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Observability of the extended Fibonacci cubes

Carol Whitehead, Norma Zagaglia Salvi
2003 Discrete Mathematics  
A Fibonacci string of order n is a binary string of length n with no two consecutive ones. The Fibonacci cube n is the subgraph of the hypercube Qn induced by the set of Fibonacci strings of order n.  ...  For positive integers i; n, with n ¿ i, the ith extended Fibonacci cube is the vertex induced subgraph of Qn for which V ( i n ) = V i n is deÿned recursively by with initial conditions V i i = Bi, V i  ...  The extended Fibonacci cubes, introduced by Wu [13] , are constructed by the same recursive relation as the Fibonacci cube, but with di erent initial conditions.  ... 
doi:10.1016/s0012-365x(02)00824-5 fatcat:2nwe4ryoonfuxip2hxrkcfz6da

On median nature and enumerative properties of Fibonacci-like cubes

Sandi Klavžar
2005 Discrete Mathematics  
Fibonacci cubes, extended Fibonacci cubes, and Lucas cubes are induced subgraphs of hypercubes defined in terms of Fibonacci strings. It is shown that all these graphs are median.  ...  Several enumeration results on the number of their edges and squares are obtained. Some identities involving Fibonacci and Lucas numbers are also presented.  ...  Observe also that 0 n = n . The Fibonacci cubes 4 and 5 , the Lucas cube 5 , and the extended Fibonacci cube 1 4 , are, together with the corresponding vertex labels, shown in Fig. 1 .  ... 
doi:10.1016/j.disc.2004.02.023 fatcat:plo2gg4cazdlrklpzkrvwrhpwe

Structure of Fibonacci cubes: a survey

Sandi Klavžar
2011 Journal of combinatorial optimization  
The Fibonacci cube Γ n is the subgraph of the n-cube induced by the binary strings that contain no two consecutive 1s.  ...  The Fibonacci dimension of a graph, studies of graph invariants on Fibonacci cubes, and related classes of graphs are also presented. Along the way some new short proofs are given.  ...  This work was supported in part by the Ministry of Science of Slovenia under the grant P1-0297.  ... 
doi:10.1007/s10878-011-9433-z fatcat:ivvuat6ujvfqrl7ihit6mwspge

A Fault-Tolerant Routing Strategy for Fibonacci-Class Cubes [chapter]

Zhang Xinhua, Peter K. K. Loh
2005 Lecture Notes in Computer Science  
Fibonacci Cubes (FCs), together with the enhanced and extended forms, are a family of interconnection topologies formed by diluting links from binary hypercube.  ...  Finally, the performance of the algorithm is presented and analyzed through software simulation, showing its feasibility.  ...  In Figure 4 , it can be observed that the average latency of regular/Enhanced/Extended Fibonacci Cubes increases when the networks dimension n is below 19.  ... 
doi:10.1007/11572961_18 fatcat:jezrjhcbm5azdiwrv7obzcpagq

On domination-type invariants of Fibonacci cubes and hypercubes

Jernej Azarija, Sandi Klavžar, Yoomi Rho, Seungbo Sim
2017 Ars Mathematica Contemporanea  
The Fibonacci cube Γ n is the subgraph of the n-dimensional cube Q n induced by the vertices that contain no two consecutive 1s.  ...  Using integer linear programming exact values for the 2packing number, connected domination number, paired domination number, and signed domination number of small Fibonacci cubes and hypercubes are obtained  ...  Acknowledgment The authors acknowledge the financial support from the Slovenian Research Agency (research code funding No. P1-0297) and from the Basic Science Research Program through  ... 
doi:10.26493/1855-3974.1172.bae fatcat:zc7x5fgylnhl3guc2dkpvktcp4

Generalized Fibonacci and Lucas cubes arising from powers of paths and cycles

P. Codara, O.M. D'Antona
2016 Discrete Mathematics  
For h=1 such a diagram is called a Fibonacci cube, and for h>1 we obtain a generalization of the Fibonacci cube.  ...  Then, we show that the number of edges of a generalized Fibonacci cube is obtained by convolution of an h-Fibonacci sequence with itself. In the second part we consider the case of cycles.  ...  Acknowledgment We are grateful to the editor for is precious work, and to the anonymous referee for a careful reading of our paper, and for his/her helpful suggestions which allowed us to improve the presentation  ... 
doi:10.1016/j.disc.2015.08.012 fatcat:lhj2hfcvcfblfpadgsqgrtqeau

Fibonacci-run graphs I: basic properties [article]

Ömer Eğecioğlu, Vesna Iršič
2021 arXiv   pre-print
They have the same number of vertices as Fibonacci cubes, but fewer edges and different connectivity properties.  ...  Among the classical models for interconnection networks are hypercubes and Fibonacci cubes.  ...  Sandi Klavžar for enabling the cooperation of the authors of this article, and his early interest in the topic. The first author would like  ... 
arXiv:2010.05518v3 fatcat:sttqzwxoz5c2xcptxmki6rjl4a

Θ-graceful labelings of partial cubes

Boštjan Brešar, Sandi Klavžar
2006 Discrete Mathematics  
The Cartesian product of -graceful partial cubes is again such and we wonder whether in fact any partial cube is -graceful.  ...  It is shown that several classes of partial cubes are -graceful, for instance even cycles, Fibonacci cubes, and (newly introduced) lexicographic subcubes.  ...  For every n i 0, the extended Fibonacci cube i n is a -graceful partial cube.  ... 
doi:10.1016/j.disc.2006.02.013 fatcat:jvwpfe7535dw7bluu5ku7els6e

