a$, for $r\ge0$. Two vertices in a graph are \textsl{cospectral} if their walk-modules are isomorphic; they are \textsl{strongly cospectral} if their walk modules are equal. (If the eigenvalues of $A$ are simple, two vertices are strongly cospectral if and only if they are cospectral.)

I will present some of the relevant theory related to these concepts, and discuss a proof that if $a$ and $b$ are strongly cospectral, there is something very close to an automorphism of the graph that swaps $a$ and $b$.