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.



if two graphs have the same distinct eigenvalues, can we conclude that they are isomorphic?
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).


