On the maximum Graovac-Pisanski index of bicyclic graphs

Jian Lu; Zhongxiang Wang; Jian Lu; Zhongxiang Wang

doi:10.3934/math.20231270

AIMS Mathematics

2023, Volume 8, Issue 10: 24914-24928. doi: 10.3934/math.20231270

Previous Article Next Article

Research article

On the maximum Graovac-Pisanski index of bicyclic graphs

Jian Lu ^,,
Zhongxiang Wang

School of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu 233030, China

Received: 28 June 2023 Revised: 16 August 2023 Accepted: 20 August 2023 Published: 25 August 2023
MSC : 05C12, 05C25

For a simple graph $ G = (V(G), E(G)) $, the Graovac-Pisanski index of $ G $ is defined as

$ GP(G) = \frac{|V(G)|}{2|{\rm{Aut}}(G)|}\sum\limits_{u\in V(G)}\sum\limits_{\alpha\in {\rm{Aut}}(G)}d_G(u,\alpha(u)), $

where $ {\rm{Aut}}(G) $ is the automorphism group of $ G $ and $ d_G(u, v) $ is the length of a shortest path between the two vertices $ u $ and $ v $ in $ G $. Obviously, $ GP(G) = 0 $ if $ G $ has no nontrivial automorphisms. Let $ B_{n}^{3, 3} $ be the graph consisting of two disjoint 3-cycles with a path of length $ n-5 $ joining them. In this article, we prove that among all those $ n $-vertex bicyclic graphs in which every edge lies on at most one cycle, $ B_{n}^{3, 3} $ has the maximum Graovac-Pisanski index.
- modified Wiener index,
- Graovac-Pisanski index,
- automorphism,
- orbit,
- bicyclic graphs
Citation: Jian Lu, Zhongxiang Wang. On the maximum Graovac-Pisanski index of bicyclic graphs[J]. AIMS Mathematics, 2023, 8(10): 24914-24928. doi: 10.3934/math.20231270

Related Papers:

Abstract

For a simple graph $ G = (V(G), E(G)) $, the Graovac-Pisanski index of $ G $ is defined as

$ GP(G) = \frac{|V(G)|}{2|{\rm{Aut}}(G)|}\sum\limits_{u\in V(G)}\sum\limits_{\alpha\in {\rm{Aut}}(G)}d_G(u,\alpha(u)), $

where $ {\rm{Aut}}(G) $ is the automorphism group of $ G $ and $ d_G(u, v) $ is the length of a shortest path between the two vertices $ u $ and $ v $ in $ G $. Obviously, $ GP(G) = 0 $ if $ G $ has no nontrivial automorphisms. Let $ B_{n}^{3, 3} $ be the graph consisting of two disjoint 3-cycles with a path of length $ n-5 $ joining them. In this article, we prove that among all those $ n $-vertex bicyclic graphs in which every edge lies on at most one cycle, $ B_{n}^{3, 3} $ has the maximum Graovac-Pisanski index.

