IMFM
Institute of Mathematics, Physics and Mechanics JADRANSKA 19, 1000 LJUBLJANA, SLOVENIA
Preprint series Vol. 49 (2011), 1136
ISSN 2232-2094
CUBE POLYNOMIAL OF FIBONACCI AND LUCAS CUBES
Sandi Klavzar Michel Mollard
Ljubljana, January 25, 2011
Cube polynomial of Fibonacci and Lucas cubes
vO 00
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a
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Sandi Klavzar* Faculty of Mathematics and Physics
u
University of Ljubljana, Slovenia and
Faculty of Natural Sciences and Mathematics University of Maribor, Slovenia e-mail: sandi.klavzar@fmf.uni-lj.si
Michel Mollard CNRS Universite Joseph Fourier Institut Fourier, BP 74 100 rue des Maths, 38402 St Martin d'Heres Cedex, France e-mail: michel.mollard@ujf-grenoble.fr
Abstract
The cube polynomial of a graph is the counting polynomial for the number of induced k-dimensional hypercubes (k > 0). We determine the cube polynomial of Fibonacci cubes and Lucas cubes, as well as the generating functions for the sequences of these cubes. Several explicit formulas for the coefficients of 00	these polynomials are obtained, in particular they can be expressed with con-
volved Fibonacci numbers. Zeros of the studied cube polynomials are explicitly determined. Consequently, the coefficients sequences of cube polynomials of Fibonacci and Lucas cubes are unimodal.
Key words: hypercubes; cube polynomials; Fibonacci cubes; Lucas cubes; generating functions; zeros of polynomials, unimodal sequences
AMS subject classifications: 05C31, 05A15, 26C10 1 Introduction
The introduction of computers in the mid of the past century and the subsequent fascinating developments opened new horizons in the development of mathematics. In particular, numerous earlier sporadic fields of mathematics flourished and transformed themselves into mature fields. Let us just mention mathematical programming, theory of error-correcting codes, theory of computing, symbolic computation,
U
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and graph theory.
* Supported by the Research Grant P1-0297 of Ministry of Higher Education, Science and Technology Slovenia.
One of the cornerstones of these developments is the binary number system. To build a related graph, it is utmost natural to let all binary strings of length n, n > 1, form the vertex set, and connect two vertices by an edge if they differ in precisely one coordinate (alias bit). The obtained graph is the n-dimensional hypercube or n-cube for short and denoted Qn. The n-cubes are ubiquitous, say as the basic model for interconnection networks, a tool for numerous applications elsewhere, and a source of challenging problems is extremal combinatorics.
In this paper we are concerned with the enumeration of hypercubes in graphs, more precisely, in Fibonacci and Lucas cubes, two families of graphs of independent interest. So the primary task here is enumerative combinatorics, but a particular goal of this paper is to expose that for this task several areas of mathematics are useful: graph theory, number theory, complex analysis, and more.
For a graph G, let cn(G) (n > 0) be the number of induced subgraphs of G isomorphic to Qn. Note that in particular c0(G) = \V(G)|, ci(G) = |L(G)|, and c2(G) is the number of induced 4-cycles. The cube polynomial, C(G,x), of G, is the 1corresponding counting polynomial, that is, the generating function
C(G, x) = £ Cn(G)xn .
VO 00
o
CSI
I
m
CD
u
a
CD u
n0
This polynomial was introduced in [4] where it was observed that it is multiplicative for the Cartesian multiplication of graphs: C(G □ H,x) = C(G,x)C(H,x) holds for any graphs G and H. Moreover, if G is a median graph, then C(G, —1) = 1 and C'(G, — 1) is the smallest n such that G isometrically embeds into Qn. The cube polynomial was generalized to a Hamming polynomial in two different ways in [3] and [7], respectively. For a survey on the cube polynomial, its extensions and related 00 results see [16]. We add here that, interestingly, the sequence {ck(G)}k>0 turned CSI out to be important in human genetics, see [1] for its use in studies of the so-called phantom mutations.
Fibonacci cubes form appealing and applicable class of graphs introduced in the first place as a model for interconnection networks [10]. These graphs later found applications in mathematical chemistry [15, 27] and elsewhere. Fibonacci cubes were studied from algorithmic [9, 23] as well as from several other points of view, see [11] and references therein. Lucas cubes [18] form a class of graphs closely related to Fibonacci cubes, in a way they can be considered as a symmetrization of Fibonacci cubes.
