Padmakar-Ivan index of power graphs with applications in silicon structures

Manju S. C; Sakander Hayat; K. Somasundaram; Rashad Ismail; Manju S. C; Sakander Hayat; K. Somasundaram; Rashad Ismail

doi:10.3934/math.20251305

AIMS Mathematics

2025, Volume 10, Issue 12: 29686-29702. doi: 10.3934/math.20251305

Previous Article Next Article

Research article Special Issues

Padmakar-Ivan index of power graphs with applications in silicon structures

1.
Department of Science and Humanities, Sree Narayana Gurukulam College of Engineering, Kolenchery, Kerala, India
2.
Mathematical Sciences, Faculty of Science, Universiti Brunei Darussalam, Jln Tungku Link, Gadong BE1410, Brunei Darussalam
3.
Department of Mathematics, Amrita School of Physical Sciences, Coimbatore, Amrita Vishwa Vidyapeetham, India
4.
Department of Mathematics, Faculty of Science and Arts, Mahayl Assir, King Khalid University, Abha 61913, Saudi Arabia

Received: 19 August 2025 Revised: 13 November 2025 Accepted: 20 November 2025 Published: 17 December 2025
MSC : 05C09, 05C35, 05C92

In this study, we focus on the Padmakar-Ivan (PI) index, a molecular descriptor that quantifies the structural characteristics of chemical networks on the basis of their vertex distances. The main objective of this work is to develop efficient approaches for computing the PI index, particularly for graph powers and complex network structures. We begin by formulating an algorithm for computing the PI index of general graphs and extend it to the $ k^{\text{th}} $ power of a graph using a distance-based framework. Furthermore, we introduce a novel clique cut method that establishes a theoretical foundation for analyzing and computing the PI index in intricate silicate and silicon-based frameworks. The proposed techniques significantly simplify and generalize existing computational procedures.
- graph indices,
- Padmakar-Ivan index,
- power of a graph
Citation: Manju S. C, Sakander Hayat, K. Somasundaram, Rashad Ismail. Padmakar-Ivan index of power graphs with applications in silicon structures[J]. AIMS Mathematics, 2025, 10(12): 29686-29702. doi: 10.3934/math.20251305

Related Papers:

Abstract

In this study, we focus on the Padmakar-Ivan (PI) index, a molecular descriptor that quantifies the structural characteristics of chemical networks on the basis of their vertex distances. The main objective of this work is to develop efficient approaches for computing the PI index, particularly for graph powers and complex network structures. We begin by formulating an algorithm for computing the PI index of general graphs and extend it to the $ k^{\text{th}} $ power of a graph using a distance-based framework. Furthermore, we introduce a novel clique cut method that establishes a theoretical foundation for analyzing and computing the PI index in intricate silicate and silicon-based frameworks. The proposed techniques significantly simplify and generalize existing computational procedures.

