On Vertex-Degree-Based Topological Indices of Cayley Digraphs of Rectangular Groups and Their Role as Molecular Descriptors
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Keywords:
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Vertex Degree, Topological Indices, Cayley Digraphs, Rectangular Proups
Abstract
Let Cay(S,A) be the Cayley digraph of a finite rectangular group S with connection set A. We derive explicit formulas for several vertex-degree–based topological indices of these digraphs, including the Randić, Zagreb, sum-connectivity, geometric–arithmetic, atom–bond connectivity, and harmonic indices. The computations are reduced to Cayley digraphs of right groups, which simplifies the analysis. The underlying graphs are shown to be isomorphic to the hydrogen–included molecular graphs of cycloalkanes CnH2n. We prove that all considered indices depend linearly on the ring size n, exhibit stable growth, and differ only in their growth rates. Comparisons between related graph models illustrate the dependence of these indices on the underlying structure and support their interpretation as molecular descriptors, with possible relevance to QSAR/QSPR studies.
References
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Gao, W. (2024). The Extremal Trees for Exponential Vertex Degree-Based Indices. MATCH Communications in Mathematical and in Computer Chemistry, 92(1); 189–204.
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Guan, C., M. Lu, W. Zeng, D. Yang, and D. Han (2020). Prediction of Standard Enthalpies of Formation Based on Hydrocarbon Molecular Descriptors and Active Subspace Methodology. Industrial and Engineering Chemistry Research, 59; 4785–4791.
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ichi Aihara, J. (1996). Bond Resonance Energies of Polycyclic Benzenoid and Non-Benzenoid Hydrocarbons. Journal of the Chemical Society, Perkin Transactions 2; 2185–2195.
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Liu, F., Y. Liang, C. Cao, and N. Zhou (2007). Theoretical Prediction of the Kovat’s Retention Index for Oxygen Containing Organic Compounds Using Novel Topological Indices. Analytica Chimica Acta, 594; 279–289.
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Meksawang, J. and S. Panma (2016). Isomorphism Conditions for Cayley Graphs of Rectangular Groups. Bulletin of the Malaysian Mathematical Sciences Society, 39; 29–41.
Monsalve, J. and J. Rada (2021). Vertex-Degree Based Topological Indices of Digraphs. Discrete Applied Mathematics, 295; 13–24.
Nupo, N. and S. Panma (2018). Independent Domination Number in Cayley Digraphs of Rectangular Groups. Discrete Mathematics, Algorithms and Applications, 10(2).
Panma, S. (2010). Characterization of Cayley Graphs of Rectangular Groups. Thai Journal of Mathematics, 8; 535–543.
Panma, S., U. Knauer, and S. Arworn (2004). On Transitive Cayley Graphs of Right (Left) Groups and of Clifford Semigroups. Thai Journal of Mathematics, 2; 183–195.
Pompe, M. and M. Novic (1999). Prediction of Gas Chromatographic Retention Indices Using Topological Descriptors. Journal of Chemical Information and Computer Sciences, 39; 59–67.
Repnjak, M. C., N. Tratnik, and P. Z. Pleteršek (2018). Predicting Melting Points of Hydrocarbons by the Graovac-Pisanski Index. Fullerenes, Nanotubes and Carbon Nanostructures, 26; 239–245.
Rouvray, D. H. and W. Tatong (1989). Novel Applications of Topological Indices. 3. Prediction of the Vapor Pressure in Polychlorinated Biphenyls. International Journal of Environmental Studies, 33; 247–257.
Ruangnai, M., S. Panma, and S. Arworn (2012). On Cayley Isomorphisms of Left and Right Groups. International Journal of Pure and Applied Mathematics, 80; 561–571.
Rücker, G. and C. Rücker (1999). On Topological Indices, Boiling Points, and Cycloalkanes. Journal of Chemical Information and Computer Sciences, 39; 788–802.
Sheridan, P. F., D. B. Adolf, A. V. Lyulin, I. Neelov, and G. R. Davies (2002). Computer Simulations of Hyperbranched Polymers: The Influence of the Wiener Index on the Intrinsic Viscosity and Radius of Gyration. Journal of Chemical Physics, 117; 7802–7812.
Skvortsova, M. I., I. I. Baskin, O. L. Slovokhotova, V. A. Palyulin, and N. S. Zefirov (1993). Inverse Problem in QSAR/QSPR Studies for the Case of Topological Indexes Characterizing Molecular Shape (Kier Indices). Journal of Chemical Information and Computer Sciences, 33; 630–634.
Trinajstić, N. (2018). Chemical Graph Theory. Routledge, Abingdon.
