Resonance Graphs on Perfect Matchings of Graphs on Surfaces

Let G be a graph embedded in surface and let the set of all faces G. For given subset $$\mathcal F$$ even (faces bounded by an cycle), resonance with respect to $${\mathcal {F}}$$ , denoted $$R(G;\mathcal F)$$ is such that its vertex perfect matchings two vertices $$M_1$$ $$M_2$$ are adjacent if only symmetric difference $$M_1\oplus M_2$$ cycle bounding some face . It has been shown plane elementary bipartite graph, inner isomorphic covering distributive lattice. However, this result does not hold general for graphs The structure properties remain unknown. In paper, we show every connected component on even-face can always into hypercube as induced subgraph. Further, Clar polynomial equal cube $$R(G;{\mathcal {F}})$$