A combinatorial interpretation of harmonic cycles

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.