Harmonic cycles for graphs
Younng-Jin Kim, Woong Kook · Linear and Multilinear Algebra · 2018
Given a finite connected graph G=(V(G),E(G)) and a basis ∂ for a hyperplane in the cycle space of G, define λ=∑Ydet(CY,∂)·CY summing over all connected spanning subgraphs Y of G such that |E(Y)|=|V(G)| with CY denoting the unique cycle in Y. We will show that λ is an element of the harmonic space ker(∂1t∂1+∂∂t) where ∂1 is the incidence matrix of G by establishing an inner product formula λ∘z=det(z,∂)k(G) for the cycles z and the tree number k(G) of G. Several examples and applications of these formulas will be given.