A Convex Quadratic Characterization of the Lovász Theta Number

In previous works an upper bound on the stability number $\alpha(G)$ of a graph G based on convex quadratic programming was introduced and several of its properties were established. The aim for this investigation is to relate theoretically this bound (usually represented by $\upsilon(G)$) with the...

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Bibliographic Details
Published inSIAM journal on discrete mathematics Vol. 19; no. 2; pp. 382 - 387
Main Authors Luz, Carlos J., Schrijver, Alexander
Format Journal Article
LanguageEnglish
Published Philadelphia Society for Industrial and Applied Mathematics 01.01.2005
Subjects
Eigenvalues
Euclidean space
Graphs
Quadratic programming
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Summary:In previous works an upper bound on the stability number $\alpha(G)$ of a graph G based on convex quadratic programming was introduced and several of its properties were established. The aim for this investigation is to relate theoretically this bound (usually represented by $\upsilon(G)$) with the well-known Lovasz $\vartheta(G)$ number. First, a new set of convex quadratic bounds on $\alpha(G)$ that generalize and improve the bound $\upsilon(G)$ is proposed. Then it is proved that $\vartheta(G)$ is never worse than any bound belonging to this set of new bounds. The main result of this note states that one of these new bounds equals $\vartheta(G)$, a fact that leads to a new characterization of the Lovasz theta number.
ISSN:0895-4801
1095-7146
DOI:10.1137/S0895480104429181