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Generation and properties of nut graphs

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HPC-UGent: the central High Performance Computing infrastructure of Ghent University
Abstract
A nut graph is a graph on at least 2 vertices whose adjacency matrix has nullity 1 and for which non-trivial kernel vectors do not contain a zero. Chemical graphs arc connected, with maximum degree at most three. We present a new algorithm for the exhaustive generation of non-isomorphic nut graphs. Using this algorithm, we determined all nut graphs up to 13 vertices and all chemical nut graphs up to 22 vertices. Furthermore, we determined all nut graphs among the cubic polyhedra up to :14 vertices and all nut fullerenes up to 250 vertices. Nut graphs are of interest in chemistry of conjugated systems, in models of electronic structure, radical reactivity and molecular conduction. The relevant mathematical properties of chemical nut graphs are the position of the zero eigenvaluc in the graph spectrum, and the dispersion in magnitudes of kernel eigenvector entries (r: the ratio of maximum to minimum magnitude of entries). Statistics are gathered on these properties for all the nut graphs generated here. We also show that all chemical nut graphs have r >= 2 and that there is at least one chemical nut graph with r = 2 for every order n >= 9 (with the exception of n = 10).
Keywords
CHEMICAL GRAPHS, SINGULAR GRAPHS, FULLERENES, EIGENVALUES, ORBITALS, CORES

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Citation

Please use this url to cite or link to this publication:

Chicago
Coolsaet, Kris, Patrick W Fowler, and Jan Goedgebeur. 2018. “Generation and Properties of Nut Graphs.” Match-communications in Mathematical and in Computer Chemistry 80 (2): 423–444.
APA
Coolsaet, K., Fowler, P. W., & Goedgebeur, J. (2018). Generation and properties of nut graphs. MATCH-COMMUNICATIONS IN MATHEMATICAL AND IN COMPUTER CHEMISTRY, 80(2), 423–444.
Vancouver
1.
Coolsaet K, Fowler PW, Goedgebeur J. Generation and properties of nut graphs. MATCH-COMMUNICATIONS IN MATHEMATICAL AND IN COMPUTER CHEMISTRY. 2018;80(2):423–44.
MLA
Coolsaet, Kris, Patrick W Fowler, and Jan Goedgebeur. “Generation and Properties of Nut Graphs.” MATCH-COMMUNICATIONS IN MATHEMATICAL AND IN COMPUTER CHEMISTRY 80.2 (2018): 423–444. Print.
@article{8591414,
  abstract     = {A nut graph is a graph on at least 2 vertices whose adjacency matrix has nullity 1 and for which non-trivial kernel vectors do not contain a zero. Chemical graphs arc connected, with maximum degree at most three. We present a new algorithm for the exhaustive generation of non-isomorphic nut graphs. Using this algorithm, we determined all nut graphs up to 13 vertices and all chemical nut graphs up to 22 vertices. Furthermore, we determined all nut graphs among the cubic polyhedra up to :14 vertices and all nut fullerenes up to 250 vertices. 
Nut graphs are of interest in chemistry of conjugated systems, in models of electronic structure, radical reactivity and molecular conduction. The relevant mathematical properties of chemical nut graphs are the position of the zero eigenvaluc in the graph spectrum, and the dispersion in magnitudes of kernel eigenvector entries (r: the ratio of maximum to minimum magnitude of entries). Statistics are gathered on these properties for all the nut graphs generated here. We also show that all chemical nut graphs have r {\textrangle}= 2 and that there is at least one chemical nut graph with r = 2 for every order n {\textrangle}= 9 (with the exception of n = 10).},
  author       = {Coolsaet, Kris and Fowler, Patrick W and Goedgebeur, Jan},
  issn         = {0340-6253},
  journal      = {MATCH-COMMUNICATIONS IN MATHEMATICAL AND IN COMPUTER CHEMISTRY},
  language     = {eng},
  number       = {2},
  pages        = {423--444},
  title        = {Generation and properties of nut graphs},
  volume       = {80},
  year         = {2018},
}

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