TITLE: Co-spectral fractional isomorphic graphs with different Laplacian spectrum
QUESTION [7 upvotes]: 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)

but

$G$ 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.

REPLY [2 votes]: EDIT: According to your link, if two graphs are cospectral with a common equitable partition, then they have cospectral complements. But this implies that they have cospectral Seidel matrices (see Theorem 3 in the first reference from your link).
So the best you can do is to not be cospectral with respect to the Laplacian nor signless Laplacian. The two graphs below satisfy this. The equitable partitions are according to the degrees.


Verification If $G$ is the top graph and $H$ the bottom graph:

Eigenvalues$(G)$=Eigenvalues$(H)\approx\{-2,-2,-1,-1,-0.73205081,1,1,2,2.73205081\}$
Their common coarsest equitable partitions are $\{\{2,3,8\},\{0,1,4,5,6,7\}\}$ for $G$ and $\{\{1,3,8\},\{0,2,4,5,6,7\}\}$ for $H$.
Eigenvalues$(L_G)\approx\{4.73\times 10^{-16},1, 1.26, 1.26,3, 4, 4, 4.73,4.73\}$ and
Eigenvalues$(L_H)\approx \{-2.08\times 10^{-16},0.585,1.26,2,3,3.41,4,4.73, 5\}$

So $G$ and $H$ indeed satisfy the required conditions.