Extremal bi-Wiener indices of unbranched catacondensed benzenoids and polyphenyl hexagonal chains

Yunfei Li; Yunfei Li

doi:10.3934/math.2026888

AIMS Mathematics

2026, Volume 11, Issue 7: 21973-21988. doi: 10.3934/math.2026888

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Research article

Extremal bi-Wiener indices of unbranched catacondensed benzenoids and polyphenyl hexagonal chains

Yunfei Li ^,

School of Accounting and Finance, Xiamen University Tan Kah Kee College, Zhangzhou, China

Received: 16 May 2026 Revised: 27 June 2026 Accepted: 10 July 2026 Published: 23 July 2026
MSC : 05C12, 05C35, 05C92

The bi-Wiener index is a recently introduced distance-based invariant for connected bipartite graphs. Let $ G $ be a connected bipartite graph with bipartition $ V(G) = X\cup Y $. The bi-Wiener index of $ G $ is defined by
$ W_b(G) = \sum\limits_{(u, v)\in X\times Y} d_G(u, v), $
where $ d_G(u, v) $ is the distance between $ u $ and $ v $ in $ G $. In this paper, we study extremal bi-Wiener indices of two families of hexagon-based molecular chains. The first family considered here is the family of unbranched catacondensed benzenoids, also called hexagonal chains, in which consecutive hexagons share an edge. If $ F_n\in \mathcal{U}_n $ is an unbranched catacondensed benzenoid with $ n $ hexagons, then
$ \frac{4n^3+36n^2-10n+15}{3} \leq W_b(F_n)\leq \frac{8n^3+18n^2+16n+3}{3}. $
For $ n\geq 3 $, the lower bound is attained uniquely by the helicene chain, and the upper bound is attained uniquely by the linear chain. The second family is the family of polyphenyl hexagonal chains, in which consecutive hexagons are joined by a cut edge. If $ G_n\in \mathcal{P}_n $ is a polyphenyl hexagonal chain with internal configuration sequence $ \ell_2, \ell_3, \ldots, \ell_{n-1}\in\{1, 2, 3\} $, then
$ W_b(G_n) = 3n^3+27n^2-15n+18\sum\limits_{k = 2}^{n-1}\ell_k(k-1)(n-k), $
where the sum is empty if $ n\leq 2 $. Consequently, among all polyphenyl hexagonal chains with $ n $ hexagons, the ortho-chain has the minimum bi-Wiener index and the para-chain has the maximum bi-Wiener index.
- bi-Wiener index,
- Wiener index,
- unbranched catacondensed benzenoid,
- hexagonal chain,
- polyphenyl hexagonal chain,
- linear chain,
- helicene chain,
- extremal graph
Citation: Yunfei Li. Extremal bi-Wiener indices of unbranched catacondensed benzenoids and polyphenyl hexagonal chains[J]. AIMS Mathematics, 2026, 11(7): 21973-21988. doi: 10.3934/math.2026888

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Abstract

The bi-Wiener index is a recently introduced distance-based invariant for connected bipartite graphs. Let $ G $ be a connected bipartite graph with bipartition $ V(G) = X\cup Y $. The bi-Wiener index of $ G $ is defined by

$ W_b(G) = \sum\limits_{(u, v)\in X\times Y} d_G(u, v), $

where $ d_G(u, v) $ is the distance between $ u $ and $ v $ in $ G $. In this paper, we study extremal bi-Wiener indices of two families of hexagon-based molecular chains. The first family considered here is the family of unbranched catacondensed benzenoids, also called hexagonal chains, in which consecutive hexagons share an edge. If $ F_n\in \mathcal{U}_n $ is an unbranched catacondensed benzenoid with $ n $ hexagons, then

$ \frac{4n^3+36n^2-10n+15}{3} \leq W_b(F_n)\leq \frac{8n^3+18n^2+16n+3}{3}. $

For $ n\geq 3 $, the lower bound is attained uniquely by the helicene chain, and the upper bound is attained uniquely by the linear chain. The second family is the family of polyphenyl hexagonal chains, in which consecutive hexagons are joined by a cut edge. If $ G_n\in \mathcal{P}_n $ is a polyphenyl hexagonal chain with internal configuration sequence $ \ell_2, \ell_3, \ldots, \ell_{n-1}\in\{1, 2, 3\} $, then

$ W_b(G_n) = 3n^3+27n^2-15n+18\sum\limits_{k = 2}^{n-1}\ell_k(k-1)(n-k), $

where the sum is empty if $ n\leq 2 $. Consequently, among all polyphenyl hexagonal chains with $ n $ hexagons, the ortho-chain has the minimum bi-Wiener index and the para-chain has the maximum bi-Wiener index.

