A Zykov algebra approach to clique propagation: Classifying clique complexes in graphs

This study applies the method of Clique Propagation ‘initially used in the context of medical imaging’ to characterize two types of clique complexes: those that exhibit ‘total clique propagation’, forming a single cohesive clique, and those that demonstrate ‘normal clique propagation’, where a coherent clique is not formed. We utilize Zykov Algebra, a graph-theoretic framework characterized by its overlay [Formula: see text], link (∗), and induction [Formula: see text] operations, and segmentation ‘sub-graphs of a graph’ to analyze these propagations through a system of clique equations. The aim is to algebraically determine whether a set of cliques conforms to one of the two propagation types. This study includes solving, graph construction, and computational applications to illustrate the practical use of this methodology. The algebraic formulations and graph equations presented reflect not only a novel approach to representing graphs but also hint at a deeper mathematical framework with broad applicability. We believe that this perspective opens new doors for theoretical and practical application in mathematics, computer science, and beyond.