Motivated by several studied of enumerative properties of Fibonacci and Lucas cubes [13, 18, 19], we consider in this paper the cube polynomial of Fibonacci and Lucas cubes and in particular extend the previous enumeration results. These results are presented in Sections 3-5.
Another motivation for our paper are the earlier investigations of zeros of cube polynomials of median graphs [5]. It was proved that C (G, x) has no zeros in [—1, to), that it always has a (real) zero in [—2, —1), and that all rational zeros are of the form —((t + 1)/t) for some integer t > 0. Moreover, if the cube polynomial C(G,x) is of degree p, then C(G,x) has a p-multiple zero if and only if G is the Cartesian
product of p trees of the same order. Note that the smallest example for this theorem is provided with the fact that C(Qp,x) = (x + 2)p. In the case of Q3-free median graphs the cube polynomial is (of course) quadratic and it was proved [6] that its zeros are always real. In Section 6 we explicitly determine the zeros of the cube polynomial of the Fibonacci and Lucas cubes. It turns out that their zeros are real (and negative) which in turn implies that these polynomials are unimodal and log-concave.
A Fibonacci string of length n is a binary string b1b2 ...bn with bibi+1 = 0 for 1 < i < n. The Fibonacci cube rn (n > 1) is the subgraph of Qn induced by the Fibonacci strings of length n. For convenience we also consider the empty string and set r0 = K1. Call a Fibonacci string b1b2 .. .bn a Lucas string if b1bn = 1. Then the Lucas cube An (n > 1) is the subgraph of Qn induced by the Lucas strings of length n. We also set A0 = K1.
Let {Fn} be the Fibonacci numbers: F0 = 0, F1 = 1, Fn = Fn-1 + Fn-2 for n > 2. It is well known that the generating function of the sequence (Fn}%=0 is
o
CO CD
U
a
CD U
2 Preliminaries
F
xn _	X
n x
Note that \V(rn)| = Fn+2. We will denote by {FFn}'^)=0 the sequence defined by Fn = Fn+1, hence
1
(FnXn) =
n2 1 — x — X2
^ n>0 c^
/ nx j —	o '
^	1 — X — X2
n>>
Let {Ln} be the Lucas numbers: L0 = 2, L1 = 1, L = Ln-1 + Ln-2 for n > 2. It is well known that the generating function of the sequence {Ln}^=0 is
V L Xn = 2 — X CO	^ LnX = 1 — X — X2 •
n>0
Note that \V(An)\ = Ln for n > 1.
Let {An}^=0 and {Bn}^=0 be two sequences of numbers, then the convolution of An and Bn is the sequence {(A*B)n}c^=0 defined by (A* B)n = n=0 AiBn-i. From the definition it is clear that the generating function of {(A* B)n}£=0 is the product
m
of those of {An}^=0 and {Bn}^=0. We will denote by A * the sequence define by
1	m	m — 1
A* = A and A * = A * A * (m > 2).
A classical question about a given sequences of numbers is whether it is unimodal, where a sequence {An}rm=0 of real numbers is unimodal if for some 0 < j < m we have a0 < a1 ■ ■ ■ < aj > aj+1 ■ ■ ■ > am. A sequence {An}mm=0 of positive reals is said to be logarithmically concave (or log-concave for short) if a2 > ai-1ai+1 for all 1 < i < m. It is easy to see that a positive log-concave sequence is unimodal,
various additional methods are known for showing the unimodality or log-concavity of a sequence [8, 22, 25]. For our purposes it is important that by a result of Newton, if {An}7m=o is a sequence of coefficients of a polynomial with negative real zeros, then {An}m=o is log-concave and therefore also unimodal. A polynomial is log-concave (resp. unimodal) if the sequence of its coefficients is log-concave (resp. unimodal).
Finally, the Cartesian product G □ H of graphs G and H has the vertex set V(G) x V(H), and (g,h) is adjacent to (g',h') if either g = g' and hh' G E(H), or gg' G E(G) and h = h'.
