Upper Bounds for the Connective Constant of Self-Avoiding Walks

We present a method for obtaining upper bounds for the connective constant of self-avoiding walks. The method works for a large class of lattices, including all that have been studied in connection with self-avoiding walks. The bound is obtained as the largest eigenvalue of a certain matrix. Numerical application of the method has given improved bounds for all lattices studied, e.g. μ < 2.696 for the square lattice, μ < 4.278 for the triangular lattice and μ < 4.756 for the simple cubic lattice.