On Irregularity Indices for Fractal and Cayley Tree Type Dendrimers
Abstract
Abstract Views: 0
Let be a simple connected (molecular) graph with and as the vertex and edge sets respectively. A graph is supposed to be regular if all vertices have equal degree, otherwise irregular. The fractal and cayley trees are irregular acyclic and connected graphs which are widely used to develop signal amplifiers for biosensors, cellular imaging and genetic engineering. The topological index (TI) serves as a mathematical function for determining numerical values of molecular graphs, aiding in the prediction of diverse physical, chemical, biological, thermodynamic, and structural properties. An irregular index, a specific type of TI, quantifies the irregularity of atomic bonding within chemical compounds represented by the graphs under analysis. This study focuses on calculating the irregularity indices for fractal dendrimers and Cayley tree-type dendrimers. A comparative analysis of the obtained indices is conducted using their numerical values and 3D visualizations. Lastly, the most efficient and consistent irregularity indices for fractal and Cayley tree dendrimers are identified and discussed.
Keywords:Topological descriptors; Irregularity indices, Fractal dendrimers and Cayley tree dendrimers.
MSC (2020) Subject Classification: 05C09; 05C92
Downloads
References
Bagley RL, Torvik PJ. Fractional calculus in the transient analysis of viscoelastically damped structures. AIAA J. 1985;23(6):918-925.
Baillie RT. Long memory processes and fractional integration in econometrics. J Econom. 1996;73(1):5-59.
Talib I, Belgacem FBM, Asif NA, Khalil H. On mixed derivatives type high dimensional multi-term fractional partial differential equations approximate solutions. In: AIP Conference Proceedings. Vol 1798, No. 1. AIP Publishing LLC; 2017:020024.
Estrada E. Randic index, irregularity and complex biomolecular networks. Acta Chim Slov. 2010;57:597-603.
Bell FK. A note on the irregularity of graphs. Linear Algebra Appl. 1992;161:45-54.
Gutman I. Irregularity of molecular graphs. Kragujev J Sci. 2016;38:71-78.
Majcher Z, Michael J. Highly irregular graphs with extreme numbers of edges. Discret Math. 1997;164:237-242.
Liu F, Zhang Z, Meng J. The size of maximally irregular graphs and maximally irregular triangle-free graphs. Graphs Comb. 2014;30:699-705.
Abdo H, Brandt S, Dimitrov D. The total irregularity of a graph. Discrete Math Theor Comput Sci. 2014;16:201-206.
Abdo H, Dimitrov D. The total irregularity of graphs under graph operations. Miskolc Math Notes. 2014;15:3-17. (Corrected page range “317” → “3-17”)
Wiener H. Structural determination of paraffin boiling points. J Am Chem Soc. 1947;69(1):17-20.
Zahid I, Aslam A, Ishaq M, Aamir M. Characteristic study of irregularity measures of some nanotubes. Can J Phys. 2019. doi:10.1139/cjp-2018-0619.
Gao W, Aamir M, Iqbal Z, Ishaq M, Aslam A. On irregularity measures of some dendrimer structures. Mathematics. 2019;7:271.
Yang B, Munir M, Rafique S, Ahmad H, Liu JB. Computational analysis of imbalance-based irregularity indices of boron nanotubes. Processes. 2019;7(10):678.
D. Dimitrov D, Rti T. Graphs with equal irregularity indices. Acta Polytech Hung. 2014;11(4):41-57.
Horoldagva B, Buyantogtokh L, Dorjsembe S, Gutman I. Maximum size of maximally irregular graphs. MATCH Commun Math Comput Chem. 2016;76(1):81-98.
Liu F, Zhang Z, Meng J. The size of maximally irregular graphs and maximally irregular triangle-free graphs. Graphs Combin. 2014;30(3):699-705.
Albertson MO. The irregularity of a graph. Ars Combin. 1997;46(1):219-225.
Dimitrov D, Abdo H. [Title missing.] Theor Comput Sci. 2014;16:6-201
Vukićević D, Graovac A. Valence connectivity versus Randić, Zagreb and modified Zagreb index: A linear algorithm to check discriminative properties of indices in acyclic molecular graphs. Croat Chem Acta. 2004;77(3):501-508.
Abdo H, Dimitrov D. The total irregularity of graphs under graph operations. Miskolc Math Notes. 2014;15(1):3-17.
Abdo H, Dimitrov D. The irregularity of graphs under graph operations. Discuss Math Graph Theory. 2014;34(2):263-278.
Gutman I. Topological indices and irregularity measures. J Bull. 2018;8:469-475.
Basak SC, Magnuson VR, Niemi GJ, Regal RR, Veith GD. Topological indices: their nature, mutual relatedness and applications. Math Model. 1987;8(1):5-300.
Imran M, Baig AQ, Khalid W. On topological indices of fractal and Cayley tree type dendrimers. Discrete Dyn Nat Soc. 2018.
Gutman I, Trinajstić N. Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons. Chem Phys Lett. 1972;17(4):528-535.
Rti T, Sharafdini R, Dregelyi-Kiss A, Haghbin H. Graph irregularity indices used as molecular descriptors in QSPR studies. MATCH Commun Math Comput Chem. 2018;79:24-509.
Manimaran A. Topological analysis of fractal binary and ternary trees. Contemp Math. 2025:715-729.
Hamanaka S, Iliasov AA, Neupert T, Bzdušek T, Yoshida T. Multifractal statistics of non-Hermitian skin effect on the Cayley tree. Phys Rev B. 2025;111(7):075162.
Pannipitiya DN. A dynamical approach to the Potts model on Cayley tree. [dissertation]. West Lafayette, IN: Purdue University; 2024.
