These states, defined as eigenvectors of the lowering operator and forming an overcomplete family, were introduced in the early papers of John R. Klauder, e.g.
HITS is an algorithm for web search that builds on the eigenvector-based methods used in algorithms and served as the full scale model for PageRank by recognizing that web pages or sites should be considered important not only if they are linked to by many others (as in PageRank), but also if they link to many others.
Cayley's formula follows from Kirchhoff's theorem as a special case, since every vector with 1 in one place, −1 in another place, and 0 elsewhere is an eigenvector of the Laplacian matrix of the complete graph, with the corresponding eigenvalue being n.