Abstract
A topological index is a numerical descriptor in mathematical chemistry and graph theory that quantifies a molecule’s structural properties without considering its three-dimensional arrangement. A crucial factor to consider when exploring topological indices is their capacity to distinguish between different structures. In light of this, the exponential vertex-degree-based topological index is put forward in the literature. The present work focuses on investigating the mathematical properties and application potential of the exponential augmented Zagreb index (EAZ). The EAZ index for a graph \(\Upsilon \) is defined as
where \(d_i\) represents the degree of a vertex \(v_i\). Crucial upper and lower bounds of EAZ for numerous classes of graphs like bipartite, unicyclic, bicyclic, chemical graph, and general graphs are derived. The bounds are computed in terms of different graph parameters including graph order, size, maximum degree, number of pendant vertices and independence number. The extremal graphs for which the bounds appear are also characterized. Moreover, the EAZ index is found to correlate well with some physico-chemical properties of octanes.
Similar content being viewed by others
Data availability
Not applicable.
References
Balachandran, S., Vetrík, T.: Exponential second Zagreb index of chemical trees. Trans. Combin. 10, 97–106 (2021)
Basak, S.C., Vracko, M.G.: Parsimony principle and its proper use/application in computer-assisted drug design and QSAR. Curr. Comput. Aided Drug Des. 16, 1–5 (2020)
Basak, S.C., Bhattacharjee, A.K.: Computational approaches for the design of mosquito repellent chemicals. Curr. Med. Chem. 27, 32–41 (2020)
Basak, S.C.: My tortuous pathway through Mathematical Chemistry and QSAR research with memories of some personal interactions and collaborations With Professors Milan Randić and Mircea Diudea. Croat. Chem. Acta 93, 247–258 (2020)
Furtula, B., Graovac, A., Vukicević, D.: Augmented Zagreb index. J. Math. Chem. 48, 370–380 (2010)
Carballosa, W., Quintana, Y., Rodríguez, J.M., Sigarreta, J.M.: Exponential topological indices: optimal inequalities and applications. J. Math. Chem. 61, 933–949 (2023)
Chen, C., Liu, M., Gu, X., Das, K.C.: Extremal augmented Zagreb index of trees with given numbers of vertices and leaves. Discrete Math. 345, 112753 (2022)
Cruz, R., Monsalve, J., Rada, J.: The balanced double star has maximum exponential second Zagreb index. J. Combin. Optim. 41, 544–552 (2021)
Cruz, R., Monsalve, J., Rada, J.: Trees with maximum exponential Randić index. Discrete Appl. Math. 283, 634–643 (2020)
Cruz, R., Rada, J.: The path and the star as extremal values of vertex-degree-based topological indices among trees. MATCH Commun. Math. Comput. Chem. 82, 715–732 (2019)
Cruz, R., Rada, J.: Extremal graphs for exponential VDB indices. Kragujev. J. Math. 46, 105–113 (2022)
Das, K.C., Mondal, S.: On neighborhood inverse sum indeg index of molecular graphs with chemical significance. Inf. Sci. 623, 112–131 (2023)
Das, K.C., Elumalai, S., Balachandran, S.: Open problems on the exponential vertex-degree-based topological indices of graphs. Discrete Appl. Math. 293, 38–49 (2021)
Das, K.C., Mondal, S., Raza, Z.: On Zagreb connection indices. Eur. Phys. J. Plus 137, 1242 (2022)
Eliasi, M.: Unicyclic and bicyclic graphs with maximum exponential second Zagreb index. Discrete Appl. Math. 307, 172–179 (2022)
Huang, Y., Liu, B., Gan, L.: Augmented Zagreb index of connected graphs. MATCH Commun. Math. Comput. Chem. 67, 483–494 (2012)
Jiang, Y., Lu, M.: Maximal augmented Zagreb index of trees with given diameter. Appl. Math. Comput. 395, 125855 (2021)
