On the function ``sandwiched'' between \(\alpha(G)\) and \(\bar\chi (G)\) (Q1378512)

Summary: A new function of a graph \(G\) is presented. Say that a matrix \(B\) that is indexed by vertices of \(G\) is feasible for \(G\) if it is real, symmetric and \(I \leq B \leq I + A(G),\) where \(I\) is the identity matrix and \(A(G)\) is the adjacency matrix of \(G\). Let \(\mathcal B(G)\) be the set of all feasible matrices for \(G\), and let \(\overline{\chi}(G)\) be the smallest number of cliques that cover the vertices of \(G\). We show that \(\alpha(G) \leq \min \{\text{rank}(B)\mid B \in \mathcal B(G)\} \leq \overline{\chi}(G)\) and that \(\alpha(G)= \min \{\text{rank }(B)\mid B \in \mathcal B(G)\}\) implies \( \alpha(G)=\overline{\chi}(G)\).