In what probability does cospectra of adjacent matrix of Cayley graph imply isomorphism of the corresponding group?
Further more,In what probability does cospectra of adjacent matrix imply isomorphism of the corresponding graphs



In what probability does cospectra of Cayley graph imply isomorphism of the corresponding group
For your second question, if by *probability* you mean $$\lim\_{n \to \infty} \frac{|S\_n|}{|G\_n|},$$ where $S\_n$ is the set of all possible spectra of simple $n$-vertex graphs, and $G\_n$ is the set of isomorphism classes of simple $n$-vertex graphs, then it is conjectured that the above probability is 1. That is, almost all graphs are determined by their spectrum. 


See my answer to this [question](https://mathoverflow.net/questions/49226/classes-of-graphs-for-which-isospectrum-implies-isomorphism/49256#49256) for more information and references.


