The characterization of graph random-walk evolution algebras

Let $G$ be a finite graph. Let $bf{A}_{RW}(G)$ denote the evolution algebra associated with the random walk on $G$, and let $bf{A}(G)$ denote the evolution algebra associated with $G$.

Graph evolution-algebra isomorphism conjecture.

$$\mathcal{A}_{RW}(G)\cong \mathcal{A}(G)$$

if and only if $G$ is regular or biregular. Moreover, if

$$\mathcal{A}_{RW}(G)\ncong \mathcal{A}(G),$$

then the only homomorphism between them is the null map.

The claim extends the corresponding criterion beyond the regular and biregular cases. The paper reports that it has been verified for several singular graphs, but states that further work is needed to establish it for all finite singular graphs.