Cube Polynomial of Fibonacci and Lucas Cubes

Klavžar, Sandi; Mollard, Michel

doi:10.1007/s10440-011-9652-4

Cited by 40 publications

(50 citation statements)

References 21 publications

(26 reference statements)

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“…We present our q -cube enumerator polynomial in Section 3 and investigate divisibility, special values, and other properties of the coefficients in Section 4. 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) . Our approach and proofs follow along the lines of the Fibonacci case treated in [8], though the q -analogues of the Lucas cube polynomials have certain interesting properties in their own right.…”

Section: Introductionsupporting

confidence: 81%

“…Certain divisibility properties of the cube polynomials for Λ n and Γ n were noted in [5]. Our results extend these and also include information about the nature of the quotients.…”

Section: Introductionsupporting

confidence: 77%

“…Many interesting results on C(Λ n , x) and h n,k in (4) appear in [5]. It is observed in [5] that for n ≥ 3 the numbers in Table 1 satisfy the recursion…”

Section: Preliminariesmentioning

confidence: 98%

“…In this paper, we consider a generalization of the hypercube enumerator polynomials of the Lucas cubes. Our polynomials involve an extra variable q (see Section 3), from which the known numerical cases given in [5] can be obtained by the specialization q = 1 .…”

Section: Preliminariesmentioning

confidence: 99%

“…Furthermore, the maximum number of disjoint hypercube subgraphs isomorphic to Q k (called subcubes) in Γ n is considered in [1,9] and it is shown that asymptotically all vertices of Γ n are covered by a maximum set of disjoint subcubes in [9]. The cube polynomial of Λ n and Γ n , which is the starting point of this paper, is studied in [5]. This is the polynomial whose coefficients enumerate subcubes of the given graph by their dimension.…”

Section: Introductionmentioning

confidence: 99%

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$q$-counting hypercubes in Lucas cubes

Saygı¹,

Eğecioǧlu²

2018

Turk J Math

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Lucas and Fibonacci cubes are special subgraphs of the binary hypercubes that have been proposed as models of interconnection networks. Since these families are closely related to hypercubes, it is natural to consider the nature of the hypercubes they contain. Here we study a generalization of the enumerator polynomial of the hypercubes in Lucas cubes, which q -counts them by their distance to the all 0 vertex. Thus, our bivariate polynomials refine the count of the number of hypercubes of a given dimension in Lucas cubes and for q = 1 they specialize to the cube polynomials of Klavžar and Mollard. We obtain many properties of these polynomials as well as the q -cube polynomials of Fibonacci cubes themselves. These new properties include divisibility, positivity, and functional identities for both families.

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Section: Introductionsupporting

confidence: 81%

“…Certain divisibility properties of the cube polynomials for Λ n and Γ n were noted in [5]. Our results extend these and also include information about the nature of the quotients.…”

Section: Introductionsupporting

confidence: 77%

“…Many interesting results on C(Λ n , x) and h n,k in (4) appear in [5]. It is observed in [5] that for n ≥ 3 the numbers in Table 1 satisfy the recursion…”

Section: Preliminariesmentioning

confidence: 98%

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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$q$-counting hypercubes in Lucas cubes

Saygı¹,

Eğecioǧlu²

2018

Turk J Math

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Asymptotic Properties of Fibonacci Cubes and Lucas Cubes

Klavžar

Mollard

2014

Ann. Comb.

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It is proved that the asymptotic average eccentricity and the asymptotic average degree of both Fibonacci cubes and Lucas cubes are (5 + √ 5)/10 and (5 − √ 5)/5, respectively. A new labeling of the leaves of Fibonacci trees is introduced and it is proved that the eccentricity of a vertex of a given Fibonacci cube is equal to the depth of the associated leaf in the corresponding Fibonacci tree. Hypercube density is also introduced and studied. The hypercube density of both Fibonacci cubes and Lucas cubes is shown to be (1 − 1/ √ 5)/ log 2 ϕ, where ϕ is the golden ratio, and the Cartesian product of graphs is used to construct families of graphs with a fixed, non-zero hypercube density. It is also proved that the average ratio of the numbers of Fibonacci strings with a 0 resp. a 1 in a given position, where the average is taken over all positions, converges to ϕ 2 , and likewise for Lucas strings.

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Vertex and Edge Orbits of Fibonacci and Lucas Cubes

et al. 2016

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The Fibonacci cube Γ n is obtained from the n-cube Q n by removing all the vertices that contain two consecutive 1s. If, in addition, the vertices that start and end with 1 are removed, the Lucas cube Λ n is obtained. The number of vertex and edge orbits, the sets of the sizes of the orbits, and the number of orbits of each size, are determined for the Fibonacci cubes and the Lucas cubes under the action of the automorphism group. In particular, the set of the sizes of the vertex orbits of Λ n is {k ≥ 1; k | n} ∪ {k ≥ 18; k | 2n}, the number of the vertex orbits of Λ n of size k, where k is odd and divides n, is equal to, and the number of the edge orbits of Λ n is equal to the number of the vertex orbits of Γ n−3 when n ≥ 5. Primitive strings, dihedral transformations and asymmetric strings are essential tools to prove these results.

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Cube Polynomial of Fibonacci and Lucas Cubes

Cited by 40 publications

References 21 publications

$q$-counting hypercubes in Lucas cubes

$q$-counting hypercubes in Lucas cubes

Asymptotic Properties of Fibonacci Cubes and Lucas Cubes

Vertex and Edge Orbits of Fibonacci and Lucas Cubes

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