Anđelić–da Fonseca–Simić–Du conjecture on cospectral connected chain graphs

Let a chain graph be a connected bipartite graph whose vertices in each part can be ordered by inclusion of their neighborhoods. Two graphs are cospectral with respect to the adjacency spectrum when their adjacency matrices have the same multiset of eigenvalues.

Anđelić–da Fonseca–Simić–Du conjecture. There do not exist non-isomorphic cospectral connected chain graphs with respect to the adjacency spectrum.

The conjecture concerns whether the adjacency spectrum determines a connected chain graph up to isomorphism. The paper states that it is disproved by examples with parameter $h=2$, while it holds for the class with $h=1$; thus the conjecture is refuted.