OOn d -Fibonacci digraphs ∗ C. Dalf´o
Departament de Matem`atica, Universitat de LleidaIgualada (Barcelona), Catalonia [email protected]
M. A. Fiol
Departament de Matem`atiques, Universitat Polit`ecnica de CatalunyaBarcelona Graduate School of MathematicsBarcelona, Catalonia [email protected]
Abstract
The d -Fibonacci digraphs F ( d, k ), introduced here, have the number of verticesfollowing generalized Fibonacci-like sequences. They can be defined both as digraphson alphabets and as iterated line digraphs. Here we study some of their nice properties.For instance, F (2 , k ) has diameter d + k − (cid:96) , with (cid:96) ∈ { k − , k − } . Moreover, it turnsout that several other numbers of F ( d, k ) (of closed l -walks, classes of vertices, etc.)also follow the same linear recurrences as the numbers of vertices of the d -Fibonaccidigraphs. Mathematics Subject Classifications:
Keywords: n -step Fibonacci number, Fibonacci graph, digraph on alphabet, de Bruijndigraph, line digraph, adjacency matrix, spectrum ∗ This research has been partially supported by AGAUR from the Catalan Government under project2017SGR1087 and by MICINN from the Spanish Government under project PGC2018-095471-B-I00. Theresearch of the first author has also been supported by MICINN from the Spanish Government underproject MTM2017-83271-R.The research of the first author has also received funding from the European Union’sHorizon 2020 research and innovation programme under the Marie Sk(cid:32)lodowska-Curiegrant agreement No 734922. a r X i v : . [ m a t h . C O ] S e p Preliminaries
Let us first introduce some basic notation and results. A digraph G = ( V, E ) consists ofa (finite) set V = V ( G ) of vertices and a set E = E ( G ) of arcs (directed edges) betweenvertices of G . As the initial and final vertices of an arc are not necessarily different, thedigraphs may have loops (arcs from a vertex to itself), and multiple arcs , that is, therecan be more than one arc from each vertex to any other. If a = ( u, v ) is an arc from u to v , then vertex u (and arc a ) is adjacent to vertex v , and vertex v (and arc a ) is adjacentfrom v . The converse digraph G is obtained from G by reversing the direction of each arc.Let G + ( v ) and G − ( v ) denote the set of arcs adjacent from and to vertex v , respectively.A digraph G is k -regular if | G + ( v ) | = | G − ( v ) | = k for all v ∈ V . As usual, we called cycle to a closed walk in which all its vertices are different.The adjacency matrix A of a digraph G = ( V, E ) is indexed by the vertices in V , andit has entries ( A ) uv = α if there are α arcs from u to v , with α ≥
0. Notice that, as weallow loops, the diagonal entries of A can be different from zero.In the line digraph LG of a digraph G , each vertex of LG represents an arc of G , thatis, V ( LG ) = { uv : ( u, v ) ∈ E ( G ) } ; and vertices uv and wz of L ( G ) are adjacent if and onlyif v = w , namely, when the arc ( u, v ) is adjacent to the arc ( w, z ) in G . The k -iteratedline digraph L k G is recursively defined as L G = G and L k G = L k − LG for k ≥