References

[1]	A. R. Ashrafi, F. Koorepazan-Moftakhar, M. V. Diudea, O. Ori, Graovac-Pisanski index of fullerenes and fullerene-like molecules, Fullerenes Nanotubes Carbon Nanostruct., 24 (2016) 779–785. https://doi.org/10.1080/1536383X.2016.1242483 doi: 10.1080/1536383X.2016.1242483
[2]	S. Bermudo, J. E. Nápoles, J. Rada, Extremal trees for the Randić index with given domination number, Appl. Math. Comput., 375 (2020), 125122. https://doi.org/10.1016/j.amc.2020.125122 doi: 10.1016/j.amc.2020.125122
[3]	F. Cataldo, O. Ori, S. Iglesias-Groth, Topological lattice descriptors of graphene sheets with fullerene-like nanostructures, Mol. Simul., 36 (2010), 341–353. https://doi.org/10.1080/08927020903483262 doi: 10.1080/08927020903483262
[4]	M. Črepnjak, M. Knor, N. Tratnik, P. Ž. Pleteršek, The Graovac-Pisanski index of connected bipartite graphs with applications to hydrocarbon molecules, Fullerenes Nanotubes Carbon Nanostruct., 29 (2021), 884–889. https://doi.org/10.1080/1536383X.2021.1910675 doi: 10.1080/1536383X.2021.1910675
[5]	M. $\check{C}$repnjak, N. Tratnik, P. $\check{Z}$. Pleter$\check{s}$ek, Predicting melting points of hydrocarbons by the Graovac-Pisanski index, Fullerenes Nanotubes Carbon Nanostruct., 26 (2018), 239–245. https://doi.org/10.1080/1536383X.2017.1386657 doi: 10.1080/1536383X.2017.1386657
[6]	K. Fathalikhani, A. Babai, S. S. Zemljič, The Graovac-Pisanski index of Sierpiński graphs, Discrete Appl. Math., 285 (2020), 30–42. https://doi.org/10.1016/j.dam.2020.05.014 doi: 10.1016/j.dam.2020.05.014
[7]	A. Graovac, T. Pisanski, On the Wiener index of a graph, J. Math. Chem., 8 (1991), 53–62. https://doi.org/10.1007/BF01166923 doi: 10.1007/BF01166923
[8]	M. Ghorbani, S. Klavžar, Modified Wiener index via canonical metric representation, and some fullerene patches, ARS Math. Contemp., 11 (2016), 247–254. https://doi.org/10.26493/1855-3974.918.0b2 doi: 10.26493/1855-3974.918.0b2
[9]	M. Hakimi-Nezhaad, M. Ghorbani, On the Graovac-Pisanski index, Kraǵujevac J. Sci., 39 (2017), 91–98. https://doi.org/10.5937/KgJSci1739091H doi: 10.5937/KgJSci1739091H
[10]	M. Knor, R. Škrekovski, A. Tepeh, Trees with the maximal value of Graovac-Pisanski index, Appl. Math. Comput., 358 (2019), 287–292. https://doi.org/10.1016/j.amc.2019.04.034 doi: 10.1016/j.amc.2019.04.034
[11]	M. Knor, J. Komorník, R. Škrekovski, A. Tepeh, Unicyclic graphs with the maximal value of Graovac-Pisanski index, ARS Math. Contemp., 17 (2019), 455–466. https://doi.org/10.26493/1855-3974.1925.57a doi: 10.26493/1855-3974.1925.57a
[12]	M. Knor, R. Škrekovski, A. Tepeh, On the difference between Wiener index and Graovac-Pisanski index, MATCH Commun. Math. Comput. Chem., 83 (2020), 109–120.
[13]	F. Koorepazan-Moftakhar, A. R. Ashrafi, Combination of distance and symmetry in some molecular graphs, Appl. Math. Comput., 281 (2016), 223–232. https://doi.org/10.1016/j.amc.2016.01.065 doi: 10.1016/j.amc.2016.01.065
[14]	F. Koorepazan-Moftakhar, A. R. Ashrafi, O. Ori, Symmetry groups and Graovac-Pisanski index of some linear polymers, Quasigroups Relat. Sys., 26 (2018), 87–102.
[15]	R. Pinal, Effect of molecular symmetry on melting temperature and solubility, Org. Biomol. Chem., 2 (2004), 2692–2699. https://doi.org/10.1039/B407105K doi: 10.1039/B407105K
[16]	M. Randic, Characterization of molecular branching, J. Am. Chem. Soc., 97 (1975), 6609–6615. https://doi.org/10.1021/ja00856a001 doi: 10.1021/ja00856a001
[17]	H. Shabani, A. R. Ashrafi, The modified Wiener index of some graph operations, ARS Math. Contemp., 11 (2016), 277–284. https://doi.org/10.26493/1855-3974.801.968 doi: 10.26493/1855-3974.801.968
[18]	H. Shabani, A. R. Ashrafi, Symmetry-moderated Wiener index, MATCH Commun. Math. Comput. Chem., 76 (2016), 3–18.
[19]	Y. Shang, The Estrada index of evolving graphs, Appl. Math. Comput., 250 (2015), 415–423. https://doi.org/10.1016/j.amc.2014.10.129 doi: 10.1016/j.amc.2014.10.129
[20]	Y. Shang, Perturbation results for the Estrada index in weighted networks, J. Phys. A, 44 (2011), 075003. https://doi.org/10.1088/1751-8113/44/7/075003 doi: 10.1088/1751-8113/44/7/075003
[21]	N. Tratnik, The Graovac-Pisanski index of zig-zag tubulenes and the generalized cut method, J. Math. Chem., 55 (2017), 1622–1637. https://doi.org/10.1007/s10910-017-0749-5 doi: 10.1007/s10910-017-0749-5
[22]	N. Tratnik, P. Ž. Pleteršek, The Graovac-Pisanski index of armchair nanotubes, J. Math. Chem., 56 (2018), 1103–1116. https://doi.org/10.1007/s10910-017-0846-5 doi: 10.1007/s10910-017-0846-5
[23]	H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc., 69 (1947), 17–20. https://doi.org/10.1021/ja01193a005 doi: 10.1021/ja01193a005

Reader Comments

Your name:*

Email:*
© 2023 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)