I'm working with signed graphs and I don't know the answer to the following question. Also, I couldn't find the answer anywhere.


Question: If we have two signed graphs with the same underlying graph and they are co-spectral, can we conclude they are switching equivalent?



Are cospectral signed graphs with identical underlying graph necessarily switching-equivalent?
No. Let $H$ be an $n\times n$ Hadamard matrix and let $X$ be the signed graph with adjacency matrix
\[
 A = \begin{pmatrix}0&H\\ H^T&0\end{pmatrix}.
\]
Then $A^2=nI$ and so the spectrum of $X$ is determined by $n$. However
if $K$ is a second $n\times n$ Hadamard matrix, then the corresponding signed graphs are switching equivalent if and only if there are monomial matrices $P$ and $Q$ with non-zero entries $\pm1$ such that $PHQ=K$.
In other words they are switching equivalent if and only if the Hadamard matrices are equivalent. If $n\ge16$, there is more than one equivalence class of Hadamard matrices.


