Morse and Lusternik-Schnirelmann for graphs

TL;DR

This paper explores the application of Morse and Lusternik-Schnirelmann theories within graph theory, establishing inequalities that connect algebraic, topological, and analytical properties of graphs.

Contribution

It formulates and reviews Morse inequalities and Lusternik-Schnirelmann inequalities specifically for graphs, linking critical points, Betti numbers, and topological invariants.

Findings

Morse inequalities relate Betti numbers and critical points in graphs.

Lusternik-Schnirelmann inequalities connect cup length, category, and critical points.

The paper provides a framework for applying classical topological theories to finite graph structures.

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.