We have the spectrum and the degree sequence of one graph.
Can we uniquely determine the graph with these given information? 



spectrum and degree sequence
No. One simple class of examples are Latin square graphs. If $L$ is an $n\times n$ Latin
square with entries from $\{1,\ldots,n\}$, the vertices of Latin square graph are the $n^2$
triples; two triples are adjacent if the agree on one of their three coordinates. This is
a regular graph of valency $3(n-1)$. In fact these graphs are strongly regular, and their
eigenvalues are $3(n-1)$, $n$ and $-3$ with respective multiplicities 1, $n-3$ and $n^2-3n+2$.
Two Latin squares give non-isomorphic graphs in they are in different main classes
(see the wikipedia article) and there are many main classes for large $n$. When $n=4$
there are two, and over a quarter of a million when $n=8$.


You can find some of the theory on line at 
 <http://www.cs.yale.edu/homes/spielman/561/lect23-09.pdf>


