Two graphs are isospectral if the combinatorial Laplacian on them has the same spectrum, equivalently, the adjacency matrix has the same the set of eigenvalues (including multiplicities). Two graphs have the same matroid if they are 2-isomorphic, that means there exists a bijection between their edge sets that preserves cycles. There are plenty of examples of isospectral graphs (that are not isomorphic) and it is fairly easy to draw examples of 2-isomorphic graphs but can both of these phenomena occur at the same time?

In general I don't see any reason why no example of this should exist. I know examples of isospectral graphs can be constructed with Seidel switching but this results in graphs with relatively large average degree and usually high connectivity. On the other hand, 3-connected graphs are uniquely determined by their matroid so any potential example cannot be 3-connected.

I am interested in connected non-weighted graphs, I don't care whether they are simple (ie have no loops or multiple edges). Thanks