Algebraic Graph Theory - Chris Godsil

Title: Can we distinguish trees by a polynomial?

Abstract:

It is well known that, in general, trees are not distinguished by their characteristic polynomials. Brendan McKay proved that no "natural" polynomial does any better than the characteristic polynomial; he also proved a tree is determined by the traces of the monomials in the adjacency matrix and the matrix of valencies of the tree. Both proofs are difficult, and I will discuss some (vague) attempts to simplify them.