Theorematics for Pauling-Wheland Resonance Theory

Ordering Theorem — Let c be a cutset in a bipartite graph G with a Kekule structure K. The difference A(G,c) between the orders of (K)ncA+ (K)ncg+ is independent of the particular Kekule structure k. 

That is, for each cutset e as defined above, the quantity A(G,c) is a type of (c-dependent) characteristic for each Kekule structure of G such that the value is independent of the Kekule structure. Proofs are given in Refs. [85,175]. An example may be of interest, such as for ovalene, for which we might consider a cut 

Theorem — Let G be a bipartite regular polymer graph G with cyclic boundary conditions and with a Kekule structure K, which has edge set (K). Then the difference 8(G,e,K) between the orders of e n (K) and of es+n (K) for translationally equivalent boundary sets e varies with a period of no more than 2. The period can only be 2 if the primitive translation interchanges starred unstarred sites. 

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