1. Itcan easily be seen that every vertex of L k G corresponds to a walk v , v , . . . , v k of length k in G , where ( v i − , v i ) ∈ E for i = 1 , . . . , k . Then, if there is one arc between pairs ofvertices and A is the adjacency matrix of G , the uv -entry of the power A k , denoted by a ( k ) uv , corresponds to the number of k -walks from vertex u to vertex v in G . The order n k of L k G turns out to be n k = jA k j (cid:62) , (1.1)where j stands for the all-1 vector. If there are multiple arcs between pairs of vertices,then the corresponding entry in the matrix is not 1, but the number of these arcs.If G is a strongly connected d -regular digraph, different from a directed cycle, withdiameter D , then its line digraph L k G is d -regular with n k = d k n vertices and has (asymp-totically optimal) diameter D + k . In fact, for a strongly connected general digraph, thefirst author [5] proved that the iterated line digraphs are always asymptotically dense. Formore details, see Harary and Norman [8], Aigner [1], and Fiol, Yebra, and Alegre [7].Given integers d ≥ k ≥
1, the de Bruijn digraph B ( d, k ) is commonly definedas a digraph on alphabet in the following way. This digraph has vertices x x . . . x k with x i ∈ [0 , d −
1] for every i = 1 , , . . . , k . Moreover, every vertex x x . . . x k is adjacent tothe vertices x . . . x k x k +1 , where x k +1 ∈ [0 , d − .1 Generalized Fibonacci numbers A proposed generalization of the well-known Fibonacci numbers is the following. Givenan integer d ≥
2, the d -step Fibonacci numbers F ( d )1 , F ( d )2 , F ( d )3 , . . . are defined through thelinear recurrence relation F ( d ) k = d (cid:88) i =1 F ( d ) k − i , (1.2)initialized with F ( d ) k = 0 for k ≤ F ( d )1 = F ( d )2 = 1. Thus, the cases d = 2 , , , . . . cor-respond to the so-called Fibonacci numbers F k , tribonacci numbers , tetrabonacci numbers ,etc., respectively. For more information, see, for example, Miles [9].In particular, the Fibonacci numbers hold the recurrence F k = F k − + F k − , which, asit is well known, is satisfied by the numbers of the form f ( k ) = aφ k + bψ k = a (cid:32) √ (cid:33) k + b (cid:32) − √ (cid:33) k , (1.3)where a and b are constants, φ = √ is the golden ratio, and ψ = φ − . Recall alsothat, from F = 0 and F = 1, we get a = − b = 1 / √
5, giving the Binet’s formula F k = (1 / √ φ k − ψ k ). d -Fibonacci digraphs on alphabets Definition 2.1.
For some given integers d ≥ and k ≥ , the d -Fibonacci digraph F ( d, k ) has vertices x = x x . . . x k , where for i = 0 , . . . , k − , x i +1 ∈ [0 , d − if x i − = 0 , and x i +1 = x i + 1 (mod d ) otherwise. Moreover, every vertex x x . . . x k is adjacent to thevertices x . . . x k x k +1 , where x k +1 ∈ [0 , d − if x k = 0 , and x k +1 = x k + 1 (mod d ) otherwise. For instance, the 2-Fibonacci digraphs F (2 , k ) with k ≤ , , , F ( d,
1) on d vertices, with d ∈ [2 , d -Fibonacci digraphs, which are easy consequences oftheir definition, are the following. Lemma 2.2.
Let F ( d, k ) be the d -Fibonacci digraph, with vertices x = x x . . . x k , for x i ∈ [0 , d − . ( i ) The out-degree of x is deg( x ) = d if x k = 0 , and deg( x ) = 1 otherwise. ( ii ) The digraph F ( d, k ) is an induced subdigraph of the de Bruijn digraph B ( d, k ) .
