We present a graph-theoretical interpretation of the recently developed interface theory of single zigzag chains N(n) in form of connectivity graphs. A remarkable property of the connectivity graphs is the possibility of constructing the full set of Clar covers of N(n) as the complete set of walks on these graphs. Connectivity graphs can be constructed in a direct form, in which the number of vertices is growing linearly with the length n of N(n), and in a reduced form, in which the number of vertices is independent of the actual length of the chain. The presented results can be immediately used for the determination of the Zhang-Zhang (ZZ) polynomials of N(n) in an easy and natural manner. Two techniques for computing the ZZ polynomial are proposed, one based on direct recursive computations and the other based on a general solution to a set of recurrence relations. Generalization of the interface theory to arbitrary benzenoid structures, the construction of associated connectivity graphs, and techniques for the computation of the associated ZZ polynomials will be presented in the near future.
In Part 1 of the current series of papers, we demonstrate the equivalence between the Zhang-Zhang polynomial ZZ(S, x) of a Kekuléan regular m-tier strip S of length n and the extended strict order polynomial E • S (n, x + 1) of a certain partially ordered set (poset) S associated with S. The discovered equivalence is a consequence of the one-to-one correspondence between the set {K} of Kekulé structures of S and the set {µ : S ⊃ A → [ n ]} of strictly order-preserving maps from the induced subposets of S to the interval [ n ]. As a result, the problems of determining the Zhang-Zhang polynomial of S and of generating the complete set of Clar covers of S reduce to the problem of constructing the set L(S) of linear extensions of the corresponding poset S and studying their basic properties. In particular, the Zhang-Zhang polynomial of S can be written in a compact form as ZZ(S, x) = |S| k=0 w∈L(S)
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