£
3 Cube polynomial of Fibonacci cubes
vd 00
In this section we determine the cube polynomial of Fibonacci cubes and read off the number of induced Qk in rn. To get a feeling we list the first few of them:
C (ro,a:) = 1 C (rux) = x + 2
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C (r2,x)	=	2x + 3
C (r3,x)	=	x2 + 5x + 5
C (r4,x)	=	3x2 + 10x + 8
C (r5,x)	=	x3 + 9x2 + 20x + 13
C (re, x)	=	4x3 + 22x2 + 38x + 21
C (r7,x)	=	x4 + 14x3 + 51x2 + 71x + 34
C (rg,x)	=	5x4 + 40x3 + 111x2 + 130x + 55
00
We next determine the generating function for the sequence of the corresponding 2cube polynomials:
£ CO
Proposition 3.1 The generating function of the sequence {C(rn,x)}£=0 is
E C(rn,x)yn = 1 + y(1 + x)
n>0	1 " y " y2(1+ x)"
Proof. Clearly, C(ro,x) = 1 and C(ri,x) = x + 2. Let n > 2 and let Xn = {x G V(rn) | x1 = 0} and Yn = {x G V(rn) \ x1 = 1}. Then Xn induces a subgraph of rn isomorphic to rn_ 1. The first two coordinates of a vertex from Yn are 10, hence Yn induces a subgraph of rn isomorphic to rn_2. Moreover, every vertex from Yn has exactly one neighbor in Xn. It now readily follows that
CD
C(rn,x)= C(rn-1,x) + (1+ x)C(rn-2,x) (n > 2) .	(1)
CD
Setting f (x,y) = En>o C(Tn,x)yn we obtain f (x,y) - 1 - y(x + 2) = y(f (x,y) -1) + y2(1 + x)f (x,y) and the result follows.	□
&	4
CD
Qn can be represented as the Cartesian product of n copies of K2. Hence the property C(G □ H,x) = C(G,x)C(H,x) immediately implies that for any n > 0,
o
a=0
C(Qn, x) = (2 + x)n = £ (n) (1 + x)a .
For Fibonacci cubes we have the following parallel result:
n
£
Theorem 3.2 For any n > 0, C(rn,x) is of degree [n+~\ and C(rn,x)^ (n"a + M(1 + x)a.
a=0 ^	'
Proof. Using Proposition 3.1 we have: 2	f (x,y) = 1+ y(1+x)
i - y — y2(i + x) = (i + y(i + x))J2(y + y2(i + x))a
a>0
= E ya (i+y(i+x))a+1
a>0
a+1 / , i\
EE r (y(i + x))by
b
a>0b=0 v
«	= ££ £ (a+OO^
a>0 b=0 c=0 ^ b '
a + i\ (n — a
a 0 n=a c=0
= EEE
&
CO From here we obtain
s
xcyn.
n — al \ c
C(rn,x) = 5: E n—an—'^
a 0 c=0
I I
n / i i \	L 2 J ,
a + , \n-a	fn — a + i
= £ n—a (i+«)n- = E c ~a ^ •
a=0	a=0
and we are done.	□
Corollary 3.3 For any n > 0, the number of induced Qk, k > 0, in rn is
LJ , . + ...
(r ) v^ (n — . + M . ck (rn) =
n
.k
i=k
Like the determination of ck(Qn) in [14], Corollary 3.3 can also be deduced directly as follows. Note first that there are precisely (n_l+1) Fibonacci strings that contain i 1's. (Indeed, writing down n — i zeros, we need to select among n — i + 1 positions where to insert ones.) For a given k-cube Q, let u be a vertex of Q that contains the largest number of 1's, say i, among the vertices of Q. Then there is set of k positions such that Q is induced by the 2k vertices obtained by varying these k bits. Now, each such vertex u give rise to Q) different induced k-cubes and the result follows.
2

CO
m
CD
fi
CD
m
fi a
CD fi cu
4 Fibonacci cubes and convolved Fibonacci numbers
We next determine, for a fixed k, the generating function of the sequence K(rn)}~0 :
CO
Theorem 4.1 Let k > 1 be a fixed integer. Then
,2k-1
Y,ck (W
y
n>0 yn>y (1—y—y2)k+1'
Proof. Since c0(rn) = \V(rn)\ = Fn+2, we have
Lc0(rn)yn =_ y _ y2 .	(2)
Ec0(rn)yn =	2
n>0 1—y—y2
C^
By a parallel argument as used in the proof of Proposition 3.1 we infer that for n > 2,
ck(rn) = ck(rn-l) + ck(rn-2) + ck-l^n-2) .	(3)
Since c1(r0) = 0 and c1(r1) = 1 a routine computation, using (3) and (2), yields
E c1(rn)yn = t^—.
