Morse and Lusternik-Schnirelmann for graphs
Oliver Knill
TL;DR
This work develops a comprehensive discrete counterpart to Morse theory and Lusternik–Schnirelmann theory for finite graphs and delta sets. By formulating graph-based Poincaré–Hopf and Gauss–Bonnet theorems, defining Morse functions, critical points, and Betti numbers via the Hodge Laplacian, and introducing cup products and category, it proves discrete analogues of the classical inequalities cup+1 ≤ cat ≤ cri and b_k-... ≤ c_k-..., linking algebraic topology with combinatorial graph structures. The framework leverages Whitney complexes, delta sets, and various graph products to establish a robust toolkit for discrete topology, including dimension theory and open-set topology on graphs. Key contributions include a full suite of discrete Morse inequalities, a discrete LS category theory for graphs, a cohomology ring structure via cup products, and detailed discussion of morphisms, refinements, and manifold-like graph classes. Collectively, the results provide a computable, structurally rich bridge between topology, geometry, and analysis in finite graphs with potential impact on network science and computational topology.
Abstract
Both Morse theory and Lusternik-Schnirelmann theory link algebra, topology and analysis in a geometric setting. The two theories can be formulated in finite geometries like graph theory or within finite abstract simplicial complexes. We work here mostly in graph theory and review the Morse inequalities b(k)-b(k-1) + ... + b(0) less of equal than c(k)-c(k-1) + ... + c(0) for the Betti numbers b(k) and the minimal number c(k) of Morse critical points of index k and the Lusternik-Schnirelmann inequalities cup+1 less or equal than cat less or equal than cri, between the algebraic cup length cup, the topological category cat and the analytic number cri counting the minimal number of critical points of a function.