Algebraic structure count of some cyclic hexagonal-square chains

Olga Bodroza-Panti]'c

Abstract: Algebraic structure count ($ASC$-value) of a bipartite graph $G$ is defined by $ASC\{G\} =\sqrt{|\det A|}$, where $A$ is the adjacency matrix of $G$. In the case of bipartite, plane graphs in which every face-boundary (cell) is a circuit of length $4s+2$ ($s=1,2,\ldots$), this number is equal to the number of the perfect matchings ($K$-value) of $G$. However, if some of the circuits are of length $4s$ ($s=1,2,\ldots$), then the problem of evaluation of $ASC$-value becomes more complicated. In this paper the algebraic structure count of the class of cyclic hexagonal-square chains is determined. An explicit combinatorial formula for $ASC$ is deduced in the special case when all hexagonal fragments are isomorphic.

Classification (MSC2000): 05C70; 05B50

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