I know that some modal formulae do classify digraphs. For example, $\Box \phi \rightarrow \Diamond \phi$ classifies all serial digraphs, i.e. digraphs such that for all vertices $v_i,$ there exists some $v_j$ such that there is an arrow from $v_i$ to $v_j$.
Is it the case that every modal formulae defines such a class, perhaps trivially small or large, of digraphs? And given an arbitrary digraph, is there a modal formula that classifies it?