On the Lovasz O-number of Almost Regular Graphs With Application to Erdos-Renyi Graphs

We consider k-regular graphs with loops, and study the Lovasz O-numbers and Schrijver O-numbers of the graphs that result when the loop edges are removed. We show that the O-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. Journal of Combinatorial Theory B, to appear]. As an application we compute the O and O numbers of certain instances of Erdos Renyi graphs. This computation exploits the graph symmetry using the methodology introduced in [E. de Klerk, D.V. Pasechnik and A. Schrijver. Reduction of symmetric semidefinite programs using the regular *-representation. Mathematical Programming B, to appear]. The computed values are strictly better than the Godsil-Newman eigenvalue bounds.