if two graphs have the same distinct eigenvalues, can we conclude that they are isomorphic? i think i saw something like the statement of the question above but i am not sure.
Given two graph G and H if they have the same characteristic polynomial and it does not have any repeated roots,then G and H are isomorphic ? 
if any one could give me the reference to the proof or even if its a true statement.
 A: Here are the edge sets for two trees on 12 vertices.
Graph 1: 
$\left[\left(0, 1\right), \left(1, 2\right), \left(2, 3\right), 
\left(2, 10\right), \left(3, 4\right), \left(3, 11\right), \left(4, 5\right), \left(5, 6\right), \left(6, 7\right), \left(7, 8\right), \left(8, 9\right)\right]$
Graph 2: $\left[\left(0, 1\right), \left(1, 2\right), \left(2, 3\right), \left(3, 4\right), \left(4, 5\right), \left(4, 9\right), \left(5, 6\right), \left(6, 7\right), \left(6, 10\right), \left(7, 8\right), \left(10, 11\right)\right]$
Both graphs have exactly two vertices of degree three, but in the first graph they are adjacent and in the second they are at distance two.
For both graphs the characteristic polynomial is $(x - 1) \cdot (x + 1) \cdot (x^{10} - 10x^{8} + 33x^{6} - 40x^{4} + 13x^{2} - 1)$ (where the degree 10 factor is irreducible).
A: No, we cannot. However, it is easy to determine if they are isomorphic or not as long as they are cospectral and have distinct eigenvalues (multiplicity 1).
Let $A,B$ are adjacency or laplacian of two graphs that you want to test. Their eigendecompositions are $A=U \Sigma U^{T}$ and  $B=V \Sigma V^{T}$. See their eigenvalues ($\Sigma$ is a diagonal matrix which keeps eigenvalues) are the same and let's suppose it has distinct values.
$S$ is a diagonal matrix that includes $\mp1$ and $P$ is a permutation matrix where each column and row there is single 1 and rest of them 0.
If you find $P$ and $S$ matrices that meets $PU=VS$ then they are isomorphic, unless they are not.
If some eigenvalues are repeated in $\Sigma$, even though you find $P$ and $S$ matrix that meets $PU=VS$, still you cannot say they are isomorphic.
check out this couse notes;
http://www.cs.yale.edu/homes/spielman/561/lect08-18.pdf
A: Check out these notes by Daniel Spielman.
