Observations on the Lovász

This paper provides new observations on the Lovász θ-function of graphs. These include a simple closed-form expression of that function for all strongly regular graphs, together with upper and lower bounds on that function for all regular graphs. These bounds are expressed in terms of the second-largest and smallest eigenvalues of the adjacency matrix of the regular graph, together with sufficient conditions for equalities (the upper bound is due to Lovász, followed by a new sufficient condition for its tightness). These results are shown to be useful in many ways, leading to the determination of the exact value of the Shannon capacity of various graphs, eigenvalue inequalities, and bounds on the clique and chromatic numbers of graphs. Since the Lovász θ-function factorizes for the strong product of graphs, the results are also particularly useful for parameters of strong products or strong powers of graphs. Bounds on the smallest and second-largest eigenvalues of strong products of regular graphs are consequently derived, expressed as functions of the Lovász θ-function (or the smallest eigenvalue) of each factor. The resulting lower bound on the second-largest eigenvalue of a