Domination on cactus chains of pentagons
Abstract
Introduction/purpose: A graph as a mathematical object occupies a special place in science. Graph theory is increasingly used in many spheres of business and scientific fields. This paper analyzes pentagonal cactus chains, a special type of graphs composed of pentagonal cycles in which two adjacent cycles have only one node in common. The aim of the research is to determine the dominant set and the dominance number on ortho and meta pentagonal cactus chains.
Methods: When the corresponding destinations are treated as graph nodes and the connections between them as branches in the graph, the complete structure of the graph is obtained, to which the laws of graph theory are applied. The vertices of the pentagon are treated as nodes of the graph and the sides as branches in the graph. By applying mathematical methods, the dominance was determined on one pentagon, then on two pentagons with a common node, and then on ortho and meta pentagonal cactus chains.
Results: The research has shown that the dominance number on the ortho chain Oh of the length h ≥ 2 is equal to the value of the expression [3h/2] while on the meta chain Mh it is equal to the value of the expression h+1, which was proven in this paper.
Conclusion: The results show that the dominant sets and the dominance numbers on ortho and meta pentagonal cactus chains are determined and explicitly expressed by mathematical expressions. They also point to the possibility of their application in the fields of science as well as in the spheres of business in which these structures appear.
References
Bakhshesh, D. 2022. Isolate Roman domination in graphs. Discrete Mathematics, Algorithms and Applications, 14(3), art.number:2150131. Available at: https://doi.org/10.1142/S1793830921501317
Balaban, A.T. 1985. Applications of graph theory in chemistry. Journal of chemical information and computer sciences, 25(3), pp.334-343. Available at: https://doi.org/10.1021/ci00047a033
Balaban, M. & Zeljković, S. 2021. HEMIJA Teorija i eksperimenti. Banja Luka, Republic of Srpska, Bosnia and Herzegovina: University of Banja Luka, Faculty of natural sciences and mathematics [online]. Available at: https://hemija.pmf.unibl.org/wp-content/uploads/2021/07/Balaban_Zeljkovic_Hemija_Teorija-i-eksperimenti.pdf (in Serbian) [Accessed: 20 February 2022]. ISBN: 978-99955-21-91-2.
Carević M.M. 2021. Dominating Number on Icosahedral-Hexagonal Network. Mathematical Problems in Engineering, art.ID:6663389. Available at: https://doi.org/10.1155/2021/6663389
Carević, M.M., Petrović, M. & Denić, N. 2020. Dominating sets on the rhomboidal cactus chains and the icosahedral network. In: 19th International Symposium INFOTEH-Jahorina, Jahorina, pp.152-157, March 18-20 [online]. Available at: https://infoteh.etf.ues.rs.ba/zbornik/2020/radovi/P-4/P-4-2.pdf [Accessed: 20 February 2022].
Farrell, E.J. 1987. Matchings in hexagonal cacti. International Journal of Mathematics and Mathematical Sciences, 10(art.ID:234184), pp.321-338. Available at: https://doi.org/10.1155/S0161171287000395
Ghergu, M. & Radulescu, V. 2012. Nonlinear PDEs: Mathematical Models in Biology, Chemistry and Population Genetics. Berlin Heidelberg: Springer-Verlag. ISBN 13: 9783642226632.
Gupta, S., Singh, M. & Madan, A.K. 2001. Applications of graph theory: Relationship of molecular connectivity index and atomic molecular connectivity index with anti-HSV activity. Journal of Molecular Structure: THEOCHEM, 571(1-3), pp.147-152. Available at: https://doi.org/10.1016/S0166-1280(01)00560-7
Gupta, S., Singh, M. & Madan, A.K. 2002. Application of Graph Theory: Relationship of Eccentric Connectivity Index and Wiener's Index with Anti-inflammatory Activity. Journal of Mathematical Analysis and Applications, 266(2), pp.259-268. Available at: https://doi.org/10.1006/jmaa.2000.7243
Hajian, M. & Rad, N.J. 2021. Fair Total Domination Number in Cactus Graphs. Discussiones Mathematicae Graph Theory, 41, pp.647-664. Available at: https://doi.org/10.7151/DMGT.2225
Hernández Mira, F.A., Parra Inza, E., Almira, J.M. S. & Vakhania, N. 2021. Properties of the Global Total k-Domination Number. Mathematics, 9(5), art.ID:480. Available at: https://doi.org/10.3390/math9050480
Husimi, K. 1950. Note on Mayers' theory of cluster integrals. The Journal of Chemical Physics, 18(5), pp.682-684. Available at: https://doi.org/10.1063/1.1747725
Klobučar,A. & Klobučar, A. 2019. Total and Double Total Domination Number on Hexagonal Grid. Mathematics, 7(11), art.number:1110. Available at: https://doi.org/10.3390/math7111110
Majstorovic, S., Doslic, T. & Klobucar, A. 2012. K-Domination on hexagonal cactus chains. Kragujevac Journal of Mathematics, 36(2), pp.335-347 [online] Available at: https://imi.pmf.kg.ac.rs/kjm/pub/13569261514726_kjom3602-17.pdf [Accessed: 20 February 2022]
Mihalić, Z. & Trinajstić, N. 1992. A graph-theoretical approach to structure-property relationships. Journal of Chemical Education, 69(9), art.ID:701. Available at: https://doi.org/10.1021/ed069p701
Riddell, R.J. 1951. Contributions to the theory of condensation. Ph.D. thesis. University of Michigan ProQuest Dissertations Publishing [online]. Available at: https://www.proquest.com/openview/4c69a76aaebdf43a91617e8dc2be8fe6/1?pq-origsite=gscholar&cbl=18750&diss=y [Accessed: 20 February 2022].
Sharma, V., Goswami, R. & Madan, A. K. 1997. Eccentric connectivity index: A novel highly discriminating topological descriptor for structure-property and structure-activity studies. Journal of chemical information and computer sciences, 37(2), pp.273-282. Available at: https://doi.org/10.1021/ci960049h
Veličković, J., Arsić, N.B. & Stošić, L.T. 2020. The Efficiency of Galvanic Wastewater Treatment Facility ‘Frad‘ in Aleksinac. Trendovi u poslovanju, 8(2), pp.78-85. Available at: https://doi.org/10.5937/trendpos2002078V
Vladimirovich, G.S. & Vasilyevich-Chernyaev, M. 2021. The experience of applying mathematical methods for analysis of the microgeneration sector in Russia. International Review, (1-2), pp.153-160. Available at: https://doi.org/10.5937/intrev2102156V
Vukičević, D. & Klobučar, A. 2007. K-Dominating sets on linear benzenoids and on the infinite hexagonal grid. Croatica Chemica Acta, 80(2), pp.187-191 [online]. Available at: https://hrcak.srce.hr/12849 [Accessed: 20 February 2022].
Zmazek, B. & Zerovnik, J. 2005. Estimating the traffic on weighted cactus networks in linear time. In: Ninth International Conference on Information Visualisation (IV'05), London, UK, pp.536-541, July 6-7. Available at: https://doi.org/10.1109/IV.2005.48
Zmazek, B. & Žerovnik, J. 2003. Computing the weighted Wiener and Szeged number on weighted cactus graphs in linear time. Croatica Chemica Acta, pp.137-143 [online]. Available at: https://hrcak.srce.hr/103089 [Accessed: 20 February 2022].
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