there are actually choices for an outward step from the starting vertex, which happens at the beginning and any number of times during the walk. The exact number of walks is trickier to compute, and is given by the formula where is the Gauss hypergeometric function. We may use this fact to bound the second largest eigenvalue of a -regular graph. Let be a -regular graph with vertices, and let be its adjacency matrix. Then is the number of closed walks of length . The number of closed walks on is at least times the number of closed walks on the Bethe lattice with degree starting at a particular vertex, as we can map the walks on the Bethe lattice to the walks on that start at a given vertex and only go back on paths that were already tread. There are often more walks on , as we can make use of cycles to create additional walks. The largest eigenvalue of is , and letting be the second largest absolute value of an eigenvalue, we have This gives . Noting that as grows, we can let grow much faster than to see that there are only finitely many -regular graphs for which the second largest absolute value of an eigenvalue is at most , for any This is a rather interesting result in the study of (n,d,λ)-graphs. Relation to Cayley graphs and Cayley trees A Bethe graph of even coordination number 2n is isomorphic to the unoriented Cayley graph of a free group of rank n with respect to a free generating set. Lattices in Lie groups Bethe lattices also occur as the discrete subgroups of certain hyperbolic Lie groups, such as the Fuchsian groups. As such, they are also lattices in the sense of a lattice in a Lie group. See also Crystal References Lattice models Trees (graph theory)