010 10 01001 000100 010 001 000000011000 1001 001001001010 0101 F (2,1) F (2,2) F (2,3) F (2,4) Figure 1: The 2-Fibonacci digraphs F (2 , k ) for k = 1 , , , iii ) The digraph F ( d, k ) contains F ( d (cid:48) , k ) as an induced subdigraph, for every d (cid:48) ≤ d . ( iv ) There is a homomorphism φ from F ( d, k ) to F ( d, k (cid:48) ) , for every k (cid:48) ≤ k . ( v ) The automorphism group of F ( d, k ) is the trivial one. Moreover, the digraph F (2 , k ) is isomorphic to its converse.Proof. ( i ) follows immediately from Definition 2.1. Similarly, ( ii ) is a direct consequence ofthe definitions of F ( d, k ) and B ( d, k ). Concerning ( iii ), notice that the vertices of F ( d, k )corresponding to the sequences x x . . . x k with x i ∈ { , d − , . . . , d − d (cid:48) + 1 } induce asubdigraph isomorphic to F ( d (cid:48) , k ). Alternatively, from the results of Section 4, note that T d (cid:48) is clearly an induced subdigraph of T d for every d (cid:48) ≤ d and, hence, the same property isinherited by F ( d (cid:48) , k ) = L k T d (cid:48) and F ( d, k ) = L k T d . To prove ( iv ), we only need to exhibitthe homorphism from F ( d, k ) to F ( d, k (cid:48) ), which is the following map on the correspondingsets of vertices φ : V ( F ( d, k )) → V ( F ( d, k (cid:48) )) x = x x . . . x k → φ ( x ) = x k − k (cid:48) +1 x k − k (cid:48) +2 . . . x k . Indeed, observe that if x → y , then φ ( x ) → φ ( y ). To prove the first part of ( v ), we onlyneed to realize that every automorphism of F ( d, k ) must send the unique cycles of lengths1 and 2 (loop and digon) to themselves. This means that vertex 00 . . . { , . . . , . . . } must be an orbit. But the only way to preservethe adjacencies between these two vertices and 00 . . . v ) is justified by the mapping x x . . . x k (cid:55)→ x k . . . x x , which isan isomorphism between F (2 , k ) and its converse F (2 , k ).4n contrast with F (2 , k ), the Fibonacci digraph F ( d, k ) with d > d > T d (cid:54)∼ = T d . However, the same approach allows toshow that most of the properties of F ( d, k ) related with d -Fibonacci numbers, are sharedby its converse F ( d, k ).To illustrate case ( ii ), in Figure 1 each Fibonacci digraph F (2 , k ) with k ≤ B ( d, k ). In partic-ular, note that F ( d,
1) has d vertices, which coincides with the order d k of the de Bruijndigraph B ( d, k ) when k = 1. In contrast, the number of vertices of F ( d, k ) is much smallerwhen k increases, as the following result shows. Proposition 2.3.
The numbers of vertices N ( d, k ) of the d -Fibonacci digraphs F ( d, k ) satisfy the same linear recurrence as the d -step Fibonacci numbers in (1.2) N ( d, k + 1) = k (cid:88) i = k − d +1 N ( d, i ) , (2.1) but now initialized with N ( d, i ) = 1 for i = d − , d − , . . . , , and N ( d,
1) = d .Proof. For j ∈ [0 , d − n kj be the number of vertices x x . . . x k of F ( d, k ) such that x k = j . Thus, n j = 1 for j ∈ [0 , d − N ( d, k ) = (cid:80) d − j =0 n kj and, from the conditions onthe digits x i , we get n k +10 = n k + n kd − ,n k +11 = n k ,n k +12 = n k + n k , ... n k +1 d − = n k + n kd − , or, in matrix form, n k +1 = (cid:16) n k +10 , n k +11 , n k +12 , . . . , n k +1 d − (cid:17) = (cid:16) n k , n k , n k , . . . , n kd − (cid:17) . . .
10 0 1 0 . . .
00 0 0 1 . . . . . .
11 0 0 0 . . . := n k R . (2.2)Then, applying recursively (2.2), n k +1 = n R k = jR k . Now, it is readily checked that thecharacteristic polynomial of the above recurrence d × d matrix R is φ ( x ) = x d − (cid:80) d − i =0 x i .5ndeed, since φ ( x ) = det( R − x I ) = det x − − − − . . . − x − . . .