West, D. B. (1996). Introduction to Graph Theory. Prentice Hall, Upper Saddle River.
Zhao, B., J. Gan, and H. Wu (2016). Redefined Zagreb Indices of Some Nano Structures. Applied Mathematics and Nonlinear Sciences, 1; 291–300.
Zhou, L. L., J. C. Jiang, Y. Pan, and Z. R. Wang (2015). A Mathematical Method for Predicting Heat of Reaction of Organic Peroxides. Journal of Loss Prevention in the Process Industries, 38; 254–259.
Basak, S. C., G. J. Niemi, and G. D. Veith (1991). Predicting Properties of Molecules Using Graph Invariants. Journal of Mathematical Chemistry, 7; 243–272.
Biggs, N. (1993). Algebraic Graph Theory. Cambridge University Press, Cambridge.
Cioslowski, J. (1985). Additive Nodal Increments for Approximate Calculation of the Total Π-Electron Energy of Benzenoid Hydrocarbons. Theoretica Chimica Acta, 68; 315–319.
Čomić, S. (2024). Vertex-Degree-Based Topological Indices of Arbitrary Sets of Graphs. Applied Mathematics and Computation, 466; 128543.
Cruz, R. (2024). A Matrix Approach to Vertex-Degree-Based Topological Indices. MATCH Communications in Mathematical and in Computer Chemistry, 92(3); 623–646.
Deng, H., Z. Tang, J. Yang, J. Yang, and M. You (2022). On the Vertex-Degree Based Invariants of Digraphs. Discrete Mathematics Letters, 9; 2–9.
Devillers, J. and A. T. Balaban, editors (2000). Topological Indices and Related Descriptors in QSAR and QSPAR. CRC Press, Boca Raton.
Eliasi, M. and D. Vukičević (2013). Comparing the Multiplicative Zagreb Indices. MATCH Communications in Mathematical and in Computer Chemistry, 69; 765–773.
Emmert-Streib, F. (2012). Statistical Modelling of Molecular Descriptors in QSAR/QSPR. Wiley, Hoboken.
Furtula, B., A. Graovac, and D. Vukičević (2010). Augmented Zagreb Index. Journal of Mathematical Chemistry, 48; 370–380.
Gao, W. (2024). The Extremal Trees for Exponential Vertex Degree-Based Indices. MATCH Communications in Mathematical and in Computer Chemistry, 92(1); 189–204.
Garcia-Domenech, R., P. Alarcon-Elbal, G. Bolas, R. Bueno Marí, F. A. Chordá-Olmos, S. A. Delacour, M. C. Mouriño, A. Vidal, and J. Gálvez (2007). Prediction of Acute Toxicity of Organophosphorus Pesticides Using Topological Indices. SAR and QSAR in Environmental Research, 18; 745–755.
Guan, C., M. Lu, W. Zeng, D. Yang, and D. Han (2020). Prediction of Standard Enthalpies of Formation Based on Hydrocarbon Molecular Descriptors and Active Subspace Methodology. Industrial and Engineering Chemistry Research, 59; 4785–4791.
Gutman, I. (2013). Degree-Based Topological Indices. Croatica Chemica Acta, 86; 351–361.
Hao, Y. and Y. Luo (2010). On the Cayley Graphs of Left (Right) Groups. Southeast Asian Bulletin of Mathematics, 34; 685–691.
ichi Aihara, J. (1996). Bond Resonance Energies of Polycyclic Benzenoid and Non-Benzenoid Hydrocarbons. Journal of the Chemical Society, Perkin Transactions 2; 2185–2195.
Jeyaraj, S., A. Princy, R. Sundararajan, S. Arumugam, and M. Venkatesan (2023). Efficient Technique for Generating Vertex-Based Topological Indices for Chemical Structures: Application to Boric Acid Sheet. ACS Omega, 8(42); 38977–38986.
Kelarev, A. V. and C. E. Praeger (2003). On Transitive Cayley Graphs of Groups and Semigroups. European Journal of Combinatorics, 24; 59–72.
Khadikar, P. V., J. Singh, and M. Ingle (2007). Topological Estimation of Aromatic Stabilities of Polyacenes and Helicenes: Modeling of Resonance Energy and Benzene Character. Journal of Mathematical Chemistry, 42; 433–446.
Khosravi, B. (2009). On Cayley Graphs of Left Groups. Houston Journal of Mathematics, 35; 745–755.
Khosravi, B. and M. Mahmoudi (2010). On Cayley Graphs of Rectangular Groups. Discrete Mathematics, 310; 804–811.
