| | | |

## The structure of $k$-Lucas cubes

#### Ömer EĞECİOĞLU [1] , Elif SAYGI [2] , Zülfükar SAYGI [3]

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.
hypercube, Fibonacci cube, Lucas cube, k-Fibonacci cube, Fibonacci number, Lucas number
• [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.
Primary Language en Mathematics Mathematics Orcid: 0000-0002-6070-761XAuthor: Ömer EĞECİOĞLUInstitution: University of California Santa BarbaraCountry: United States Orcid: 0000-0001-8811-4747Author: Elif SAYGIInstitution: HACETTEPE UNIVERSITYCountry: Turkey Orcid: 0000-0002-7575-3272Author: Zülfükar SAYGI (Primary Author)Institution: TOBB UNIVERSITY OF ECONOMICS AND TECHNOLOGYCountry: Turkey Tübitak 117R032 Publication Date : June 7, 2021
 Bibtex @research article { hujms750244, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2021}, volume = {50}, pages = {754 - 769}, doi = {10.15672/hujms.750244}, title = {The structure of \$k\$-Lucas cubes}, key = {cite}, author = {Eğecioğlu, Ömer and Saygı, Elif and Saygı, Zülfükar} } APA Eğecioğlu, Ö , Saygı, E , Saygı, Z . (2021). The structure of $k$-Lucas cubes . Hacettepe Journal of Mathematics and Statistics , 50 (3) , 754-769 . DOI: 10.15672/hujms.750244 MLA Eğecioğlu, Ö , Saygı, E , Saygı, Z . "The structure of $k$-Lucas cubes" . Hacettepe Journal of Mathematics and Statistics 50 (2021 ): 754-769 Chicago Eğecioğlu, Ö , Saygı, E , Saygı, Z . "The structure of $k$-Lucas cubes". Hacettepe Journal of Mathematics and Statistics 50 (2021 ): 754-769 RIS TY - JOUR T1 - The structure of $k$-Lucas cubes AU - Ömer Eğecioğlu , Elif Saygı , Zülfükar Saygı Y1 - 2021 PY - 2021 N1 - doi: 10.15672/hujms.750244 DO - 10.15672/hujms.750244 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 754 EP - 769 VL - 50 IS - 3 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.750244 UR - https://doi.org/10.15672/hujms.750244 Y2 - 2020 ER - EndNote %0 Hacettepe Journal of Mathematics and Statistics The structure of $k$-Lucas cubes %A Ömer Eğecioğlu , Elif Saygı , Zülfükar Saygı %T The structure of $k$-Lucas cubes %D 2021 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 50 %N 3 %R doi: 10.15672/hujms.750244 %U 10.15672/hujms.750244 ISNAD Eğecioğlu, Ömer , Saygı, Elif , Saygı, Zülfükar . "The structure of $k$-Lucas cubes". Hacettepe Journal of Mathematics and Statistics 50 / 3 (June 2021): 754-769 . https://doi.org/10.15672/hujms.750244 AMA Eğecioğlu Ö , Saygı E , Saygı Z . The structure of $k$-Lucas cubes. Hacettepe Journal of Mathematics and Statistics. 2021; 50(3): 754-769. Vancouver Eğecioğlu Ö , Saygı E , Saygı Z . The structure of $k$-Lucas cubes. Hacettepe Journal of Mathematics and Statistics. 2021; 50(3): 754-769. IEEE Ö. Eğecioğlu , E. Saygı and Z. Saygı , "The structure of $k$-Lucas cubes", Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 3, pp. 754-769, Jun. 2021, doi:10.15672/hujms.750244

Authors of the Article