A lower bound on the solution to the traveling salesman problem is provided, which is expressed in terms of eigenvalues related to the distance matrix for the problem. This bound has many interesting properties such as transforming appropriately under affine distance transformations and is notably tight for various families of traveling salesman problems with arbitrarily many cities. It is also computed for some real world traveling salesman problems. The eigenvalues in the bound are further related to the Schoenberg criterion from Euclidean geometry. Graph theoretic applications to the Hamiltonian cycle and path problems are given by the fact that the new bound entails necessary graph eigenvalue conditions for a graph to be Hamiltonian or traceable. Various non-trivial families of Cayley graphs saturate these Hamiltonicity conditions, thus in a sense providing almost counterexamples to the famous conjecture that all Cayley graphs are Hamiltonian.