All Pairs of Pentagons in Leapfrog Fullerenes Are Nice
Abstract
:1. Introduction
2. Definitions and Preliminary Results
3. Main Results
4. Open Questions and Concluding Remarks
Funding
Conflicts of Interest
References
- Austin, S.J.; Fowler, P.W.; Hansen, P.; Monolopoulos, D.E.; Zheng, M. Fullerene isomers of C60. Kekulé counts versus stability. Chem. Phys. Lett. 1994, 228, 478–484. [Google Scholar] [CrossRef]
- Došlić, T. Cyclical edge-connectivity of fullerene graphs and (k,6)-cages. J. Math. Chem. 2003, 33, 103–112. [Google Scholar] [CrossRef]
- Kardoš, F.; Škrekovski, R. Cyclic edge-cuts in fullerene graphs. J. Math. Chem. 2008, 44, 121–132. [Google Scholar] [CrossRef]
- Kutnar, K.; Marušič, D. On cyclic edge-connectivity of fullerenes. Discrete Appl. Math. 2008, 156, 1661–1669. [Google Scholar] [CrossRef] [Green Version]
- Qi, Z.; Zhang, H. A note on the cyclical edge-connectivity of fullerene graphs. J. Math. Chem. 2008, 43, 134–140. [Google Scholar] [CrossRef]
- Došlić, T. Nice pairs of odd cycles in fullerene graphs. J. Math. Chem. 2020, 58, 2204–2222. [Google Scholar] [CrossRef]
- de Carvalho, M.H.; Lucchesi, C.L.; Murty, U.S.R. Optimal ear decomposition of matching covered graphs. J. Combin. Theory Ser. B 2002, 85, 59–93. [Google Scholar] [CrossRef] [Green Version]
- de Carvalho, M.H.; Lucchesi, C.L.; Murty, U.S.R. On a conjecture of Lovász concerning bricks I. The Characteristic of a Matching Covered Graph. J. Combin. Theory Ser. B 2002, 85, 94–136. [Google Scholar] [CrossRef] [Green Version]
- de Carvalho, M.H.; Lucchesi, C.L.; Murty, U.S.R. On a conjecture of Lovász concerning bricks II. Bricks of finite characteristic. J. Combin. Theory Ser. B 2002, 85, 137–180. [Google Scholar] [CrossRef] [Green Version]
- de Carvalho, M.H.; Lucchesi, C.L.; Murty, U.S.R. The perfect matching polytope and solid bricks. J. Combin. Theory Ser. B 2004, 92, 319–324. [Google Scholar] [CrossRef] [Green Version]
- de Carvalho, M.H.; Lucchesi, C.L.; Murty, U.S.R. Graphs with independent perfect matchings. J. Graph Theory 2005, 48, 19–50. [Google Scholar] [CrossRef]
- de Carvalho, M.H.; Lucchesi, C.L.; Murty, U.S.R. How to build a brick. Discrete Math. 2006, 306, 2383–2410. [Google Scholar] [CrossRef] [Green Version]
- Harary, F. Graph Theory; Addison-Wesley: Reading, MA, USA, 1969. [Google Scholar]
- Lovász, L.; Plummer, M.D. Matching Theory; North-Holland: Amsterdam, The Netherlands, 1986. [Google Scholar]
- Grünbaum, B.; Motzkin, T.S. The number of hexagons and the simplicity of geodesics on certain polyhedra. Can. J. Math. 1963, 15, 744–751. [Google Scholar] [CrossRef]
- Fowler, P.W.; Manolopoulos, D.E. An Atlas of Fullerenes; Clarendon Press: Oxford, UK, 1995. [Google Scholar]
- Schwerdtfeger, P.; Wirz, L.N.; Avery, J. The topology of fullerenes. WIRE: Comput. Mol. Sci. 2015, 5, 96–145. [Google Scholar] [CrossRef] [PubMed]
- Petersen, J. Die Theorie der regulären graphs. Acta Math. 1891, 15, 193–220. [Google Scholar] [CrossRef]