00 0 x − . . . . . . − − . . . x , we can expand the determinant relative to the first line to get φ ( x ) = ( x − x d − − x d − − x d − − · · · − d − × ( d −
1) submatrix haveonly one transversal with nonzero product ( x − x d − , − x d − , x d − , etc.) Then, from R k − d φ ( R ) = O , where O is the all-0 matrix, we get R k = k − (cid:88) i = k − d R i (2.3)or, multiplying both terms by the vector n = j , jR k = n k +1 = k − (cid:88) i = k − d jR i = k − (cid:88) i = k − d n i +1 = k (cid:88) i = k − d +1 n i . (2.4)Hence, N ( d, k + 1) = d − (cid:88) j =0 n k +1 j = d − (cid:88) j =0 k (cid:88) i = k − d +1 n ij = k (cid:88) i = k − d +1 d − (cid:88) j =0 n ij = k (cid:88) i = k − d +1 N ( d, i ) , as claimed. Besides, to show that the recurrence can be initialized with the d values N ( d, i ) = 1 for i = d − , d − , . . . , N ( d,
1) = d , we need to show that N ( d,
2) = N ( d,
1) + d − N ( d,
3) = N ( d,
2) + N ( d,
1) + d −
2, . . . , N ( d, d ) = (cid:80) d − i =1 N ( d, i ) + 1. Withthis aim, note first that N ( d, k ) = (cid:80) d − j =0 n kj = n k j (cid:62) = jR k − j (cid:62) for k = 1 , . . . , d , so thatwe first compute the vectors u k − = R k − j (cid:62) for k = 0 , , . . . , k + 1 to get: u = j (cid:62) = (1 , , , ( d ) . . ., , (cid:62) , u = Rj (cid:62) = ( d, , , ( d − . . . , , (cid:62) , u = R j (cid:62) = R ( u ) (cid:62) = ( d + ( d − , , ( d − . . . , , d ) (cid:62) , u = R j (cid:62) = R ( u ) (cid:62) = (2 d + ( d −
1) + ( d − , , ( d − . . . , , d, d + ( d − (cid:62) , ... u d = R d j (cid:62) = R ( u d − ) (cid:62) = (2 d − d + 2 d − ( d −
1) + · · · + 1 , d, d + ( d − , . . . , d − d + 2 d − ( d −
1) + · · · + 2) (cid:62) . n n n n n
16 8 12 14 15 n
31 16 24 28 30 n
61 31 47 55 59 n
120 61 92 108 116Table 1: The vectors n k , for k = 0 , . . . ,
7, with entries n kj being the numbers of vertices x x . . . x k of F (5 , k ) such that x k = j ∈ [0 , k = 1 , . . . , d , the sum of all entries of u k − equals the first entry u k of u k . Consequently, N ( d, k ) = j ( u k − ) (cid:62) = u k , so giving N ( d,
2) = u = d + ( d −
1) = N ( d,
1) + d − N ( d,
3) = u = 2 d + ( d −
1) + d − N ( d,
2) + N ( d,
1) + d − N ( d, d ) = u d = 2 d − d + 2 d − ( d −
1) + · · · + 1 = d − (cid:88) i =1 N ( d, i ) + 1 , as required.Notice that, from (2.4), we proved that not only the total number of vertices of F ( d, k ),but also those vertices whose sequences end with a given digit j ∈ [0 , d −
1] satisfy thesame recurrence as the d -step Fibonacci numbers in (1.2). For example, for d = 5, Table1 shows the vectors n k , for k = 0 , . . . ,
7, with entries being such number of sequences.Then, we can observe the claimed recurrence n k +1 j = n kj + · · · + n k − j , for k ≥
5, by lookingat each j -th column of the formed array. Although a similar (although more involved) study for general d can be done, we concen-trate here in the case d = 2, where we simply refer to Fibonacci digraphs F ( k ). The reasonis that, from Proposition 2.3, the numbers N ( k ) = N (2 , k ) of vertices of the (2-)Fibonaccidigraphs F ( k ) = F (2 , k ) are N (1) = 2 , N (2) = 3 , N (3) = 5 , N (4) = 8 , N (5) = 13 , N (6) = 21 , . . .