Johnson, C., Sankar, R.: Graph energy and topological descriptors of zero divisor graph associated with commutative ring. J. Appl. Math. Comput. 69, 2641–2656 (2023)
Liu, M., Pang, S., Belardo, F., Ali, A.: The k-apex trees with minimum augmented Zagreb index. Discrete Math. 346, 113390 (2023)
Liu, J.B., Zheng, Y.Q., Peng, X.B.: The statistical analysis for Sombor indices in a random polygonal chain networks. Discrete Appl. Math. 338, 218–233 (2023)
Liu, M., Cheng, K., Furtula, B.: Minimum augmented Zagreb index of \(c\)-cyclic graphs. Discrete Appl. Math. 295, 32–38 (2021)
Liu, H., You, L., Chen, H., Tang, Z.: On the first three minimum Mostar indices of tree-like phenylenes. J. Appl. Math. Comput. 68, 3615–3629 (2022)
Moon, S., Park, S.: Bounds for the geometric-arithmetic index of unicyclic graphs. J. Appl. Math. Comput. 69, 2955–2971 (2023)
Mondal, S., Das, K.C.: Zagreb connection indices in structure property modelling. J. Appl. Math. Comput. 69, 3005–3020 (2023)
Mondal, S., Das, K.C.: On the Sanskruti index of graphs. J. Appl. Math. Comput. 69, 1205–1219 (2023)
Mondal, S., Das, K.C.: Degree-based graph entropy in structure-property modeling. Entropy 25, 1092 (2023)
Milovanović, E., Milovanović, I., Jamil, M.: Some properties of the Zagreb indices. Filomat 32, 2667–2675 (2018)
Milovanović, I., Milovanović, E., Altindag, S.B.B., Matejić, M.: McClelland-type upper bounds for graph energy. MATCH Commun. Math. Comput. Chem. 88, 141–155 (2022)
Nithya, P., Elumalai, S., Balachandran, S., Mondal, S.: Smallest ABS index of unicyclic graphs with given girth. J. Appl. Math. Comput. 69, 3675–3692 (2023)
Rada, J.: Exponential vertex-degree-based topological indices and discrimination. MATCH Commun. Math. Comput. Chem. 82, 29–41 (2019)
Randić, M., Trinajstić, N.: In search for graph invariants of chemical interest. J. Mol. Struct. 300, 551–571 (1993)
Shanmukha, M.C., Basavarajappa, N.S., Usha, A., Shilpa, K.C.: Novel neighbourhood redefined first and second Zagreb indices on carborundum structures. J. Appl. Math. Comput. 66, 263–276 (2021)
Stein, W.A.: Sage Mathematics Software (Version 6.8), The Sage Development Team, http://www.sagemath.org (2015)
Siddiqui, M.K., Imran, M., Iqbal, M.A.: Molecular descriptors of discrete dynamical system in fractal and Cayley tree type dendrimers. J. Appl. Math. Comput. 61, 57–72 (2019)
Sun, X., Gao, Y., Du, J., Xu, L.: Augmented Zagreb index of trees and unicyclic graphs with perfect matchings. Appl. Math. Comput. 335, 75–81 (2018)
Shao, Y., Gao, W.: Complete characterization of chemical trees with maximal Augmented Zagreb index. J. Appl. Math. Comput. 69, 3851–3870 (2023)
Wang, H., Kang, L.: Further properties on the degree distance of graphs. J. Combin. Optim. 31, 427–446 (2016)
Wang, H., Hua, H., Wang, M.: Comparative study of distance-based graph invariants. J. Appl. Math. Comput. 64, 457–469 (2020)
Wiener, H.: Structural determination of paraffin boiling points. J. Am. Chem. Soc. 69, 17–20 (1947)
Xu, C., Horoldagva, B., Buyantogtokh, L.: The exponential second Zagreb index of \((n,\, m)\)-graphs. Mediterr. J. Math. 20, 181 (2023)
Zhao, J., Liu, J.B., Hayat, S.: Resistance distance-based graph invariants and the number of spanning trees of linear crossed octagonal graphs. J. Appl. Math. Comput. 63, 1–27 (2020)
Acknowledgements
K. C. Das is supported by National Research Foundation funded by the Korean government (Grant No. 2021R1F1A1050646). This work was supported by the Post-doctoral Research Program of Sungkyunkwan University (2023).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Das, K.C., Mondal, S. & Huh, Dy. On the exponential augmented Zagreb index of graphs. J. Appl. Math. Comput. 70, 839–865 (2024). https://doi.org/10.1007/s12190-023-01982-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-023-01982-5