References

[1]	N. Trinajstić, Chemical graph theory, 2 Eds., Boca Raton: CRC Press, 1992. https://doi.org/10.1201/9781315139111
[2]	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
[3]	P. V. Khadikar, On a novel structural descriptor PI, Natl. Acad. Sci. Lett., 23 (2000), 113–118.
[4]	P. V. Khadikar, S. Karmarkar, V. K. Agrawal, A novel PI index and its applications to QSPR/QSAR studies, J. Chem. Inf. Comput. Sci., 41 (2001), 934–949. https://doi.org/10.1021/ci0003092 doi: 10.1021/ci0003092
[5]	M. H. Khalifeh, H. Yousefi-Azari, A. R. Ashrafi, Vertex and edge PI indices of Cartesian product graphs, Discrete Appl. Math., 156 (2008), 1780–1789. https://doi.org/10.1016/j.dam.2007.08.041 doi: 10.1016/j.dam.2007.08.041
[6]	A. Ilić, N. Milosavljević, The weighted vertex PI index, Math. Comput. Model., 57 (2013), 623–631. https://doi.org/10.1016/j.mcm.2012.08.001 doi: 10.1016/j.mcm.2012.08.001
[7]	A. R. Ashrafi, M. Ghorbani, M. Jalali, The vertex PI and Szeged indices of an infinite family of fullerenes, J. Theor. Comput. Chem., 7 (2008), 221–231. https://doi.org/10.1142/S0219633608003757 doi: 10.1142/S0219633608003757
[8]	P. E. John, P. V. Khadikar, J. Singh, A method of computing the PI index of benzenoid hydrocarbons using orthogonal cuts, J. Math. Chem., 42 (2007), 37–45. https://doi.org/10.1007/s10910-006-9100-2 doi: 10.1007/s10910-006-9100-2
[9]	M. J. Nadjafi-Arani, G. H. Fath-Tabar, A. R. Ashrafi, Extremal graphs with respect to the vertex PI index, Appl. Math. Lett., 22 (2009), 1838–1840. https://doi.org/10.1016/j.aml.2009.07.005 doi: 10.1016/j.aml.2009.07.005
[10]	C. Gopika, J. Geetha, K. Somasundaram, Weighted PI index of tensor product and strong product of graphs, Discrete Math. Algorithms Appl., 13 (2021), 2150019. https://doi.org/10.1142/S1793830921500191 doi: 10.1142/S1793830921500191
[11]	S. C. Manju, J. Geetha, K. Somasundaram, PI and weighted PI indices for powers of paths, cycles, and their complements, J. Intel. Fuzzy Syst. Appl. Eng. Tech., 44 (2022), 1439–1452. https://doi.org/10.3233/JIFS-221436 doi: 10.3233/JIFS-221436
[12]	S. C. Manju, K. Somasundaram, PI index of bicyclic graphs, Commun. Comb. Optim., 9 (2024), 425–436. https://doi.org/10.22049/cco.2023.27817.1360 doi: 10.22049/cco.2023.27817.1360
[13]	M. S. Chithrabhanu, K. Somasundaram, Padmakar-Ivan index of some types of perfect graphs, Discrete Math. Lett., 9 (2022), 92–99. https://doi.org/10.47443/dml.2021.s215 doi: 10.47443/dml.2021.s215
[14]	S. C. Manju, K. Somasundaram, Y. L. Shang, A new method for computing the vertex PI index with applications to special classes of graphs, AKCE Int. J. Graphs Comb., 22 (2025), 117–124. https://doi.org/10.1080/09728600.2024.2424317 doi: 10.1080/09728600.2024.2424317
[15]	X. H. An, B. Wu, The Wiener index of the $k$th power of a graph, Appl. Math. Lett., 21 (2008), 436–440. https://doi.org/10.1016/j.aml.2007.03.025 doi: 10.1016/j.aml.2007.03.025
[16]	W. J. Zhang, B. Wu, X. H. An, The hyper-Wiener index of the $k$th power of a graph, Discrete Math. Algorithms Appl., 3 (2011), 17–23. https://doi.org/10.1142/S1793830911000973 doi: 10.1142/S1793830911000973
[17]	G. F. Su, L. M. Xiong, I. Gutman, Harary index of the $k$-th power of a graph, Appl. Anal. Discrete Math., 7 (2013), 94–105.
[18]	K. C. Das, M. Imran, T. Vetrík, General Sombor index of graphs and trees, J. Discrete Math. Sci. Cryptogr., 28 (2025), 101–111. http://dx.doi.org/10.47974/JDMSC-1918 doi: 10.47974/JDMSC-1918
[19]	S. Rouhani, M. Habibi, M. A. Mehrpouya, On Sombor index of extremal graphs, J. Discrete Math. Appl., 9 (2024), 335–344. https://doi.org/10.22061/jdma.2024.11328.1101 doi: 10.22061/jdma.2024.11328.1101
[20]	J. C. Hernández, J. M. Rodríguez, O. Rosario, J. M. Sigarreta, Extremal problems on the general Sombor index of a graph, AIMS Math., 7 (2022), 8330–8343. http://dx.doi.org/10.3934/math.2022464 doi: 10.3934/math.2022464

Reader Comments

Your name:*

Email:*
© 2025 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)