On the function "sandwiched" between $\alpha(G)$ and $\overline{\chi}(G)$

Abstract

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 $I \le B \le I + A(G),$ where $I$ is the identity matrix and $A(G)$ is the adjacency matrix of $G$. Let $\cal B(G)$ 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 $$ \alpha(G) \le \min \{ {\rm rank }(B)\ \vert B \in \cal B(G)\} \le \overline{\chi}(G) $$ and that $\alpha(G)= \min \{\ {\rm rank }(B)\ \vert B \in \cal B(G)\}$ implies $ \alpha(G)=\overline{\chi}(G).$