I am looking for two undirected graphs $G$ and $H$ of the same order (i.e., they have the same number of vertices) such that $G$ and $H$ are
- cospectral (i.e., their adjacency matrices $A_G$ and $A_H$ have the same multiset of eigenvalues)
- fractional isomorphic (i.e., there exists a doubly stochastic matrix $S$ such that $A_G\cdot S=S\cdot A_H$, or equivalentlty, $G$ and $H$ have a common equitable partition)
- $G$ and $H$ are not cospectral with regards to their Laplacians (signed or otherwise) or Seidel matrix.
The context for this question can be found in an earlier post Orthogonal similarity of adjacency matrices of graphs which are cospectral and have a common equitable partition.