The structure of $k$-Lucas cubes

Year 2021, , 754 - 769, 07.06.2021

Ömer Eğecioğlu , Elif Saygı , Zülfükar Saygı

https://doi.org/10.15672/hujms.750244

Cited By: 2

Abstract

Fibonacci cubes and Lucas cubes have been studied as alternatives for the classical hypercube topology for interconnection networks. These families of graphs have interesting graph theoretic and enumerative properties. Among the many generalization of Fibonacci cubes are $k$-Fibonacci cubes, which have the same number of vertices as Fibonacci cubes, but the edge sets determined by a parameter $k$. In this work, we consider $k$-Lucas cubes, which are obtained as subgraphs of $k$-Fibonacci cubes in the same way that Lucas cubes are obtained from Fibonacci cubes. We obtain a useful decomposition property of $k$-Lucas cubes which allows for the calculation of basic graph theoretic properties of this class: the number of edges, the average degree of a vertex, the number of hypercubes they contain, the diameter and the radius.

Keywords

hypercube, Fibonacci cube, Lucas cube, k-Fibonacci cube, Fibonacci number, Lucas number

Project Number

117R032

References

• [1] B. Brešar, S. Klavžar and R. Škrekovski, The cube polynomial and its derivatives: the case of median graphs, Electron. J. Combin. 10, #R3, 2003.
• [2] Ö. Eğecioğlu, E. Saygı and Z. Saygı, k-Fibonacci cubes: A family of subgraphs of Fibonacci cubes, Int. J. Found. Comput. Sci. 31 (5), 639–661, 2020.
• [3] S. Gravier, M. Mollard, S. Špacapan and S.S. Zemljič, On disjoint hypercubes in Fibonacci cubes, Discrete Appl. Math. 190-191, 50–55, 2015.
• [4] W.-J. Hsu, Fibonacci cubes–a new interconnection technology, IEEE Trans. Parallel Distrib. Syst. 4 (1), 3–12, 1993.
• [5] A. Ilić, S. Klavžar and Y. Rho, Generalized Fibonacci cubes, Discrete Math. 312, 2–11, 2012.
• [6] A. Ilić, S. Klavžar and Y. Rho, Generalized Lucas cubes, Appl. Anal. Discrete Math. 6 (1), 82–94, 2012.
• [7] C. Kimberling, The Zeckendorf array equals the Wythoff array, The Fibonacci Quar- terly 33, 3–8, 1995.
• [8] S. Klavžar, Structure of Fibonacci cubes: a survey, J. Comb. Optim. 25, 505–522, 2013.
• [9] S. Klavžar and M. Mollard, Cube polynomial of Fibonacci and Lucas cube, Acta Appl. Math. 117, 93–105, 2012.
• [10] S. Klavžar and M. Mollard, Asymptotic properties of Fibonacci cubes and Lucas cubes, Ann. Comb. 18, 447–457, 2014.
• [11] M. Mollard, Non covered vertices in Fibonacci cubes by a maximum set of disjoint hypercubes, Discrete Appl. Math. 219, 219–221, 2017.
• [12] E. Munarini, Pell graphs, Discrete Math. 342 (8), 2415–2428, 2019.
• [13] E. Munarini, C.P. Cippo and N. Zagaglia Salvi, On the Lucas cubes, Fibonacci Quart. 39, 12–21, 2001.
• [14] E. Saygı and Ö. Eğecioğlu, Counting disjoint hypercubes in Fibonacci cubes, Discrete Appl. Math. 215, 231–237, 2016.
• [15] E. Saygı and Ö. Eğecioğlu, q-cube enumerator polynomial of Fibonacci cubes, Discrete Appl. Math. 226, 127–137, 2017.
• [16] E. Saygı and Ö. Eğecioğlu, q-counting hypercubes in Lucas cubes, Turk. J. Math. 42, 190–203, 2018.
• [17] X. Wang, X. Zhao and H. Yao, Structure and enumeration results of matchable Lucas cubes, Discrete Appl. Math. 277, 263–279, 2020.