Möbius inversion on a poset of a graph and its acyclic subgraphs
In the frame of the cluster expansion concept, a molecular property \(X(G)\) is formally expanded into a sum of contributions from all molecular fragments \(G'\) of \(G\), determining cluster expansion in chemistry. For expressing the resonance energy in molecules the Möbius function for the partially ordered set of a connected cyclic graph \(G\) and its acyclic subgraphs \(G'\) is useful. Here the authors determine this Möbius function using terms of graph theory as follows: \[ \mu[G',G]=-\prod_{v_i\in G-G'}(1-d_i), \] where \(d_i\) denotes the number of edges by which \(v_i\) is linked to vertices in \(G'\).
scientific article; zbMATH DE number 897414