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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.

##### MSC:
 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) 90C35 Programming involving graphs or networks
SeDuMi
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##### References:
 [1] Bannai, E.; Hao, S.; Song, S.-Y., Character tables of the association schemes of finite orthogonal groups acting on the nonisotropic points, J. combin. theory ser. A, 54, 2, 164-200, (1990) · Zbl 0762.20005 [2] Brown, W.G., On graphs that do not contain a thomsen graph, Canad. math. bull., 9, 281-285, (1966) · Zbl 0178.27302 [3] Cvetković, D.M.; Doob, M.; Sachs, H., Spectra of graphs, (1980), Academic Press Inc. [Harcourt Brace Jovanovich Publishers] New York · Zbl 0458.05042 [4] Delsarte, P., An algebraic approach to the association schemes of coding theory, Philips res. rep. suppl., 10, vi+97, (1973) · Zbl 1075.05606 [5] Erdós, P.; Rényi, A.; Sós, V.T., On a problem of graph theory, Studia sci. math. hungar., 1, 215-235, (1966) · Zbl 0144.23302 [6] Füredi, Z., Graphs without quadrilaterals, J. combin. theory ser. B, 34, 2, 187-190, (1983) · Zbl 0505.05038 [7] Füredi, Z., On the number of edges of quadrilateral-free graphs, J. combin. theory ser. B, 68, 1, 1-6, (1996) · Zbl 0858.05063 [8] de Klerk, E.; Pasechnik, D.V.; Schrijver, A., Reduction of symmetric semidefinite programs using the regular *-representation, Math. program. B, 109, 2-3, 613-624, (2007) · Zbl 1200.90136 [9] Gatermann, K.; Parrilo, P.A., Symmetry groups, semidefinite programs, and sum of squares, J. pure appl. algebra, 192, 95-128, (2004) · Zbl 1108.13021 [10] Godsil, C.D.; Newman, M.W., Eigenvalue bounds for independent sets, J. combin. theory B, 98, 4, 721-734, (2008) · Zbl 1156.05041 [11] Hirschfeld, J.W.P., Projective geometry over finite fields, (1998), Clarendon Press Oxford · Zbl 0899.51002 [12] Lovász, L., On the Shannon capacity of a graph, IEEE trans. inform. theory, 25, 1-7, (1979) · Zbl 0395.94021 [13] Lovász, L.; Schrijver, A., Cones of matrices and set-functions and 0-1 optimization, SIAM J. optim., 1, 2, 166-190, (1991) · Zbl 0754.90039 [14] Mubayi, D.; Williford, J., On the independence number of the erdős-Rényi and projective norm graphs and a related hypergraph, J. graph theory, 56, 2, 113-127, (2007) · Zbl 1128.05040 [15] Parsons, T.D., Graphs from projective planes, Aequationes math., 14, 1-2, 167-189, (1976) · Zbl 0323.05116 [16] Schrijver, A., A comparison of the delsarte and lovász bounds, IEEE trans. inform. theory, 25, 425-429, (1979) · Zbl 0444.94009 [17] Sturm, J.F., Using sedumi 1.02, a MATLAB toolbox for optimization over symmetric cones, Optim. methods softw., 11-12, 625-653, (1999) · Zbl 0973.90526
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