A characterization of the weighted version of McEliece-Rodemich-Rumsey-Schrijver number based on convex quadratic programming

Abstract

For any graph $G,$ Luz and Schrijver \cite{LuzSchrijver} introduced a characterization of the Lov\'{a}sz number $\vartheta(G)$ based on convex quadratic programming. A similar characterization is now established for the weighted version of the number $\vartheta^{\prime}(G),$ independently introduced by McEliece, Rodemich, and Rumsey \cite{McElieceetal} and Schrijver \cite{Schrijver1}. Also, a class of graphs for which the weighted version of $\vartheta^{\prime}(G)$ coincides with the weighted stability number is characterized.