Calculating the number of Hamilton cycles in layered polyhedral graphs

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  • cmm4.1142

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We describe a method for computing the number of Hamilton cycles in cubic polyhedral graphs. The Hamilton cycle counts are expressed in terms of a finite-state machine, and can be written as a matrix expression. In the special case of polyhedral graphs with repeating layers, the state machines become cyclic, greatly simplifying the expression for the exact Hamilton cycle counts, and let us calculate the exact Hamilton cycle counts for infinite series of graphs that are generated by repeating the layers. For some series, these reduce to closed form expressions, valid for the entire infinite series. When this is not possible, evaluating the number of Hamiltonian cycles admitted by the series' k-layer member is found by computing a (k - 1)th matrix power, requiring O(log(2)(k)) matrix-matrix multiplications. We demonstrate our technique for the two infinite series of fullerene nanotubes with the smallest caps. In addition to exact closed form and matrix expressions, we provide approximate exponential formulas for the number of Hamilton cycles.

Original languageEnglish
Article numbere1142
JournalComputational and Mathematical Methods in Medicine
Issue number4
Number of pages15
Publication statusPublished - Jul 2021

    Research areas

  • cubic graph, finite-state machines, fullerene graph, Hamilton cycle, layered graph, COMPLEXITY

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