Calculating the number of Hamilton cycles in layered polyhedral graphs

Research output: Contribution to journal › Journal article › Research › peer-review

Standard

Calculating the number of Hamilton cycles in layered polyhedral graphs. / Wirz, Lukas N.; Schwerdtfeger, Peter; Avery, James.

In: Computational and Mathematical Methods in Medicine, Vol. 3, No. 4, e1142, 07.2021.

Research output: Contribution to journal › Journal article › Research › peer-review

Harvard

Wirz, LN, Schwerdtfeger, P & Avery, J 2021, 'Calculating the number of Hamilton cycles in layered polyhedral graphs', Computational and Mathematical Methods in Medicine, vol. 3, no. 4, e1142. https://doi.org/10.1002/cmm4.1142

APA

Wirz, L. N., Schwerdtfeger, P., & Avery, J. (2021). Calculating the number of Hamilton cycles in layered polyhedral graphs. Computational and Mathematical Methods in Medicine, 3(4), [e1142]. https://doi.org/10.1002/cmm4.1142

Vancouver

Wirz LN, Schwerdtfeger P, Avery J. Calculating the number of Hamilton cycles in layered polyhedral graphs. Computational and Mathematical Methods in Medicine. 2021 Jul;3(4). e1142. https://doi.org/10.1002/cmm4.1142

Author

Wirz, Lukas N. ; Schwerdtfeger, Peter ; Avery, James. / Calculating the number of Hamilton cycles in layered polyhedral graphs. In: Computational and Mathematical Methods in Medicine. 2021 ; Vol. 3, No. 4.

Bibtex

@article{6561567b808846aa94ba107675a9b90a,
title = "Calculating the number of Hamilton cycles in layered polyhedral graphs",
abstract = "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.",
keywords = "cubic graph, finite-state machines, fullerene graph, Hamilton cycle, layered graph, COMPLEXITY",
author = "Wirz, {Lukas N.} and Peter Schwerdtfeger and James Avery",
year = "2021",
month = jul,
doi = "10.1002/cmm4.1142",
language = "English",
volume = "3",
journal = "Computational and Mathematical Methods in Medicine",
issn = "1748-670X",
publisher = "Hindawi Publishing Corporation",
number = "4",

}

RIS

TY - JOUR

T1 - Calculating the number of Hamilton cycles in layered polyhedral graphs

AU - Wirz, Lukas N.

AU - Schwerdtfeger, Peter

AU - Avery, James

PY - 2021/7

Y1 - 2021/7

N2 - 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.

AB - 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.

KW - cubic graph

KW - finite-state machines

KW - fullerene graph

KW - Hamilton cycle

KW - layered graph

KW - COMPLEXITY

U2 - 10.1002/cmm4.1142

DO - 10.1002/cmm4.1142

M3 - Journal article

VL - 3

JO - Computational and Mathematical Methods in Medicine

JF - Computational and Mathematical Methods in Medicine

SN - 1748-670X

IS - 4

M1 - e1142

ER -

ID: 276266696