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On the Lovász \(\vartheta\)-number of almost regular graphs with application to Erdős-Rényi graphs. (English) Zbl 1200.05163
Summary: We consider \(k\)-regular graphs with loops, and study the Lovász \(\vartheta\)-numbers and Schrijver \(\vartheta'\)-numbers of the graphs that result when the loop edges are removed. We show that the \(\vartheta\)-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. Comb. Theory, Ser. B 98, No. 4, 721–734 (2008; Zbl 1156.05041)].
As an application we compute the \(\vartheta\) and \(\vartheta'\) numbers of certain instances of Erdős-Rényi graphs. This computation exploits the graph symmetry using the methodology introduced in [|it E. de Klerk, D.V. Pasechnik and A. Schrijver, “Reduction of symmetric semidefinite programs using the regular \(*\)-representation,” Math. Program. 109, No. 2-3 (B), 613–624 (2007; Zb; 05131080].
The computed values are strictly better than the Godsil-Newman eigenvalue bounds.

05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
90C35 Programming involving graphs or networks
Full Text: DOI
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