Knauer, U. (2011). Algebraic Graph Theory. W. de Gruyter, Berlin.
Korinth, G., T. Wellner, K. H. Schaller, and H. Drexler (2012). Potential of the Octanol-Water Partition Coefficient (Log P) to Predict the Dermal Penetration Behaviour of Amphiphilic Compounds in Aqueous Solutions. Toxicology Letters, 215; 49–53.
Liu, F., Y. Liang, C. Cao, and N. Zhou (2007). Theoretical Prediction of the Kovat’s Retention Index for Oxygen Containing Organic Compounds Using Novel Topological Indices. Analytica Chimica Acta, 594; 279–289.
Lučić, B., N. Trinajstić, and B. Zhou (2009). Comparison Between the Sum-Connectivity Index and Product Connectivity Index for Benzenoid Hydrocarbons. Chemical Physics Letters, 475; 146–148.
Maallah, R. L. and A. J. M. Khalaf. Randic and General Randic Indices of Unicyclic Graphs. Saudi Journal of Engineering and Technology.
Meksawang, J. and S. Panma (2016). Isomorphism Conditions for Cayley Graphs of Rectangular Groups. Bulletin of the Malaysian Mathematical Sciences Society, 39; 29–41.
Monsalve, J. and J. Rada (2021). Vertex-Degree Based Topological Indices of Digraphs. Discrete Applied Mathematics, 295; 13–24.
Nupo, N. and S. Panma (2018). Independent Domination Number in Cayley Digraphs of Rectangular Groups. Discrete Mathematics, Algorithms and Applications, 10(2).
Panma, S. (2010). Characterization of Cayley Graphs of Rectangular Groups. Thai Journal of Mathematics, 8; 535–543.
Panma, S., U. Knauer, and S. Arworn (2004). On Transitive Cayley Graphs of Right (Left) Groups and of Clifford Semigroups. Thai Journal of Mathematics, 2; 183–195.
Pompe, M. and M. Novic (1999). Prediction of Gas Chromatographic Retention Indices Using Topological Descriptors. Journal of Chemical Information and Computer Sciences, 39; 59–67.
Repnjak, M. C., N. Tratnik, and P. Z. Pleteršek (2018). Predicting Melting Points of Hydrocarbons by the Graovac-Pisanski Index. Fullerenes, Nanotubes and Carbon Nanostructures, 26; 239–245.
Rouvray, D. H. and W. Tatong (1989). Novel Applications of Topological Indices. 3. Prediction of the Vapor Pressure in Polychlorinated Biphenyls. International Journal of Environmental Studies, 33; 247–257.
Ruangnai, M., S. Panma, and S. Arworn (2012). On Cayley Isomorphisms of Left and Right Groups. International Journal of Pure and Applied Mathematics, 80; 561–571.
Rücker, G. and C. Rücker (1999). On Topological Indices, Boiling Points, and Cycloalkanes. Journal of Chemical Information and Computer Sciences, 39; 788–802.
Sheridan, P. F., D. B. Adolf, A. V. Lyulin, I. Neelov, and G. R. Davies (2002). Computer Simulations of Hyperbranched Polymers: The Influence of the Wiener Index on the Intrinsic Viscosity and Radius of Gyration. Journal of Chemical Physics, 117; 7802–7812.
Skvortsova, M. I., I. I. Baskin, O. L. Slovokhotova, V. A. Palyulin, and N. S. Zefirov (1993). Inverse Problem in QSAR/QSPR Studies for the Case of Topological Indexes Characterizing Molecular Shape (Kier Indices). Journal of Chemical Information and Computer Sciences, 33; 630–634.
Trinajstić, N. (2018). Chemical Graph Theory. Routledge, Abingdon.
West, D. B. (1996). Introduction to Graph Theory. Prentice Hall, Upper Saddle River.
Zhao, B., J. Gan, and H. Wu (2016). Redefined Zagreb Indices of Some Nano Structures. Applied Mathematics and Nonlinear Sciences, 1; 291–300.
Zhou, L. L., J. C. Jiang, Y. Pan, and Z. R. Wang (2015). A Mathematical Method for Predicting Heat of Reaction of Organic Peroxides. Journal of Loss Prevention in the Process Industries, 38; 254–259.
Authors
Pongpipat, D., & Nupo, N. (2026). On Vertex-Degree-Based Topological Indices of Cayley Digraphs of Rectangular Groups and Their Role as Molecular Descriptors. Science and Technology Indonesia, 11(2), 732–741. https://doi.org/10.26554/sti.2026.11.2.732-741
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