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Let we consider cayley graph $G=cayley(Z_2^n,S)$ which S is a subset of $Z_2^n$. If we consider a set of spectrum for this graph which satisfies all relations for cayley graph like $\sum_i \lambda_i^2=2E(G)$ and all other relations, Can we find the adjacency matrix or graph of that?

Any reference or hint would be appreciated

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If I understood correctly, you do not have any information about $S$. In general, the answer to your question is no. The group $Z_2^n$ is not $Cay-DS$ in general. So, for suitable $n$, you can find two subsets $S$ and $S'$ of $Z_2^n$ in such a way that $Cay(Z_2^n,S)$ is cospectral with $Cay(Z_2^n,S')$, but these graphs are not isomorphic. So, we can not identify the set $S$ and so its adjacency matrix in general. You can see our paper:

"Groups all whose undirected Cayley graphs are determined by their spectra", J. Algebra Appl., 15(9) (2016),1650175, 15 pp.

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