010 10 01001 000100 010 001 0000 00011000 1001 001001001010 0101
Figure 2: The four first Fibonacci graphs as subgraphs of the hypercubes.which corresponds to the standard Fibonacci sequence F , F , F , F , F , F . . . , see againFigure 1. Indeed, it is known that the number of binary sequences of length k withoutconsecutive 1’s is the Fibonacci number F k +2 . For example, among the 16 binary sequencesof length k = 4, there are F = 8 without consecutive 1’s. Namely, 0000, 0001, 0010, 0100,0101, 1000, 1001, and 1010, which are the vertices of F (2 ,
4) in Figure 1. Indeed, suchbinary sequences also correspond to the vertices of the (undirected)
Fibonacci graphs thatare induced subgraphs of the k -cubes. So, two vertices are adjacent when their labels differexactly in one digit. In Figure 2, there are represented the four first Fibonacci graphs.For more information, see Hsu, Page, and Liu [10].Now, let us show a result on the lengths of the cycles in the Fibonacci digraphs. Moreprecisely, we prove that F ( k ) is semi-pancyclic. Proposition 3.1.
For every k ≥ , let (cid:96) = 2 k − if k is odd and (cid:96) = 2 k − if k is even.Then, the Fibonacci digraph F ( k ) is (1 , (cid:96) ) -pancyclic, that is, it contains a cycle of everylength , , . . . , (cid:96) .Proof. A p ( > -periodic vertex of F ( k ) has 1’s in the positions i ( ≤ k ), i + p , i + 2 p ,. . . Then, by cyclically shifting at the left the corresponding sequence, but keeping theperiodicity, such a vertex gives rise to a cycle of length p . For instance in F (7), vertex0001000 gives the 4-cycle0001000 → → → → . The other cycles (of lengths k + 1 , k + 2 , . . . , (cid:96) ) go either through the vertex = 00 . . . = 00 . . .
01. In both cases, if we look at the successive sequences of the cycleas the rows of an array, the entries 1 form a number q = 1 , , . . . , (cid:98) k/ (cid:99) of anti-diagonals,as shown in Table 2 for k = 7 and q = 1 , ,
3. We label the corresponding cycles with theprefixes [ , q ] and [ , q ], respectively. Then, summarizing, we have the following cases: • The vertex gives a cycle of length 1 (a loop). • The p -periodic vertices give cycles of length p for p = 2 , , . . . , k .8 x x x x x x x x x x x x x x x x x x x x F (7). • The [ , q ]-cycles, containing vertex , have length k + 2 q − q = 1 , , . . . , (cid:98) k/ (cid:99) .(For q = 1, the cycle of length k is also obtained in the previous case for p = k .) • The [ , q ]-cycles, containing vertex , have lengths k + 2 q − q = 1 , , . . . , (cid:98) k/ (cid:99) .In particular, for q = (cid:98) k/ (cid:99) , the [ , (cid:98) k/ (cid:99) ]-cycle has length (cid:96) = k + 2 (cid:98) k/ (cid:99) − ∈{ k − , k − } , as required.This completes the proof. d -Fibonacci digraphs as iterated line digraphs The following result shows that the d -Fibonacci digraphs can also be constructed as iter-ated line digraphs. Let T d be the digraph with set of vertices Z d and arcs (0 , i ) for every i ∈ Z d , and arcs ( i, i + 1) for every i = Z d \
0. Thus, T d has d vertices and 2 d − D = d −
1. As examples, seeFigure 3.The adjacency matrix A of T d , indexed by the vertices 0 , , . . . , d −
1, has first row j ,the all-1 vector, and i -th row the unit vector e i +1 , for i = 1 , , . . . , d − d ). Then, A coincides with the recurrence matrix R in (2.2) and,hence, the entries of the powers of A satisfy the recurrence( A k +1 ) uv = k (cid:88) i = k − d +1 ( A i ) uv (4.1)for k ≥ d . 9 T T T T Figure 3: The digraphs T d = F ( d,
1) for d = 2 , , , d -Fibonacci digraphs can also be defined asiterated line digraphs of T d . Proposition 4.1.
