Computational Simulation of Equitable Topological Graphs

Hanady F.  Khudhair; Mohammed A. Abdlhusein

Authors

Hanady F. Khudhair Department of Mathematics, College of Education for Pure Sciences, University of Thi-Qar, Thi-Qar, Iraq
Mohammed A. Abdlhusein Department of Mathematics, College of Education for Women, Shatrah University, Thi-Qar, Iraq

Keywords:

Equitable Topological Graph, Topological Graphs, Computational Simulation, Graph Theory, Python, Monte Carlo Methods

Abstract

This study presents a computational simulation of the equitable topological graph G_q, which is constructed from a finite non-empty set endowed with the discrete topology. The vertices of the graph correspond to all non-empty proper subsets of the underlying set, while adjacency is defined according to a complementary-cardinality rule. The main objective of this work is to provide computational verification of the theoretical properties previously established for this class of graphs. An exact python-based algorithm was developed to generate the graph and compute its principal parameters, including the order, size, degree distribution, number of connected components, clique number, radius, and diameter. In addition, growth behavior was analyzed for increasing values of n, and monte carlo sampling techniques were employed to estimate graph size in cases where exact construction becomes computationally expensive. The computational results show complete agreement with the theoretical formulas and structural decomposition of the equitable topological graph. The study confirms that vertex degrees depend solely on subset cardinality and that the graph is connected only for small values of n, becoming disconnected for larger values. Furthermore, the simulations verify the decomposition of G_q into complete bipartite components and, in the even case, an additional complete graph corresponding to the middle cardinality class. Monte carlo experiments produced highly accurate estimates with very small relative errors, demonstrating the effectiveness of sampling methods for large-scale instances. The findings highlight the usefulness of computational techniques in validating theoretical results and investigating the structural properties of topological graphs when direct analytical or manual construction becomes impractical.

References

M. K. Idan and M. A. Abdlhusein, “Constructed discrete topological space from certain topological graphs,” Int. J. Acad. Appl. Res., vol. 7, no. 7, pp. 22–27, 2023.

Z. N. Jwair and M. A. Abdlhusein, “Constructing new topological graph with several properties,” Iraqi J. Sci., vol. 64, no. 6, pp. 2991–2999, Jun. 2023, doi: 10.24996/ijs.2023.64.6.27.

Z. M. Khalil and M. A. Abdlhusein, “New form of discrete topological graphs,” AIP Conf. Proc., vol. 3282, no. 1, Art. no. 040026, 2025, doi: 10.1063/5.0266555.

R. Balakrishnan and K. Ranganathan, A Textbook of Graph Theory. New York, NY, USA: Springer, 2012.

Z. M. Kalil, M. A. Abdlhusein, and M. R. Farahani, “Some dominating applications on discrete topological graphs,” J. Educ. Pure Sci., vol. 15, no. 1, pp. 40–46, 2025, doi: 10.32792/jeps.v15i1.678.

Z. N. Jwair and M. A. Abdlhusein, “Some dominating results of the topological graph,” Int. J. Nonlinear Anal. Appl., vol. 14, no. 2, pp. 133–140, 2023, doi: 10.22075/ijnaa.2022.6404.

M. K. Idan and M. A. Abdlhusein, “Different types of dominating sets of the discrete topological graph,” Int. J. Nonlinear Anal. Appl., vol. 14, no. 1, pp. 101–108, 2023, doi: 10.22075/ijnaa.2022.6452.

M. K. Idan and M. A. Abdlhusein, “Some dominating results of the join and corona operations between discrete topological graphs,” Int. J. Nonlinear Anal. Appl., vol. 14, no. 5, pp. 235–242, 2023, doi: 10.22075/ijnaa.2022.6795.

V. R. Kulli and S. C. Sigarkanti, “Inverse domination in graphs,” Natl. Acad. Sci. Lett., vol. 14, pp. 473–475, 1991.

E. J. Cockayne, R. M. Dawes, and S. T. Hedetniemi, “Total domination in graphs,” Networks, vol. 10, pp. 211–219, 1980.

G. van Rossum and F. L. Drake, Python 3 Reference Manual. Scotts Valley, CA, USA: CreateSpace, 2009.

A. A. Hagberg, D. A. Schult, and P. J. Swart, “Exploring network structure, dynamics, and function using NetworkX,” in Proc. 7th Python in Science Conf. (SciPy 2008), Pasadena, CA, USA, 2008, pp. 11–15.

J. D. Hunter, “Matplotlib: A 2D graphics environment,” Comput. Sci. Eng., vol. 9, no. 3, pp. 90–95, May–Jun. 2007, doi: 10.1109/MCSE.2007.55.

T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms, 4th ed. Cambridge, MA, USA: MIT Press, 2022.

R. Y. Rubinstein and D. P. Kroese, Simulation and the Monte Carlo Method, 3rd ed. Hoboken, NJ, USA: Wiley, 2016, doi: 10.1002/9781118631980.