Harmonic-Arithmetic‎ ‎Index‎ ‎of‎ ‎Unicyclic‎ ‎Graphs‎ ‎with given Girth and Connected Graphs with Minimum Degree

Document Type : Research Paper

Authors

1 Department of Mathematics‎, ‎College of Engineering and Technology‎, ‎Faculty of Engineering and Technology‎, ‎SRM Institute of‎ ‎Science and Technology‎, ‎Kattankulathur‎, ‎Tamil Nadu 603203‎, ‎India

2 Department of Mathematics, School of Arts, Sciences, Humanities, and Education, SASTRA Deemed University, Thanjavur 613401, India

10.22052/ijmc.2025.257071.2022

Abstract

‎Let G be the finite‎, ‎simple‎, ‎and connected graph with a vertex set as V(G) and an edge set as E(G)‎. ‎The harmonic-arithmetic index of graph G is defined as $HA(G) = \sum\limits_{\rho\phi \in E(G)} {\dfrac{{4{d_\rho}{d_\phi}}}{{{{({d_\rho}‎ + ‎{d_\phi})}^2}}}}$ where $d_\rho$ denotes the degree of the vertex $\rho$ and $\rho\phi$ denotes the edge‎. ‎Let $U_{\eta,\mathfrak{g}}$ be the set of unicyclic graphs with $\eta$ vertices and given girth g‎. ‎Let $G_{\eta,\delta}$ be the set of simple connected graphs with $\eta$ vertices with minimum degree $\delta$‎. ‎In this article‎, ‎we present the maximum and second-maximum harmonic-arithmetic index of unicyclic graphs with a given girth and determine their corresponding graphs‎. ‎The obtained results remain valid when the analysis is confined to the class of chemical unicyclic graphs‎. ‎Further‎, ‎we obtain extremal graphs in $G_{\eta‎, ‎\delta}$ for which the HA index reaches its smallest value‎, ‎or we provide a lower bound‎, ‎for $\delta \geq\left\lceil \delta_0 \right\rceil$‎, ‎with $\delta_0 = p_0(\eta-1)$‎, ‎where $p_0 \approx 0.23606$ is the distinct positive root of the expression p^2‎ + ‎4p‎ -‎1 =0‎. ‎We demonstrate that the extremal graphs are regular graphs of degree $\delta$ when $\delta$ or $\eta$ is even‎.

