On generalized KKT points of the Motzkin-Straus program

In 1965, Motzkin and Straus established a profound connection between the clique number of a graph and the global maxima of a quadratic program defined on the standard simplex. Since then, a line of active and intensive research has been yielding heuristics and bounds concerning the maximum clique problem, thanks to the discoveries pertaining local/global solutions of the Motzkin-Straus program. However, the Karush-Kuhn-Tucker (KKT) points thereof have received little to no attention in the literature. In this work, a parameterized version of the Motzkin-Straus program is discussed, and some results about its KKT points are obtained. What emerges is a connection between a generalized notion of KKT point and some regular structures contained in the graph.