In this article, we derive the closed formulas for a few standard graphs for $(GH)$ and $(GS)$ indices and then go on to calculate the lowest and the greatest general harmonic index, as well as the general sum-connectivity index, for various graphs that correspond to their total graphs.
📌 3 Maddede Bu Makale
🎯
Amaç & Hipotez:In this paper, we discuss the concept of total graph and computed some topological indices.
📊
Temel Bulgular & Veri:We define the general harmonic $(GH)$ index and general sum connectivity $(GS)$ index for graph $\Theta$ regarding incident vertex-edge degrees as: $H^{\alpha}(\Theta)=\sum_{e\acute{u}}\big(\frac{2}{\aleph_{\acute{u}}+\aleph_{e}}\big)^{\alpha}$ and $\hat{\chi} ^{\alpha}(\Theta)=\sum_{e\acute{u}}(\aleph_{\acute{u}}+\aleph_{e})^{\alpha}$, where $\alpha$ is any real number.
💡
Sonuç & Katkı:In this article, we derive the closed formulas for a few standard graphs for $(GH)$ and $(GS)$ indices and then go on to calculate the lowest and the greatest general harmonic index, as well as the general sum-connectivity index, for various graphs that correspond to their total graphs.
🔬Tür:Research Article
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:general harmonic index · general sum connectivity index · incident
Abstract
In this paper, we discuss the concept of total graph and computed some topological indices. If $\Theta$ is a simple graph, then the elements of $\Theta$ are the vertices $\Theta_V$ and edges $\Theta_E$. For $ e=u\acute{u}\in \Theta_E$, the vertex $u$ and edge $e$, as well as $\acute{u}$ and $e$, are incident. We define the general harmonic $(GH)$ index and general sum connectivity $(GS)$ index for graph $\Theta$ regarding incident vertex-edge degrees as: $H^{\alpha}(\Theta)=\sum_{e\acute{u}}\big(\frac{2}{\aleph_{\acute{u}}+\aleph_{e}}\big)^{\alpha}$ and $\hat{\chi} ^{\alpha}(\Theta)=\sum_{e\acute{u}}(\aleph_{\acute{u}}+\aleph_{e})^{\alpha}$, where $\alpha$ is any real number. In this article, we derive the closed formulas for a few standard graphs for $(GH)$ and $(GS)$ indices and then go on to calculate the lowest and the greatest general harmonic index, as well as the general sum-connectivity index, for various graphs that correspond to their total graphs.
Yang, H., Zhang, D., Hanif, M. F., Hanif, M. F., Siddiqui, M. K., Manzoor, S. (2024). On some bounds of degree based topological indices for total graphs. Hacettepe Journal of Mathematics and Statistics, Volume: 53 Issue: 6. https://doi.org/10.15672/hujms.1240245
D. Ami´c, D. Beslo, B. Lucic, S. Nikolic and N. Trinajstic, The vertex-connectivity index revisited, J. Chem. Inf. Comput. Sci. 38 (5), 819-822, 1998.
B. Bollobas and P. Erdös, Graphs of extremal weights, Ars Combin. 50, 225-233, 1998.
D. Bonchev, Chemical graph theory: introduction and fundamentals, CRC Press, 1, 1-200, 1991.
K.C. Das and I. Gutman Some properties of the second Zagreb index, MATCH Commun. Math. Comput. Chem. 52 (1), 104-112, 2004.
Z. Du, B. Zhou, B and N. Trinajsti´c, Minimum sum-connectivity indices of trees and unicyclic graphs of a given matching number, J. Math. Chem. 47 (2), 842-855, 2010.
Z. Du, B. Zhou, B and N. Trinajsti´c, On the general sum-connectivity index of trees, Appl. Math. Lett. 24 (3), 402-405, 2011.
E. Estrada, L. Torres, L. Rodriguez and I. Gutman, An atom-bond connectivity index: modelling the enthalpy of formation of alkanes Indian Journal of Chemistry, 37A, 849- 855, 1998.
S. Fajtlowicz, On conjectures of Graffiti-II Congr. Numer, 60, 187-197, 1987.
O. Favaron, M. Maheo and J.F. Sacle, Some eigenvalue properties in graphs (conjectures of GraffitiII), Discrete Math. 111 (1-3), 197-220, 1993.
