On the generalized cospectral mates of chain graphs of rank 4

Source abstract

A graph is called a chain graph if it is bipartite and the neighborhoods of the vertices in each color class are linearly ordered by inclusion. We study the generalized spectral determination of chain graphs of adjacency rank $4$. We first show that generalized cospectral chain graphs have the same number of isolated vertices, reducing the problem to the connected case. For connected rank-$4$ chain graphs, we derive four explicit equations in the cell parameters that are necessary and sufficient for generalized cospectrality. These equations reduce to a quadratic equation in a single parameter. As a consequence, every connected rank-$4$ chain graph has at most one generalized cospectral mate, and the possible mate can be reconstructed explicitly. We further obtain a simple sufficient condition for a rank-$4$ chain graph to be determined by its generalized spectrum (DGS) and, in particular, prove that every double star graph is DGS. Finally, we construct an explicit infinite family of non-double-star rank-$4$ chain graphs that are DGS but not determined by their adjacency spectra, with both the graphs and their complements being connected.