On the Lovász theta-number of almost regular graphs with application to Erdos-Rényi graphs

We consider k-regular graphs with loops, and study the Lovász ϑ-numbers and Schrijver ϑ′-numbers of the graphs that result when the loop edges are removed. We show that the ϑ-number dominates a recent eigenvalue upper bound on the stability number due to Godsil and Newman [C.D. Godsil and M.W. Newman. Eigenvalue bounds for independent sets, J. Combin. Theory B 98 (4) (2008) 721–734].