$ l_1 $-embeddability of pentagon-based open-ended nanotubes and its applications

Guangfu Wang; Chunxiao Xu; Wenchao Cong; Guangfu Wang; Chunxiao Xu; Wenchao Cong

doi:10.3934/math.2026532

AIMS Mathematics

2026, Volume 11, Issue 5: 12934-12960. doi: 10.3934/math.2026532

Previous Article Next Article

Research article Special Issues

$ l_1 $-embeddability of pentagon-based open-ended nanotubes and its applications

School of Mathematics and Information Sciences, Yantai University, Yantai 264005, China

Received: 20 January 2026 Revised: 15 April 2026 Accepted: 28 April 2026 Published: 11 May 2026
MSC : 05C09, 05C12, 05C92

A graph is $ l_1 $-embeddable if it admits a binary addressing such that the Hamming distance between the binary addresses is, up to scale, the distance in the graph between the corresponding vertices. The zigzag pentagon-based nanotubes and armchair pentagon-based nanotubes are two new materials with the tube shapes. In order to easily calculate their chemical indices based on distances, we showed that only five types $ \mathcal{I}(p, 1) $, $ \mathcal{I}(1, q, \theta) $ with $ q\geq 5 $, $ I(1, 3, \theta) $, $ I(2, 3, \zeta) $ and $ I(2, 5, \zeta) $ zigzag pentagon-based nanotubes were $ l_1 $-embeddable. Finally, we gave the precise value of the Wiener index and the hyper-Wiener index of these five zigzag pentagon-based nanotubes.
- $ l_{1} $-graph,
- zigzag pentagon-based nanotubes,
- armchair pentagon-based nanotubes,
- the Wiener index
Citation: Guangfu Wang, Chunxiao Xu, Wenchao Cong. $ l_1 $-embeddability of pentagon-based open-ended nanotubes and its applications[J]. AIMS Mathematics, 2026, 11(5): 12934-12960. doi: 10.3934/math.2026532

Related Papers:

Abstract

A graph is $ l_1 $-embeddable if it admits a binary addressing such that the Hamming distance between the binary addresses is, up to scale, the distance in the graph between the corresponding vertices. The zigzag pentagon-based nanotubes and armchair pentagon-based nanotubes are two new materials with the tube shapes. In order to easily calculate their chemical indices based on distances, we showed that only five types $ \mathcal{I}(p, 1) $, $ \mathcal{I}(1, q, \theta) $ with $ q\geq 5 $, $ I(1, 3, \theta) $, $ I(2, 3, \zeta) $ and $ I(2, 5, \zeta) $ zigzag pentagon-based nanotubes were $ l_1 $-embeddable. Finally, we gave the precise value of the Wiener index and the hyper-Wiener index of these five zigzag pentagon-based nanotubes.