- Došlić, T. On some structural properties of fullerene graphs. J. Math. Chem. 2002, 31, 187–195. [Google Scholar] [CrossRef]
- Li, H.; Zhang, H. The isolated-pentagon rule and nice substructures in fullerenes. Ars Math. Contemp. 2018, 15, 487–497. [Google Scholar] [CrossRef] [Green Version]
- Ye, D.; Zhang, H. On k-resonant fullerene graphs. SIAM J. Discrete Math. 2009, 23, 1023–1044. [Google Scholar] [CrossRef] [Green Version]
- Diudea, M.V.; Stefu, M.; John, P.E.; Graovac, A. Generalized operations on maps. Croat. Chem. Acta 2006, 79, 355–362. [Google Scholar]
- Došlić, T. Leapfrog fullerenes have many perfect matchings. J. Math. Chem. 2008, 44, 1–4. [Google Scholar] [CrossRef]
- Došlić, T. Finding more perfect matchings in leapfrog fullerenes. J. Math. Chem. 2009, 45, 1130–1136. [Google Scholar] [CrossRef]
- King, R.B.; Diudea, M.V. The chirality of icosahedral fullerenes: A comparison of the tripling, (leapfrog), quadrupling (chamfering) and septupling (capra) transformations. J. Math. Chem. 2006, 39, 597–604. [Google Scholar] [CrossRef]
- Thurston, W.P. Shapes of polyhedra and triangulations of the sphere. Geom. Topol. Mono. 1998, 1, 511–549. [Google Scholar]
- Cioslowski, J. Note on the asymptotic isomer count of large fullerenes. J. Math. Chem. 2014, 52, 1–5. [Google Scholar] [CrossRef] [Green Version]
- Došlić, T.; Dehkordi, M.T.; Fath-Tabar, G.H. Packing stars in fullerenes. J. Math. Chem. 2020, 58, 2223–2244. [Google Scholar] [CrossRef]
- Nagy, P.; Ehlich, R.; Biró, L.P.; Gyulai, J. Y-branching of single walled carbon nanotubes. Appl. Phys. A 2000, 70, 481–483. [Google Scholar] [CrossRef]
- Astakhova, T.Y.; Vinogradov, G.A. Fullerene notation and isomerization operations. Fullerene Sci. Tech. 1997, 5, 1545–1562. [Google Scholar] [CrossRef]
- Sabirov, D.S.; Ori, O. Skeletal rearrangements of the C240 fullerene: Efficient topological descriptors for monitoring Stone-Wales transformations. Mathematics 2020, 8, 968. [Google Scholar] [CrossRef]
- Ori, O.; Cataldo, F. Moving pentagons on nanocones. Fuller. Nanotub. Carbon Nanostruct. 2020, 28, 732–736. [Google Scholar] [CrossRef]
- Diudea, M.V.; Nagy, C.L.; Ursu, O.; Balaban, T.S. C60 dimers revisited. Fuller. Nanotub. Carbon Nanostruct. 2003, 11, 245–255. [Google Scholar] [CrossRef]
- Gadomski, A. Three types of computational soft-matter problems revisited, an own-selection-based opinion. Front. Phys. 2014, 2, 36. [Google Scholar] [CrossRef] [Green Version]
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Došlić, T. All Pairs of Pentagons in Leapfrog Fullerenes Are Nice. Mathematics 2020, 8, 2135. https://doi.org/10.3390/math8122135
Došlić T. All Pairs of Pentagons in Leapfrog Fullerenes Are Nice. Mathematics. 2020; 8(12):2135. https://doi.org/10.3390/math8122135
Chicago/Turabian StyleDošlić, Tomislav. 2020. "All Pairs of Pentagons in Leapfrog Fullerenes Are Nice" Mathematics 8, no. 12: 2135. https://doi.org/10.3390/math8122135
APA StyleDošlić, T. (2020). All Pairs of Pentagons in Leapfrog Fullerenes Are Nice. Mathematics, 8(12), 2135. https://doi.org/10.3390/math8122135