I am looking for two undirected graphs $G$ and $H$ of the same order (i.e., they have the same number of vertices) with adjacency matrices $A\_G$ and $A\_H$, respectively, such that


* $G$ and $H$ are *fractional isomorphic*, that is, there exists a doubly stochastic matrix $S$ such that $A\_G\cdot S=S\cdot A\_H$ holds (or equivalently, $G$ and $H$ have a common equitable partition),
* **But** $A\_G$ has eigenvalue $0$ whereas $A\_H$ does not.


The context of this question is an investigation in which spectral properties are invariant under fractional isomorphisms.



Two fractionally isomorphic graphs but only one having eigenvalue zero
As mentioned in the comments, it is sufficient to find two regular graphs $G$ and $H$ of the same order and same degree, $G$ having zero as eigenvalue whereas $H$ does not.


* Let $G$ consist of the disjoint union of a $3$-cycle and a $4$-cycle, with eigenvalues $$\{-2.0000,-1.0000,-1.0000,-0.0000,0.0000,2.0000,2.0000\}$$
* Let $H$ consists of a $7$-cycle with eigenvalues
$$
\{-1.8019,-1.8019,-0.4450,-0.4450,1.2470,1.2470,2.0000\}.
$$


Both graphs have order $7$ and are regular of degree $2$ and exhibit the desired eigenvalue property.


Perhaps there are smaller examples out there, but these suffice as answer to my question.


