Perfect Matching of Fractal Honeycomb Meshes

Year 2019, Volume: 4 Issue: 1, 38 - 46, 01.06.2019

Burhan Selçuk , Ayşe Nur Altıntaş Tankül

Abstract

In this paper, we investigated Hamiltonian properties of fractal honeycomb meshes which are created in two different ways using 2-bit gray code. We presented the structure of honeycomb meshes and examined the fractal properties of them and got perfect matching of labeling of nodes in Fractal Honeycomb Meshes for any dimension. Network topology is an illustration of nodes and their connections. There are different types of network topologies and bus, ring, star, tree, mesh, tori and hypercube topologies are the most commonly known network topologies. In this paper, we used honeycomb pattern to construct network topology in fractal-like structure with two variants of honeycomb meshes and examined their Hamilton properties.

Keywords

Fractal honeycomb meshes, Interconnection network, Hamilton graph, Perfect matching, Gray code

Perfect Matching of Fractal Honeycomb Meshes

Abstract

In this paper, we investigated Hamiltonian properties of fractal honeycomb meshes which are created in two different ways using 2-bit gray code. We presented the structure of honeycomb meshes and examined the fractal properties of them and got perfect matching of labeling of nodes in Fractal Honeycomb Meshes for any dimension. Network topology is an illustration of nodes and their connections. There are different types of network topologies and bus, ring, star, tree, mesh, tori and hypercube topologies are the most commonly known network topologies. In this paper, we used honeycomb pattern to construct network topology in fractal-like structure with two variants of honeycomb meshes and examined their Hamilton properties.

Keywords

Fractal honeycomb meshes, Interconnection network, Hamilton graph, Perfect matching, Gray code

References

• Hales T. C. (2001) The Honeycomb Conjecture. Discrete and Computational Geometry 25(1): 1-22.
• Nocetti F. G., Stojmenovic I., Zhang J. (2002) Addressing and Routing in Hexagonal Networks with Applications for Tracking Mobile Users and Connection Rerouting in Cellular Networks. IEEE Trans. Parallel Distrib. Syst. 13(9): 963-971.
• Carle J., Myoupo J. F., Seme D. (1999) All-to-all Broadcasting Algorithms on Honeycomb Networks and Applications. Parallel Process. Lett. 9(4): 539-550.
• Manuel P., Rajan B., Rajasingh I., M C. M. (2008) On Minimum Metric Dimension of Honeycomb Networks. J. Discret. Algorithms 6(1): 20-27.
• Lester L. N., Sandor J. (1985) Computer Graphics on a Hexagonal Grid. Comput. Graph 8(4): 401-409.
• Rajan B., William A., Grigorious C., Stephen S. (2012) On Certain Topological Indices of Silicate , Honeycomb and Hexagonal Networks. J. Comp. Math. Sci 3(5): 530-535.
• Boudjemai A., Amri R., Mankour A., Salem H., Bouanane M. H., Boutchicha D. (2012) Modal Analysis and Testing of Hexagonal Honeycomb Plates Used for Satellite Structural Design. Mater. Des. 35: 266–275.
• Engelmary G. C., Cheng M., Bettinger C. J., Borenstein J. T., Langer R., Freed L. E. (2008) Accordion-like Honeycombs for Tissue Engineering of Cardiac Anisotropy. Nature Materials 7: 1003-1010.
• Mandelbrot B. (2004) Fractals and Chaos : The Mandelbrot Set and Beyond, Springer Science & Business Media, United States of America
• Kluge T. (2000) Fractals in Nature and Applications. https://kluge.in-chemnitz.de/documents/fractal/node2.html. Accessed 13 December 2018.
• Joye Y. (2005) Evolutionary and Cognitive Motivations for Fractal Art in Art and Design Education. International Journal of Art and Design Education 2(2): 175-185.
• Keller J., Chen S. (1989) Texture Description and Segmentation Fractal Geometry. Computer Vison, Graphics, and Image Proccessing 45: 150–166.
• Karci A., Selçuk B. (2014) A new hypercube variant : Fractal Cubic Network Graph. Engineering Science and Technology, An International Journal 18(1): 32-41.
• Meier J., Reiter C. A. (1996) Fractal Representations of Cayley Graphs. Caos & Graphics. 20(1): 163-170.
• Warchalowski W., Krawczyk M. J. (2017) Line Graphs for Fractals. Commun. Nonlinear Sci. Numer. Simul. 44: 506-512.
• Fiala J., Hubička J., Long Y., Nešetřil J. (2017) Fractal Property of the Graph Homomorphism Order. Eur. J. Comb. 66: 101–109.
• Jaggard D. L., Bedrosian S. D. (1987) A Fractal-Graph Approach to Large Networks. Proceeding of IEEE 75 (7): 966–968.
• Montiel M. E., Aguado A. S., Zaluska E. (1995) Topology in Fractals. Caos, Solitons & Fractals 7(8): 1187-1201
• Brown J. I., Hickman C. A., Nowakowski R. J. (2003) The Independence Fractal of a Graph. J. Comb. Theory 87: 209-230.
• Ejov V., Filar J. A., Lucas S. K., Zograf P. (2007) Clustering of Spectra and Fractals of Regular Graphs. J. Math. Anal. Appl. 333: 236-246.
• Science N., Phenomena C., Komjáthy J., Simon K. (2011) Generating Hierarchial Scale-Free Graphs from Fractals. Chaos , Solitons Fractals 44: 651–666, 2011.