Pebble Games and Cospectral Graphs

Two graphs are cospectral if their respective adjacency matrices have the same multiset of eigenvalues. Cospectrality is an equivalence relation on graphs with many interesting facets. Isomorphic graphs are necessarily cospectral but the converse is not always true, thus cospectrality is a coarser relation than graph isomorphism. Here we study cospectrality from the point of view of pebble games, which are certain type of combinatorial games used in Finite Model Theory to capture the notion of elementary equivalence induced by various logic.