On the Lovász δ-number of almost regular graphs with application to Erdős-Rényi graphs

Authors:
E. de Klerk;M. W. Newman;D. V. Pasechnik;R. Sotirov
Affiliations:
Department of Econometrics and Operations Research, Tilburg University, The Netherlands;School of Mathematical Sciences at Queen Mary, University of London, UK;School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore;Department of Econometrics and Operations Research, Tilburg University, The Netherlands
Venue:
European Journal of Combinatorics
Year:
2009

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Abstract

We consider k-regular graphs with loops, and study the Lovász -numbers and Schrijver '-numbers of the graphs that result when the loop edges are removed. We show that the -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]. As an application we compute the and ' 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, Math. Program. B 109 (2-3) (2007) 613-624]. The computed values are strictly better than the Godsil-Newman eigenvalue bounds.