Cube Polynomial of Fibonacci and Lucas Cubes

Sandi Klavžar, Michel Mollard
2011 Acta Applicandae Mathematicae - An International Survey Journal on Applying Mathematics and Mathematical Applications  
We determine the cube polynomial of Fibonacci cubes and Lucas cubes, as well as the generating functions for the sequences of these cubes.  ...  Zeros of the studied cube polynomials are explicitly determined. Consequently, the coefficients sequences of cube polynomials of Fibonacci and Lucas cubes are unimodal.  ...  Hence understanding the cube polynomial of Fibonacci cubes and how it behaves under the Cartesian multiplication, one can obtain all the corresponding results for the extended Fibonacci cubes as well.  ... 
doi:10.1007/s10440-011-9652-4 fatcat:2c4kow6xkjg7zmt3ekvhaojrqq

On the Wiener index of generalized Fibonacci cubes and Lucas cubes

Sandi Klavžar, Yoomi Rho
2015 Discrete Applied Mathematics  
In this paper the Wiener index of Q d (1 s ) and the Wiener index of Q d ( ↽ 1 s ) are expressed as functions of the order of the generalized Fibonacci cubes.  ...  The generalized Fibonacci cube Q d (f ) is the graph obtained from the d-cube Q d by removing all vertices that contain a given binary word f as a factor; the generalized Lucas cube Q d ( ↽ f ) is obtained  ...  the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology grant 2011-00253195.  ... 
doi:10.1016/j.dam.2015.02.002 fatcat:263xpci5hvhupc7zj3hzq7go7i

Exploration of Fibonacci Function

K. Raja Rama Gandhi
2012 Bulletin of Mathematical Sciences and Applications  
Also, I defined the limit value of Fibonacci function, which is closed to 1.618... where x tends to infinity. Including, Fibonacci sum as well.  ...  In this paper, I define Fibonacci function (probably unknown) on Real number field, for all x∈R, F:R → R,э F (x+n) = ∫nF(x+1)+∫n-1F(x).  ...  Nicolas, whose encouragement, guidance and support from the initial to the final level enabled me to develop an understanding of the subject.  ... 
doi:10.18052/www.scipress.com/bmsa.1.57 fatcat:7rphko7ctvbllejbc6mkndjdoa

Recursive fault-tolerance of Fibonacci cube in hypercubes

Petr Gregor
2006 Discrete Mathematics  
If we remove from Fibonacci cube the vertices with 1 both in the first and the last position, we obtain Lucas cube.  ...  We consider the problem of determining the minimum number of vertices in n-dimensional hypercube whose removal leaves no subgraph isomorphic to m-dimensional Fibonacci cube.  ...  Acknowledgment The author wishes to thank V. Koubek for his encouragement and helpful discussions and P. Kolman and anonymous referee for their comments.  ... 
doi:10.1016/j.disc.2004.09.017 fatcat:jauwe5zphbeoxpjav5rcgf46sq

Daisy cubes and distance cube polynomial

Sandi Klavžar, Michel Mollard
2018 European journal of combinatorics (Print)  
Daisy cubes are partial cubes that include Fibonacci cubes, Lucas cubes, and bipartite wheels.  ...  Then the daisy cube Q n (X) is introduced as the subgraph of Q n induced by the intersection of the intervals I(x, 0 n ) over all x ∈ X.  ...  The Fibonacci cube Γ n , n ≥ 1, is the subgraph of Q n induced by the Fibonacci words of length n. A Fibonacci word u 1 . . . u n is a Lucas word if in addition u 1 · u n = 0 holds.  ... 
doi:10.1016/j.ejc.2018.02.019 fatcat:gayjzhdgiff3dp6yyykv5eajom

Daisy cubes and distance cube polynomial [article]

Sandi Klavžar, Michel Mollard
2017 arXiv   pre-print
Daisy cubes are partial cubes that include Fibonacci cubes, Lucas cubes, and bipartite wheels.  ...  Then the daisy cube Q n (X) is introduced as the sub-graph of Q n induced by the intersection of the intervals I(x, 0 n) over all x ∈ X.  ...  The Fibonacci cube Γ n , n ≥ 1, is the subgraph of Q n induced by the Fibonacci words of length n. A Fibonacci word u 1 . . . u n is a Lucas word if in addition u 1 · u n = 0 holds.  ... 
arXiv:1705.08674v1 fatcat:yddy62q7zbcfhptqqqtabpgpky

$q$-counting hypercubes in Lucas cubes

Elif SAYGI, Ömer EĞECİOĞLU
2018 Turkish Journal of Mathematics  
Acknowledgment This work was done while the first author was visiting Koç Lab at the Department of Computer Science, University of California Santa Barbara.  ...  The authors would like to thank the anonymous reviewers for their valuable comments.  ...  We note that many of the results presented here extend those of Klavžar and Mollard [5] , as C(Λ n , x; q) is a refinement of the cube polynomial C(Λ n , x) .  ... 
doi:10.3906/mat-1605-2 fatcat:a5sgbbln4rfb7odvdetbsrzpbi
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