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