References

[1]	J. A. Bondy, U. S. R. Murty, Graph Theory, New York: Springer, 2008.
[2]	H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc., 69 (1947), 17–20.
[3]	R. Todeschini, V. Consonni, Handbook of Molecular Descriptors, Weinheim: Wiley-VCH, 2000.
[4]	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.
[5]	X. Chen, S. Karimi, K. Xu, M. Lewinter, E. Choi, A. Delgado, et al., Extremal trees with respect to bi-Wiener index, Bull. Malays. Math. Sci. Soc., 47 (2024), 156. https://doi.org/10.1007/s40840-024-01757-6 doi: 10.1007/s40840-024-01757-6
[6]	S. Huang, S. Li, M. Zhang, On the extremal Mostar indices of hexagonal chains, MATCH Commun. Math. Comput. Chem., 84 (2020), 249–271.
[7]	I. Gutman, Extremal hexagonal chains, J. Math. Chem., 12 (1993), 197–210. https://doi.org/10.1007/BF01164635 doi: 10.1007/BF01164635
[8]	H. Y. Deng, The anti-forcing number of hexagonal chains, MATCH Commun. Math. Comput. Chem., 58 (2007), 675–682.
[9]	A. A. Dobrynin, E. Estaji, Wiener index of certain families of hexagonal chains, J. Appl. Math. Comput., 59 (2019), 245–256. https://doi.org/10.1007/s12190-018-1177-9 doi: 10.1007/s12190-018-1177-9
[10]	T. Došlić, A. Graovac, O. Ori, Eccentric connectivity index of hexagonal belts and chains, MATCH Commun. Math. Comput. Chem., 65 (2011), 745–752.
[11]	P. V. Khadikar, P. P. Kale, N. V. Deshpande, S. Karmarkar, V. K. Agrawal, Novel PI indices of hexagonal chains, J. Math. Chem., 29 (2001), 143–150. https://doi.org/10.1023/A:1010931213729 doi: 10.1023/A:1010931213729
[12]	L. Z. Zhang, On the ordering of a class of hexagonal chains with respect to Merrifield–Simmons index, J. Syst. Sci. Compl., 13 (2000), 219–224.
[13]	L. Z. Zhang, F. J. Zhang, Extremal hexagonal chains concerning $k$-matchings and $k$-independent sets, J. Math. Chem., 27 (2000), 319–329. https://doi.org/10.1023/A:1018875823127 doi: 10.1023/A:1018875823127
[14]	S. J. Xu, H. P. Zhang, Generalized Hosoya polynomials of hexagonal chains, J. Math. Chem., 43 (2008), 852–863. https://doi.org/10.1007/s10910-007-9234-x doi: 10.1007/s10910-007-9234-x
[15]	C. Q. Xiao, H. Y. Chen, Kekulé structures of square–hexagonal chains and the Hosoya index of caterpillar trees, Discrete Math., 339 (2016), 506–510. https://doi.org/10.1016/j.disc.2015.09.018 doi: 10.1016/j.disc.2015.09.018
[16]	T. Došlić, I. Martinjak, R. Škrekovski, S. T. Spužević, I. Zubac, Mostar index, J. Math. Chem., 56 (2018), 2995–3013. https://doi.org/10.1007/s10910-018-0928-z
[17]	Y. Bai, B. Zhao, P. Zhao, Extremal Merrifield–Simmons index and Hosoya index of polyphenyl chains, MATCH Commun. Math. Comput. Chem., 62 (2009), 649–656.
[18]	W. L. Yang, Hosoya and Merrifield–Simmons indices in random polyphenyl chains, Acta Math. Appl. Sin. Engl. Ser., 37 (2021), 485–494. https://doi.org/10.1007/s10255-021-1026-8 doi: 10.1007/s10255-021-1026-8
[19]	H. Deng, Wiener indices of spiro and polyphenyl hexagonal chains, Math. Comput. Model., 55 (2012), 634–644. https://doi.org/10.1016/j.mcm.2011.08.037 doi: 10.1016/j.mcm.2011.08.037
[20]	H. Y. Deng, Z. Tang, Kirchhoff indices of spiro and polyphenyl hexagonal chains, Util. Math., 95 (2014), 113–128.
[21]	Y. Yang, H. Liu, H. Wang, H. Fu, Subtrees of spiro and polyphenyl hexagonal chains, Appl. Math. Comput., 268 (2015), 547–560. https://doi.org/10.1016/j.amc.2015.06.094 doi: 10.1016/j.amc.2015.06.094
[22]	Z. Raza, J. L. G. Guirao, G. Bassioni, The comparative analysis of two molecular indices in random polyphenyl and spiro chains, Math. Biosci. Eng., 19 (2022), 12500–12517. https://doi.org/10.3934/MBE.2022583 doi: 10.3934/MBE.2022583
[23]	K. Deng, S. Li, Extremal Mostar indices of tree-like polyphenyls, Int. J. Quant. Chem., 121 (2021), e26602. https://doi.org/10.1002/qua.26602 doi: 10.1002/qua.26602

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