The d -Fibonacci digraph F ( d, k ) coincides with the ( k − -iterated linedigraph of T d , that is F ( d, k ) = L k − T d , for k ≥ , with F ( d,
1) = L T d = T d .Proof. We know that the vertices of L k − T d correspond to the walks of length k − T d . But, according to Definition 2.1, such walks are in correspondence with the sequencesof length k defining the vertices of F ( d, k ). Moreover, the adjacencies in L k − T d are thesame as in F ( d, k ).As a consequence of the last proposition and the proof of Proposition 2.3, we have thefollowing result. Proposition 4.2.
Let F ( d, k ) be the d -Fibonacci digraph with N = N ( d, k ) vertices givenby Proposition 2.3. Let A ( k ) and A be, respectively, the adjacency matrices of F ( d, k ) and T d . ( i ) The diameter of the d -Fibonacci digraph F ( d, k ) is D = k + d − . ( ii ) The eigenvalues of F ( d, k ) are the d zeros of the polynomial p ( x ) = x d − x d − − x d − − · · · − (or, alternatively, the d zeros different from 1 of the polynomial q ( x ) = x d +1 − x d + 1 ) plus N − d zeros. ( iii ) Fo any given d, k ≥ , the total number of closed l -walks C l ( d, k ) in F ( d, k ) satisfiesthe same linear recurrence as the d -step Fibonacci numbers in (1.2) , initiated with C l ( d, k ) = tr A l for l = 0 , . . . , d − .Proof. ( i ) Since the diameter of T d is D = d − ii ) Since A = R , the characteristic polynomial φ ( x ) of T d is φ d ( x ) = x d − (cid:80) d − r =0 x r .Hence, from the results in Balbuena, Ferrero, Marcote, and Pelayo [2], the charac-teristic polynomial of F ( d, k ) = L k T d is ψ ( x ) = x N − d φ ( x ), which gives the result.10 iii ) From ( ii ), the nonzero eigenvalues of F ( d, k ) and T ( d ) coincide. Then, for k ≥
1, thetotal numbers of closed walks of length l , with l ≥
1, in F ( d, k ) and in T d coincidebecause tr A ( k ) l = tr A l . But A coincides with the recurrence matrix R in (2.2), sothat, from (4.1) and l ≥ d , C l +1 ( d, k ) = tr A l +1 = l (cid:88) i = l − d +1 tr A i = l (cid:88) i = l − d +1 C i ( d, k ) , (4.2)and C l ( d, k ) = tr A k for l = 0 , . . . , d − d = 2), (4.2) becomes the version of1.3 for the number of closed walks in F (2 , k ). Namely, C l (2 , k ) = φ l + ψ l = (cid:32) √ (cid:33) l + (cid:32) − √ (cid:33) l , (4.3)initiated with C (2 , k ) = 2 and C (2 , k ) = 1. Compare (4.3) with the Binet’s formula F l = (1 / √ φ l − ψ l ).In fact, from (4.1) and the fact that every closed walk of length l in T d gives a closedwalk of the same length in F ( d, k ), and vice versa, we can prove that, for any given j, d ∈ [0 , d − C j ( d, k ) of closed walks in the digraphs F ( d, k ) for k ≥ d follow the same recurrence of the d -step Fibonacci numbers. What is more, the same holdsfor the total number of walks in F ( d, k ), which go from the vertices of type x x . . . j tothe vertices of type x x . . . j (cid:48) for any given j, j (cid:48) ∈ [0 , d − d = 2, this is aconsequence of the following known formula for the powers of the adjacency matrix of T ,as a particular case of (2.3), A k = (cid:18) (cid:19) k = (cid:18) (cid:19) k − + (cid:18) (cid:19) k − = (cid:18) F k +1 F k F k F k − (cid:19) for k ≥ A k ) = F k corresponds to the number of closed walks of length k rooted at vertex in the digraph T is cited in On-line Encyclopedia of Integer SequencesA000045 [11]. 11 eferences [1] M. Aigner, On the linegraph of a directed graph, Math. Z. (1967) 56–61.[2] C. Balbuena, D. Ferrero, X. Marcote, and I. Pelayo, Algebraic properties of a digraphand its line digraph,
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