^0	(1 — y — y2)2
To rich the conclusion for k > 2, apply induction, the fact that ck(r0) = ck(r1) = 0, and (3).	□
Corollary 4.2 Let k > 0. Then the number of induced Qk in rn is
_ fc+i
ck (rn) = Fn-2k+1 .
Proof. Sinc^n>0(i?nyn) = J-—2 , we haveZn>0(Fn* yn) = (1_y-1y2)k+i . So the
1 ~k+1 -y_y'
2k —1 _fc + 1
yn in the expansion of (1_°y_y2)k+1 is Fn_2k+1. The assertion follows by Theorem 4.1.
□ V V
coefficient at yn in the expansion of (1_y_ly2)k+1 is Fn* and hence the coefficient at
Note that Corollary 4.2 for k = 0 reads as \V(rn)| = c0(rn) = F*+1 = Fn+1 = Fn+2.
— m
The numbers Fn* are known as convolved Fibonacci numbers and have been
10 studied earlier, for the first time probably in [20]. Bergum and Hoggatt [2] considered
h	~ m
the array Rn,m for m > 1,n > 1 whose m column is the sequence {i7n*_1}jj=1. Thus
ck(rn) = Rn_2k+2,k+1 and the coefficients of C(rn,x) appear as rising diagonal of
this array:
L S+1 J
C (rn,x)= E Rn_2k+2,k+1 xk. k=0
They also proved that for any n > 1, Em>1 Rn,mxm_1 = Ni_X)n where
vO
L ^ J
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u
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Nn(x) = E Rn_2k,k+1 (-1)
1)kxk.
k=0
Furthermore, they studied the properties of the polynomial Nn(x) that differs from C(rn_2,x) by the alternate sign of coefficients. The same array appears also in [17,
/ \	_ r
12] with the notation Fn+1 — F,*; see [21] for additional references. (Note that we selected the notation with reference to convolution because it can be also used for Lucas cubes; see below.)
Using the properties of convolved Fibonacci numbers, we can state further results on the number of cubes in Fibonacci cubes. For instance, since the convolved Fibonacci number can be written as sums of products of Fibonacci numbers, see [17, page 1] (this fact can also be deduced directly from the definition of convolution):
C^
Corollary 4.3 For any n > 1,
£ CO
il+i2 + -+ifc + l="-2fc+1
HH
Corollary 4.3 for k = 1 reduces to
n_1	n
|E(rn)| = c1(rn) = E Fi+1Fj+1 = E Fi+1Fn_i = ^ FiFn_i+1 ,
i=0 i=1
i+j = n-1
a result obtain in [13, Proposition 3], while for the squares S(rn) of rn it gives:
CD
|S(rn)| = c2(rn) = E Fi+1 Fj+1Fk+1 .
i+j+k=n-3
ck(rn) = E Fil+1Fi2 + 1 ' ' ' Fifc+l+1
Another way of looking at Corollary 4.3 is:
b ti
ti
VD CO
0 fi
o
1
CO ^
CO CO
CO CD
$H
CD CO
u
a
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Corollary 4.4 For any n > 1,
n-2k+2
Ck (rn) = £ Fjck-1(rn-i-1) .
i=1
The case k = 2 of Corollary 4.4 reduces to [13, Proposition 6].
5 Lucas cubes
In this section we consider Lucas cubes and present results parallel to the results for Fibonacci cubes from Sections 3 and 4. Proofs are similar as in the previous sections hence we omit most of the details but point out the differences. The first polynomials are:
C(A0, x)	=1
C (A1,x)	=1
C (A2,x)	= 2x + 3
C (A3,x)	= 3x + 4
C (A4, x)	= 2x2 + 8x + 7
C (A5,x)	= 5x2 + 15x + 11
C (Aa,x)	= 2x3 + 15x2 + 30x + 18
C (Ar,x)	= 7x3 + 35x2 + 56x + 29
C (Ag, x)	= 2x4 + 24x3 + 80x2 + 104x + 47
We begin with the generating function for the sequence of the cube polynomials of Lucas cubes:
Proposition 5.1 The generating function of the sequence {C(An,x)}£=0 is
1 + y2(1 + x)
EC (An, x)yr"
n0
1 - y - y2(1 + x)'
Proof. Let n > 3 and partition the vertex set of An into vertices that start with 0 and those that start with 1. The latter vertices are then of the form 10...0. Similarly as in the proof of Proposition 3.1 we now get that for n > 3,
C(An, x) = C(rn-1,x) + (1 + x)C(rn-3, x) .
