A combinatorial interpretation of harmonic cycles

TL;DR

This paper provides a combinatorial interpretation of harmonic cycles in discrete harmonic forms using concepts like cycletree, unicyclization, and winding number, linking combinatorics with harmonic analysis on CW-complexes.

Contribution

It introduces new combinatorial concepts and proves properties that relate harmonic cycles to combinatorial structures, offering a novel interpretation of harmonic cycles.

Findings

Defined the concepts of cycletree, unicyclization, winding number map, and standard harmonic cycle.

Proved properties relating winding number and inner product with the harmonic cycle.

Established that the standard harmonic cycle is indeed a harmonic cycle, providing a combinatorial perspective.

Abstract

In this paper, we will investigate a harmonic cycle (discrete harmonic form). With a CW-complex, we can construct the combinatorial Laplacian operator. The kernel of the operator is the harmonic space, the set of harmonic cycles, and is isomorphic to its homology due to combinatorial Hodge theory. We will introduce four concepts; cycletree, unicyclization, winding number map, and standard harmonic cycle. A cycletree is the disjoint union of a spanning tree and an edge on a given graph. A unicyclization consists of a graph and information substituting for faces. Then, we will define the winding number map and the standard harmonic cycle, and prove related properties. Finally, we will show a relation between the winding number and the inner product with the standard harmonic cycle. Then, we will see the standard harmonic cycle is actually a harmonic cycle. In other words, we give a combinatorial interpretation on a harmonic cycle.