PI Index of Bicyclic Graphs

Document Type : Original paper

Authors

1 Department of Mathematics, School of Physical Sciences, Kochi, Amrita Vishwa Vidyapeetham, India

2 Department of Mathematics, Amrita School of Physical Sciences, Coimbatore, Amrita Vishwa Vidyapeetham, India

Abstract

The PI index of a graph $G$ is given by $PI(G)=\sum_{e\in E(G)}(\left|V(G)\right|-N_G(e))$, where $N_G(e)$ is  the number of equidistant vertices for the edge $e$. Various topological indices of bicyclic graphs have already been calculated. In this paper, we obtained the exact value of the PI index of bicyclic graphs. We also explore the extremal graphs among all bicyclic graphs with respect to the PI index. Furthermore, we calculate the PI index of a cactus graph and determine the extremal values of the PI index among cactus graphs.

Keywords

Main Subjects

References

[1] A.R. Ashrafi and A. Loghman, PI index of zig-zag polyhex nanotubes, MATCH Commun. Math. Comput. Chem. 55 (2006), no. 2, 447–452.
[2] M.S. Chithrabhanu, J. Geetha, and K. Somasundaram, PI and weighted PI indices for powers of paths, cycles, and their complements, J. Intelligent & Fuzzy Sys. 44 (2023), no. 1, 1439–1452.
[3] M.S. Chithrabhanu and 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
[4] C. Gopika, J. Geetha, and K. Somasundaram, Weighted PI index of tensor product and strong product of graphs, Discrete Math. Algorithms Appl. 13 (2021), no. 3, Article ID: 2150019.
https://doi.org/10.1142/S1793830921500191
[5] A. Ilić and N. Milosavljević, The weighted vertex PI index, Math. Comput. Model. 57 (2013), no. 3-4, 623–631.
https://doi.org/10.1016/j.mcm.2012.08.001
[6] G. Indulal, L. Alex, and I. Gutman, On graphs preserving PI index upon edge removal, J. Math. Chem. 59 (2021), no. 7, 1603–1609.
https://doi.org/10.1007/s10910-021-01255-1
[7] P.V. Khadikar, On a novel structural descriptor PI, National Academy Science Letters 23 (2000), no. 7/8, 113–118.
[8] P.V. Khadikar, Padmakar-Ivan index in nanotechnology, Iran. J. Math. Chem. 1 (2010), no. 1, 7–42.
https://doi.org/10.22052/ijmc.2010.5133
[9] P.V. Khadikar, S. Karmarkar, and V.K. Agrawal, A novel PI index and its applications to QSPR/QSAR studies, J. Chem. Inform. Comput. Sci. 41 (2001), no. 4, 934–949.
https://doi.org/10.1021/ci0003092
[10] M.H. Khalifeh, H. Yousefi-Azari, and A.R. Ashrafi, Vertex and edge PI indices of cartesian product graphs, Discrete Appl. Math. 156 (2008), no. 10, 1780–1789.
https://doi.org/10.1016/j.dam.2007.08.041
[11] G. Ma and Q. Bian, Correction of the paper “Bicyclic graphs with extremal values of PI index”, Discrete Appl. Math. 207 (2016), 132–133.
https://doi.org/10.1016/j.dam.2016.02.016
[12] G. Ma, Q. Bian, and J. Wang, Bounds on the PI index of unicyclic and bicyclic graphs with given girth, Discrete Appl. Math. 230 (2017), 156–161.
https://doi.org/10.1016/j.dam.2017.06.011
[13] G. Ma, Q. Bian, and J. Wang, The weighted vertex PI index of bicyclic graphs, Discrete Appl. Math. 247 (2018), 309–321.
https://doi.org/10.1016/j.dam.2018.03.087
[14] G. Ma, Q. Bian, and J. Wang, The maximum PI index of bicyclic graphs with even number of edges, Inform. Process. Lett. 146 (2019), 13–16.
https://doi.org/10.1016/j.ipl.2019.02.001
[15] Ž.K. Vukićević and D. Stevanović, Bicyclic graphs with extremal values of PI index, Discrete Appl. Math. 161 (2013), no. 3, 395–403.
https://doi.org/10.1016/j.dam.2012.09.015
[16] H. Wiener, Structural determination of paraffin boiling points, J. American Chem. Soc. 69 (1947), no. 1, 17–20.
https://doi.org/10.1021/ja01193a005
Communications in Combinatorics and Optimization
Volume 9, Issue 3
September 2024
Pages 425-436

Files

Share

How to cite

Statistics