I am wondering if the following pair of cospectral graphs was previously known.
As far as I know, it was previously unknown whether it is determined by ts spectrum. On Wolfram Mathowrld, in the table describing properties of the graph, it simply has a "?" in the field for "determined by spectrum". However, I have found a cospectral (with respect to adjacency matrices) mate for this graph, which is displayed below (graph6 string: 'M?????rrAiTOX_eO?' ):
I checked in Sage and you cannot go from one to the other through Godsil-McKay switching (EDIT: I made a mistake here, you can get them from GM switching as @KrystalGuo points out below). They are also cospectral with respect to Laplacians, signless Laplacians, and normalized Laplacians. Though maybe this follows from adjacency cospectrality since they are biregular bipartite graphs? For what it is worth, both graphs are induced subgraphs of the Hoffman graph, which is the unique cospectral mate of the 4-cube. The 4-cube also contains the rhombic dodecahedron as an induced subgraph but not the cospectral mate. They are also both edge-transitive.