TITLE: cospectral graphs
QUESTION [7 upvotes]: The simple connected graph $G$ has $n$ vertices and we have:
1) $|E(G)|‎\geq‎ \frac{n(n-1)}{3}$
2) we have the spectrum and degree sequence of $G$
3) $Spectrum(G)=Spectrum(H)$
Is $G \cong ‎H$?

REPLY [2 votes]: Connected strongly regular graphs with parameters $(v,k,\lambda)$ have eigenvalue $k$ appearing once and two other eigenvalues with prescribed multiplicity.  In general there are many non-isomorphic graphs for a fixed parameter.  There are two graphs with parameters $(16,6,2)$ and 15 with parameters $(25,12,5)$.  The number of non isomorphic graphs can grow dramatically....
All strongly regular graphs with parameter $(v,k,\lambda)$ are regular (so the degree sequence is obvious) and are cospectral.  Either the graph or its complement meets your bound on the number of edges.
Andries Brouwer maintains a nice webpage on strongly regular graphs here.  (Check out the number of $(36,15,6)$ strongly regular graphs!)