We determine exactly the expected number of hamilton cycles in the random graph obtained by starting with n isolated vertices and adding edges at random until each vertex degree is at least two. This complements recent work of Cooper and Frieze. There are similar results concerning expected numbers for example of perfect matchings, spanning trees, hamilton paths and directed hamilton cycles.

We study a Fermi Hamilton operator K̂ which does not commute with the number operator N̂ . The eigenvalue problem and the Schrödinger equation is solved. Entanglement is also discussed. Furthermore the Lie algebra generated by the two terms of the Hamilton operator is derived and the Lie algebra generated by the Hamilton operator and the number operator is also classified.