References

[1]	S. Zhang, J. Zhou, Q. Wang, X. Chen, Y. Kawazoe, P. Jena, Penta-graphene: a new carbon allotrope, Proc. Natl. Acad. Sci. USA, 112 (2015), 2372–2377. https://doi.org/10.1073/pnas.1416591112 doi: 10.1073/pnas.1416591112
[2]	C. Zhong, Y. Chen, Z. Yu, Y. Xie, H. Wang, S. A. Yang, et al. Three-dimensional pentagon carbon with a genesis of emergent fermions, Nat. Commun., 8 (2017), 15641. https://doi.org/10.1038/ncomms15641 doi: 10.1038/ncomms15641
[3]	D. Liao, J. Zhang, S. Wang, Z. Zhang, A. Cortijo, M. A. H. Vozmediano, et al., Visualizing the topological pentagon states of a giant $C_540$ metamaterial, Nat. Commun., 15 (2024), 9644. https://doi.org/10.1038/s41467-024-53819-9 doi: 10.1038/s41467-024-53819-9
[4]	Y. Liu, L. Yuan, W. Chi, W. Han, J. Zhang, H. Pang, et al., Cairo pentagon tessellated covalent organic frameworks with mcm topology for near-infrared phototherapy, Nat. Commun., 15 (2024), 7150. https://doi.org/10.1038/s41467-024-50761-8 doi: 10.1038/s41467-024-50761-8
[5]	A. Gupta, I. Newman, Y. Rabinovich, Alistair Sinclair, cuts, trees and $l_1$-embeddings of graphs, Combinatorica, 24 (2004), 233–269. https://doi.org/10.1007/s00493-004-0015-x doi: 10.1007/s00493-004-0015-x
[6]	Y. Aumann, Y. Rabani, An $O(log k)$ approximate min-cut max-flow theorem and approximation algorithm, SIAM J. Comput., 27 (1998), 291–301. https://doi.org/10.1137/S009753979428598 doi: 10.1137/S009753979428598
[7]	A. Blum, G. Konjevod, R. Ravi, S. Vempala, Semi-definite relaxations for minimum bandwidth and other vertex-ordering problems, Theor. Comput. Sci., 235 (2000), 25–42. https://doi.org/10.1016/S0304-3975(99)00181-4 doi: 10.1016/S0304-3975(99)00181-4
[8]	P. Assouad, M. Deza, Espaces métriques plongeables dans un hypercube: aspects combinatoires, Ann. Discrete Math., 8 (1980), 197–210. https://doi.org/10.1016/S0167-5060(08)70874-4 doi: 10.1016/S0167-5060(08)70874-4
[9]	S. Shpectorov, On scale embeddings of graphs into hypercubes, Eur. J. Combin., 14 (1993), 117–130. https://doi.org/10.1006/eujc.1993.1016 doi: 10.1006/eujc.1993.1016
[10]	V. Chepoi, M. Deza, V. Grishukhin, Clin d'oeil on $l_1$-embeddable planar graphs, Discrete Appl. Math., 80 (1997), 3–19. https://doi.org/10.1016/S0166-218X(97)00066-8 doi: 10.1016/S0166-218X(97)00066-8
[11]	D. Ž. Djoković, Distance-preserving subgraphs of hypercubes, J. Combin. Theory Ser. B, 14 (1973), 263–267. https://doi.org/10.1016/0095-8956(73)90010-5 doi: 10.1016/0095-8956(73)90010-5
[12]	I. Blake, J. Gilchrist, Addresses for graphs, IEEE Trans. Inf. Theory, 19 (1973), 683–688. https://doi.org/10.1109/TIT.1973.1055087 doi: 10.1109/TIT.1973.1055087
[13]	H. Zhang, G. Wang, Embeddability of open-ended carbon nanotubes in hypercubes, Comput. Geom., 43 (2010), 524–534. https://doi.org/10.1016/j.comgeo.2009.12.001 doi: 10.1016/j.comgeo.2009.12.001
[14]	G. Wang, C. Li, $l_1$-embeddability under the edge-gluing operation on some nonbipartite graphs, Discrete Appl. Math., 375 (2025), 79–84. https://doi.org/10.1016/j.dam.2025.05.036 doi: 10.1016/j.dam.2025.05.036
[15]	S. Klavžar, I. Gutman, B. Mohar, Labeling of benzenoid systems which reflects the vertex-distance relations, J. Chem. Inf. Comput. Sci., 35 (1995), 590–593. https://doi.org/10.1021/ci00025a030 doi: 10.1021/ci00025a030
[16]	S. J. Cyvin, B. N. Cyvin, J. Brunvoll, E. Brendsdal, F. Zhang, X. Guo, et al., Theory of polypentagons, J. Chem. Inf. Comput. Sci., 33 (1993), 466–474. https://doi.org/10.1021/ci00013a027 doi: 10.1021/ci00013a027
[17]	M. Deza, M. I. Shtogrin, Embeddings of chemical graphs in hypercubes, Math. Notes, 68 (2000), 295–305. https://doi.org/10.1007/BF02674552 doi: 10.1007/BF02674552
[18]	H. Zhang, S. Xu, None of the coronoid systems can be isometrically embedded into a hypercube, Discrete Appl. Math., 156 (2008), 2817–2822. https://doi.org/10.1016/j.dam.2007.11.010 doi: 10.1016/j.dam.2007.11.010
[19]	A. Deza, M. Deza, V. Grishukhin, Fullerenes and coordination polyhedra versus half-cube embeddings, Discrete Math., 192 (1998), 41–80. https://doi.org/10.1016/S0012-365X(98)00065-X doi: 10.1016/S0012-365X(98)00065-X
[20]	M. Marcušanu, The classification of $l_1$-embeddable fullerenes, Ph. D. thesis, Bowling Green State University, 2007.
[21]	M. Deza, P. W. Fowler, M. Shtogrin, Version of zones and zigzag structure in icosahedral fullerenes and icosadeltahedra, J. Chem. Inf. Comput. Sci., 43 (2003), 595–599. https://doi.org/10.1021/ci0200669 doi: 10.1021/ci0200669
[22]	H. Wiener, Structural determination of paraffin boiling points, J. Amer. Chem. Soc., 69 (1947), 17–20. https://doi.org/10.1021/ja01193a005 doi: 10.1021/ja01193a005