W. Gao, H. Wu, M.K. Siddiqui, and A.Q. Baig, Study of biological networks using graph theory Saudi J. Biol. Sci. 25 (6), 1212-1219, 2018.
X. Zhang, X. Wu, S. Akhter, M.K. Jamil, J.B. Liu and M.R. Farahani, Edge-version atom-bond connectivity and geometric arithmetic indices of generalized bridge molecular graphs Symmetry, 10 (12), 751–786, 2018.
X. Zhang, H.M. Awais, M. Javaid, M. and M.K. Siddiqui, Multiplicative Zagreb indices of molecular graphs, J. Chem. 5, 1-19, 2019.
I. Gutman and N. Trinajsti´c, Graph theory and molecular orbitals., Total -electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17, 535-538, 1972.
I. Gutman and B. Furtula, Recent results in the theory of Randi index, University, Faculty of Science. 6, 1-282, 2008.
V.R. Kulli, On K Banhatti indices of graphs, J. Comput. Math Sci. 7 (4), 213-218, 2016.
V.R. Kulli, On K Banhatti indices and K hyper-Banhatti indices of V-Phenylenic nanotubes and nanotorus, J. Comput. Math Sci. 7 (6), 302-307, 2016.
X. Zhang, A. Rauf, M. Ishtiaq, M.K. Siddiqui and M.H. Muhammad, On Degree Based Topological Properties of Two Carbon Nanotubes, Polycyclic Aromatic Compounds, 10, 22-35, 2020.
X. Zhang, H. Jiang, J.B. Liu and Z. Shao, The cartesian product and join graphs on edge-version atom-bond connectivity and geometric arithmetic indices, Molecules, 23 (7), 1-17, 2018.
V.R. Kulli, On K Banhatti indices and K hyper-Banhatti indices of V-Phenylenic nanotubes and nanotorus, J. Comput. Math Sci. 7 (6), 302-307, 2016.
V.R. Kulli, New K Banhatti topological indices, International J. Fuzzy Math. Arch. 12 (1), 29-37, 2017.
X. Li and Y. Shi, A survey on the Randi´c index, MATCH Commun. Math. Comput. Chem, 59 (1), 127-156, 2008.
M. Randic Characterization of molecular branching, J. Am. Chem. Soc. 97 (23), 6609-6615, 1975.
M.K. Siddiqui, M. Imran and A. Ahmad, On Zagreb indices, Zagreb polynomials of some nanostar dendrimers, Appl. Math. Comput. 280, 132-139, 2016.
J.B. Liu, C. Wang, S. Wang and B. Wei, Zagreb indices and multiplicative Zagreb indices of eulerian graphs, Bull. Malays. Math. Sci. Soc. 42 (1), 67-78, 2019.
J.B. Liu, J. Zhao, H. He and Z. Shao, Valency-based topological descriptors and structural property of the generalized sierpiski networks, J. Stat. Phys. 177 (6), 1131-1147, 2019.
X. Zhang, M. Naeem, A.Q. Baig and M.A. Zahid, Study of Hardness of Superhard Crystals by Topological Indices, J. Chem. 10, 7-20, 2021.
X. Zhang, M.K. Siddiqui, S. Javed, L. Sherin, F. Kausar and M.H. Muhammad, Physical analysis of heat for formation and entropy of Ceria Oxide using topological indices, Comb. Chem. High Throughput Screen, 25 (3), 441-450, 2022.
J.B. Liu, J. Zhao, J. Min and J. Cao, The Hosoya index of graphs formed by a fractal graph, Fractals, 27 (8), 195-215, 2019.
H.Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1), 17-20, 1947.
L. Yan, W. Gao and J. Li, General harmonic index and general sum connectivity index of polyomino chains and nanotubes, J. Comput. Theor. Nanoscience, 12 (10), 3940-3944, 2015.
L. Zhong, The harmonic index for graphs, Appl. Math. Lett. 25 (3), 561-566, 2012.
L. Zhong, The harmonic index on unicyclic graphs. Ars Combin. 104, 261-269, 2012.
B. Zhou and N. Trinajstic, On a novel connectivity index, J. Math. Chem. 46 (4), 1252-1270, 2009.
B. Zhou and N. Trinajstic, On general sum-connectivity index, J. Math. Chem. 47 (1), 210-218, 2010.
Z. Zhu and H. Lu, On the general sum-connectivity index of tricyclic graphs, J. Appl. Math. Comput. 51 (1-2), 177-188, 2016.