(4)
To complete the proof combine Proposition 3.1 with (4) and the initial conditions C(Ao,x) = C(Ai,x) = 1, C(A2,x) = 2x + 3.	□
We note in passing that considering degree zero coefficients in (4) yields the well known relation Ln = Fn+1 + Fn-1.
£
CO CO
CO CD
Jh
a
CD Jh
Theorem 5.2 For any n > 3, C(An,x) is of degree |_nJ and
L n J
C(An, x) =
a=0
2, n — a\ (n — a + i aa
(i + x)a
Proof. Rewrite (i) as C(rn__,x) = C(rn-2,x) + (i + x)C(rn_3,x) and subtract this equality from (4) to obtain
C(An,x) = 2C(rn_1,x) — C(rn_2,x) (n > 3) .	(5)
fi
The result now follows from Theorem 3.2.	□
Corollary 5.3 Let n > i. Then the number of induced Qk, k > i, in An is
L 2 J
ck(An) =
E
i= k
2, n — . \ jn — . + i .

o
We continue with the generating function of the sequence {ck(An)}^=0, where k is a fixed integer.
Theorem 5.4 Let k > i be a fixed integer. Then
o
Ck (An) = Ck (rn_l) + Ck (rn_3) + Ck_l(rn_3) .
(A )yn y2k(2 — y)
.LCk (An)y = (i — y — y2)k+1
n>0
Proof. From (4) we deduce that for n > 3
Then using Theorem 4.i the result follows for k > 2. Furthermore using (2) it is easy to verify that
y2(2 — y)
ECi(An)y^ (1 2)2 '
(i — y — y2)2
n0
Corollary 5.5 Let k > i. Then the number of induced Qk in An is
Ck (An) = (L * F*)n_2k .
□
~k 2- _
Proof. The generating function of L*F* is __ y 2 ■ _ 2)k . Then arguing similarly
y y (	y y )
as in the proof of Corollary 4.2 the result follows.	□
Proceeding along the same lines as at the end of Section 4 we have:
k
o io
Corollary 5.6 For any k > 1,
CO CO
fi a
CD fi cu
n-2k
ck(An) = E Fi2+1 . . . Fik+1+1 = E ^ick_1(rn_i_3) .
i1>i2 •■■■•ik+1 i=0 i1+i2+-----+ik+1=n — 2k
The first equality of Corollary 5.6 reduces for k = 1 to [13, Proposition 7], while the second equality for k = 2 was obtained in [13, Proposition 8].
6 Roots and unimodality of C(rn,x) and C(An,x)
To present Binet-like formulas for C(rn,x) and C(An,x), the following recursion is useful:
CO
Lemma 6.1 For any n > 3, C(An, x) = C(An_1, x) + (1 + x)C(An_2,x).
i—I
Proof. Immediate consequence of relations (1) and (5).	□
o
(i) C (rn,x) =
Theorem 6.2 Let x = —5/4. Then
+	2 /1 — ^5 + 4x\ra+2'
(n > 0)
^ (ii) C(An, x) = ^-2-) +[-2-) (n > 1).
Proof. Theorem is proved by standard methods for second order linear recursion, or by induction: for (i) use (1) together with initial conditions and for (ii) apply Lemma 6.1 together with initial conditions.	□
When x = —5/4 we obtain by continuity (or with the standard method when the roots of the characteristic polynomial are equals) that
n+2
C (^ —5/4) = ^ 2
and
C(An, —5/4) = -_I (n > 1). 2
Theorem 6.2 leads to numerous divisibility consequences, some are collected in the next statement.
Corollary 6.3 (i) For any p > 1, C(L2p,x) = C(rp_1,x)C(Ap+1,x).