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References

[1] J. A. Bondy and U. S. R. Murty, Graph Theory, Springer, New York, 2008.
[2] G. Chartrand, L. Lesniak and P. Zhang, Graphs and Digraphs, Wadsworth and Brooks/Cole, California, 2016.
[3] F. Harary, Graph Theory, Addison-Wesley, Boston, 1969.
[4] N. Trinajstic, Chemical Graph Theory, CRC Press, Boca Raton, 1992.
[5] A. V. Sadybekov and V. Katritch, Computational approaches streamlining drug discovery, Nature 616 (2023) 673–685, https://doi.org/10.1038/s41586-023-05905-z.
[6] S. Wagner and H. Wang, Introduction to Chemical Graph Theory, CRC Press, Boca Raton, New York, 2018.
[7] A. Ali and D. Dimitrov, On the extremal graphs with respect to bond incident degree indices, Discrete Appl. Math. 238 (2018) 32–40, https://doi.org/10.1016/j.dam.2017.12.007.
[8] D. Vukicevic and J. Durdevic, Bond additive modelling 10. Upper and lower bounds of bond incident degree indices of catacondensed fluoranthenes, Chem. Phys. Lett. 515 (2011) 186–189, https://doi.org/10.1016/j.cplett.2011.08.095.
[9] I. Gutman, Degree-based topological indices, Croat. Chem. Acta. 86 (2013) 351–361, http://dx.doi.org/10.5562/cca2294.
[10] B. Hollas, The covariance of topological indices that depend on the degree of a vertex, MATCH Commun. Math. Comput. Chem. 54 (2005) 177–187.
[11] H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1947) 17–20, https://doi.org/10.1021/ja01193a005 .
[12] S. Hayat, M. Imran and J.-B. Liu, An efficient computational technique for degree and distance based topological descriptors with applications, IEEE Access 7 (2019) 32276–32296, https://doi.org/10.1109/ACCESS.2019.2900500.
[13] S. Fajtlowicz, On conjectures of Graffiti-II, Congr. Numer. 60 (1987) 187–197.
[14] D. Vukicevic and M. Gasperov, Bond additive modelling 1. Adriatic Indices, Croat. Chem. Acta 83 (2010) 243–260.
[15] M. Ghorbani, S. Zangi and N. Amraei, New results on symmetric division deg index, J. Appl. Math. Comput. 65 (2021) 161–176, https://doi.org/10.1007/s12190-020-01386-9.
[16] A. M. Albalahi, A. Ali, A. M. Alanazi, A. A. Bhatti and A. E. Hamza, Harmonic-Arithmetic Index of (Molecular) Trees, Contrib. Math. 7 (2023) 41–47, https://doi.org/10.47443/cm.2023.008.
[17] A. Ali, E. Milovanovic, S. Stankov, M. Matejic and I Milovanovic, Inequalities involving the harmonic-arithmetic index, Afr. Mat. 35 (2024) #46, https://doi.org/10.1007/s13370-024-01183-8.
[18] R. Kalpana and L. Shobana, Harmonic-arithmetic index: Lower bound for n-order trees with fixed number of pendant vertices and monogenic semigroup for graph operations, Int. J. Math. Math. Sci. 2025 (2025) #5576652, https://doi.org/10.1155/ijmm/5576652 .
[19] K. Ramesh and S. Loganathan, Harmonic-arithmetic index: Trees with maximum degrees and comparative analysis of antidrugs, Symmetry, 17 (2025) #167, https://doi.org/10.3390/sym17020167.
[20] G. Ding, L. Zhong and X. Wang, Harmonic-arithmeticiIndex of unicyclic graphs, Iranian J. Math. Chem. 16 (2025) 93 – 108, https://doi.org/ 10.22052/IJMC.2024.255486.1909.
[21] P. D. Namidass and S. Loganathan, Harmonic–arithmetic index for the generalized mycielskian graphs and graphenes with curvilinear regression models of Benzenoid hydrocarbons, Comput. Math. Method 2025 (2025) #6402353, https://doi.org/10.1155/cmm4/6402353.
[22] G. Caporossi, I. Gutman and P. Hansen, Variable neighborhood search for extremal graphs: IV: Chemical trees with extremal connectivity index, Comput. Chem. 23 (1999) 469–477, https://doi.org/10.1016/S0097-8485(99)00031-5 .
[23] Lj. Pavlovic and I. Gutman, Graphs with extremal connectivity index, Novi Sad J. Math. 31 (2001) 53–58.
[24] S. Nagarajan and A. Vijaya, Maximum modified Sombor index of unicyclic graphs with given girth, Iranian J. Math. Chem. 15 (2024) 117–122, https://doi.org/10.22052/IJMC.2023.253810.1780.
[25] P. Nithya, S. Elumalai, S. Balachandran, A. Ali, Z. Raza and A. A. Attiya, On unicyclic graphs with a given girth and their minimum symmetric division deg index, Discrete Math. Lett. 13 (2024) 135–142, https://doi.org/10.47443/dml.2024.123.
[26] L. Zhong and Q. Cui, The harmonic index for unicyclic graphs with given girth, Filomat 29 (2015) 673–686, https://doi.org/10.2298/FIL1504673Z.
[27] T. Divnic, M. Milivojevic and L. Pavlovic, Extremal graphs for the geometricarithmetic index with given minimum degree, Discrete Appl. Math. 162 (2014) 386–390, https://doi.org/10.1016/j.dam.2013.08.001.
Iranian Journal of Mathematical Chemistry
Volume 17, Issue 1
In Progress
March 2026
Pages 35-51

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