[23]	H. Hosoya, Topological index: a newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons, Bull. Chem. Soc. Jpn., 44 (1971), 2332–2339. https://doi.org/10.1246/bcsj.44.2332 doi: 10.1246/bcsj.44.2332
[24]	A. A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees: theory and applications, Acta Appl. Math., 66 (2001), 211–249. https://doi.org/10.1023/A:1010767517079 doi: 10.1023/A:1010767517079
[25]	B. Mohar, T. Pisanski, How to compute the Wiener index of a graph, J. Math. Chem., 2 (1988), 267–277. https://doi.org/10.1007/BF01167206 doi: 10.1007/BF01167206
[26]	V. Chepoi, S. Klavžar, The Wiener index and the szeged index of benzenoid systems in linear time, J. Chem. Inf. Comput. Sci., 37 (1997), 752–755. https://doi.org/10.1021/ci9700079 doi: 10.1021/ci9700079
[27]	M. Randić, Novel molecular descriptor for structure–property studies, Chem. Phys. Lett., 211 (1993), 478–483. https://doi.org/10.1016/0009-2614(93)87094-J doi: 10.1016/0009-2614(93)87094-J
[28]	D. J. Klein, I. Lukovits, I. Gutman, On the definition of the hyper-Wiener index for cycle-containing structures, J. Chem. Inf. Comput. Sci., 35 (1995), 50–52. https://doi.org/10.1021/ci00023a007 doi: 10.1021/ci00023a007
[29]	K. Xu, M. Liu, I. Gutman, B. Furtula, A survey on graphs extremal with respect to distance-based topological indices, Match Commun. Math. Comput. Chem., 71 (2014), 461–508.
[30]	I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. Total $\varphi$-electron energy of alternant hydrocarbons, Chem. Phys. Lett., 17 (1972), 535–538. https://doi.org/10.1016/0009-2614(72)85099-1 doi: 10.1016/0009-2614(72)85099-1
[31]	I. Gutman, B. Ruščić, N. Trinajstić, C. F. Wilcox, Graph theory and molecular orbitals. XII. Acyclic polyenes, J. Chem. Phys., 62 (1975), 3399–3405. https://doi.org/10.1063/1.430994 doi: 10.1063/1.430994
[32]	H. Rahbani, H. A. Ahangar, M. A. Henning, New upper bounds on Zagreb indices with given domination number, Appl. Math. Comput., 514 (2026), 129815. https://doi.org/10.1016/j.amc.2025.129815 doi: 10.1016/j.amc.2025.129815
[33]	G. Wang, Z. Xiong, L. Chen, $l_1$-embeddability of shifted quadrilateral cylinder graphs, Graphs Combin., 39 (2023), 129. https://doi.org/10.1007/s00373-023-02725-w doi: 10.1007/s00373-023-02725-w
[34]	G. Wang, Y. Liu, The edge-Wiener index of zigzag nanotubes, Appl. Math. Comput., 377 (2020), 125191. https://doi.org/10.1016/j.amc.2020.125191 doi: 10.1016/j.amc.2020.125191
[35]	T. Ghosh, B. Mandal, Wiener indices of zigzag single walled carbon nanotubes and related nanotories, Chem. Phys., 572 (2023), 111973. https://doi.org/10.1016/j.chemphys.2023.111973 doi: 10.1016/j.chemphys.2023.111973
[36]	G. Wang, Y. Liu, J. Wei, J. Liu, The edge-hyper-Wiener index of zigzag single-walled nanotubes, Polyc. Aromat. Comp., 43 (2023), 1509–1523. https://doi.org/10.1080/10406638.2022.2030764 doi: 10.1080/10406638.2022.2030764
[37]	H. J. Bandelt, V. Chepoi, Decomposition and $l_{1}$-embedding of weakly median graphs, Eur. J. Combin., 21 (2000), 701–714. https://doi.org/10.1006/eujc.1999.0377 doi: 10.1006/eujc.1999.0377
[38]	M. Deza, V. Grishukhin, M. I. Shtogrin, Scale-isometric polytopal graphs in hypercubes and cubic lattices: polytopes in hypercubes and Zn, Imperial College Press, 2004. https://doi.org/10.1142/p308
[39]	G. Wang, H. Zhang, $l_1$-embeddability under the edge-gluing operation on graphs, Discrete Math., 313 (2013), 2115–2118. https://doi.org/10.1016/j.disc.2013.04.032 doi: 10.1016/j.disc.2013.04.032
[40]	M. E. Tylkin, On Hamming geometry of unitary cubes, Dokl. Akad. Nauk SSSR, 134 (1960), 1037–1040.
[41]	M. Deza, M. Laurent, Geometry of cuts and metrics, Springer, 1997. https://doi.org/10.1007/978-3-642-04295-9
[42]	J. H. Koolen, V. Moulton, D. Stevanović, The structure of spherical graphs, Eur. J. Combin., 25 (2004), 299–310. https://doi.org/10.1016/S0195-6698(03)00116-1 doi: 10.1016/S0195-6698(03)00116-1
[43]	V. Chepoi, On distances in benzenoid systems, J. Chem. Inf. Comput. Sci., 36 (1996), 1169–1172. https://doi.org/10.1021/ci9600869 doi: 10.1021/ci9600869
[44]	M. Deza, M. Laurent, $l_1$-rigid graphs, J. Algebraic Combin., 3 (1994), 153–175. https://doi.org/10.1023/A:1022441506632 doi: 10.1023/A:1022441506632
[45]	M. Deza, S. Shpectorov, Recognition of the $l_1$-graphs with complexity $O(nm)$, or football in a hypercube, Eur. J. Combin., 17 (1996), 279–289. https://doi.org/10.1006/eujc.1996.0024 doi: 10.1006/eujc.1996.0024
[46]	I. N. Cangul, A. Saleh, A. Alqesmah, H. Alashwali, Entire Wiener index of graphs, Commun. Combin. Optim., 7 (2022), 227–245. https://doi.org/10.22049/CCO.2021.27234.1217 doi: 10.22049/CCO.2021.27234.1217

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)