(ii)	For any n,k > 0, C(rn, x) divides C(rk(n+2)+n, x).
(iii)	For any n,k > 1, C(An,x) divides C(Akn,x).
Proof. Set f (x) = i+f+M and g(x) = _fH*.
(i)	By Theorem 6.2,
i
UO	C(r2p,x) = ^ [f2p+2(x) - g2p+2(x)
= ^=+= [f^1 (x) - gp+1 (x)] [fp+1 (x) + gp+1 (x)] = C(Tp_i,x)C(Ap+i,x).
$
(ii)	Suppose C(rra,xo) = 0. Then by Theorem 6.2, fn+2(xo) = gn+2(x0). It follows that f (k+1)(n+2)(x0) = g(k+1)(n+2)(x0) which in turn implies that
C (rk(n+2)+2,x0) = 0.
(iii)	C(An,x0) = 0 means that fn(x0) = —gn(x0). Therefore f kn(x0) = -gkn(x0) VD which in turn implies C(Akn, x0) =0.	□
CO
In order to establish unimodality of coefficients of Fibonacci and Lucas polynomials, we next explicitly determine its zeros.
o
CO
u
a
CD U
Proposition 6.4 Let n > 1. Then the roots of C(Tn,x) and C(An,x) are
n + 1
ta"2 (&)+ 5 r = 12
C^	4	'	' '
o
c^
and
tan2(^) +5
---—:---, r = 0,1,...
n
L2J
1
4
respectively. In particular, all the roots are negative reals.
CSI
Proof. By Theorem 6.2, roots of C(rn,x) are determined with (1 + ^5 + 4x)n+2 = (1 - ^5 + 4x)n+2 .
CO
Suppose x £ R. Then for a root, 5x + 4 < 0 must hold. Writing ^5 + 4x = iy we need to solve (1 + iy)n+2 = (1 — iy)n+2. Setting z = 1 + iy = \z\(cos 0 + i sin 0) this reduces to sin(n + 2)0 = 0 so that y = tan 0 = tan(). Hence the zeros. Note that the above zeros are all different because tan2(x) is strictly increasing on [0,^/2).
The degree of C(rn,x) is [nf1] thus we conclude that all zeros are negative reals.
Similar computation gives the zeros of C(An,x).	□
CD
The sequence of coefficients of a polynomial whose zeros are all negative reals is log-concave and a positive log-concave sequence is unimodal [8, 22, 25]. Hence:
CO
Corollary 6.5 For any n > 0, the sequences of coefficients of C(rn) and C(An) are log-concave and unimodal.
2
7 Concluding remark
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For n > i > 0, the ith extended Fibonacci cube of order n, rin, was introduced in [26] as follows. Let Bi be the set of all binary strings of length i. Then the vertex set Vn of rn is defined recursively by Vi+2 = OVJ^+i U 10V,i, with initial conditions V/ = Bi, Vii+1 = Bi+1. Note that then ri = Qi and ri+1 = Qi+1. Observe also that ri = rn. It was proved in [24] (see also [13]), that rin is isomorphic to rn_ □ Qi. Hence understanding the cube polynomial of Fibonacci cubes and how it behaves under the Cartesian multiplication, one can obtained all the corresponding results for the extended Fibonacci cubes as well.
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[3] B. Bresar, P. Dorbec, S. Klavzar, and M. Mollard. Hamming polynomials and their partial derivatives. European J. Combin., 28(4):1156-1162, 2007.
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[5] B. Bresar, S. KlavZar, and R. Skrekovski. Roots of cube polynomials of median
graphs. J. Graph Theory, 52(1):37-50, 2006.
[6]	B. Bresar, S. KlavZar, and R. Skrekovski. On cube-free median graphs. Discrete Math., 307(3-5):345-351, 2007.
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[9]	P. Gregor. Recursive fault-tolerance of Fibonacci cube in hypercubes. Discrete Math., 306(13):1327-1341, 2006.
[10]	W.-J. Hsu. Fibonacci cubes—a new interconnection technology. IEEE Trans. Parallel Distrib. Syst., 4(1):3-12, 1993.
[11]	A. Ilic, S. KlavZar, and Y. Rho. Generalized Fibonacci cubes. To appear in Discrete Math., 2011.
10
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