Chain graphs are not spectrally determined

Abstract

Two non isomorphic graphs $G$ and $H$ are said to be cospectral, if they share the same adjacency eigenvalues.
In this paper, we use a recursive procedure for computing the characteristic polynomial of chain graphs in order to obtain infinitely many pairs of connected cospectral chain graphs.
This result disproves a conjecture posed in [Czechoslovak Math. J. 70 (4) (2020), 1125--1138]. In addition we construct infinite families of bipartite graphs sharing the same spectrum.