On $d$-Fibonacci digraphs

The $d$-Fibonacci digraphs $F(d,k)$, introduced here, have the number of vertices following generalized Fibonacci-like sequences. They can be defined both as digraphs on alphabets and as iterated line digraphs. Here we study some of their nice properties. For instance, $F(2,k)$ has diameter $d+k-2$ and is semi-pancyclic, that is, it has a cycle of every length between 1 and $\ell$, with $\ell\in\{2k-2,2k-1\}$. Moreover, it turns out 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 $d$-Fibonacci digraphs.


Preliminaries
Let us first introduce some basic notation and results. A digraph G = (V, E) consists of a (finite) set V = V (G) of vertices and a set E = E(G) of arcs (directed edges) between vertices of G. As the initial and final vertices of an arc are not necessarily different, the digraphs may have loops (arcs from a vertex to itself), and multiple arcs, that is, there can 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 adjacent from 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 , and it has entries (A) uv = α if there are α arcs from u to v, with α ≥ 0. Notice that, as we allow 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, that is, V (LG) = {uv : (u, v) ∈ E(G)}; and vertices uv and wz of L(G) are adjacent if and only if v = w, namely, when the arc (u, v) is adjacent to the arc (w, z) in G. The k-iterated line digraph L k G is recursively defined as L 0 G = G and L k G = L k−1 LG for k ≥ 1. It can easily be seen that every vertex of L k G corresponds to a walk v 0 , v 1 , . . . , v k of length k in G, where (v i−1 , v i ) ∈ E for i = 1, . . . , k. Then, if there is one arc between pairs of vertices 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 , 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, with diameter D, then its line digraph L k G is d-regular with n k = d k n vertices and has (asymptotically optimal) diameter D + k. In fact, for a strongly connected general digraph, the first author [5] proved that the iterated line digraphs are always asymptotically dense. For more details, see Harary and Norman [8], Aigner [1], and Fiol, Yebra, and Alegre [7].
Given integers d ≥ 2 and k ≥ 1, the de Bruijn digraph B(d, k) is commonly defined as a digraph on alphabet in the following way. This digraph has vertices For the concepts and results on digraphs not presented here, see, for instance, Bang-Jensen and Gutin [3], Chartrand and Lesniak [4] or Diestel [6].

Generalized Fibonacci numbers
A proposed generalization of the well-known Fibonacci numbers is the following. Given an integer d ≥ 2, the d-step Fibonacci numbers F Thus, the cases d = 2, 3, 4, . . . correspond 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−1 + F k−2 , which, as it is well known, is satisfied by the numbers of the form where a and b are constants, φ = 1+ is the golden ratio, and ψ = φ −1 . Recall also that, from F 0 = 0 and F 1 = 1, we get a = −b = 1/ √ 5, giving the Binet's formula For instance, the 2-Fibonacci digraphs F (2, k) with k ≤ 4 and 2, 3, 5, 8 vertices, are shown in Figure 1, whereas the 1-Fibonacci digraphs F (d, 1) on d vertices, with d ∈ [2,5], are depicted in Figure 3.
Some simple properties of the d-Fibonacci digraphs, which are easy consequences of their definition, are the following.
is isomorphic to its converse.
Proof. (i) follows immediately from Definition 2.1. Similarly, (ii) is a direct consequence of the definitions of F (d, k) and B(d, k). Concerning (iii), notice that the vertices of F (d, k) corresponding to the sequences Alternatively, from the results of Section 4, note that T d is clearly an induced subdigraph of T d for every d ≤ d and, hence, the same property is inherited by To prove (iv), we only need to exhibit the homorphism from F (d, k) to F (d, k ), which is the following map on the corresponding sets of vertices To prove the first part of (v), we only need to realize that every automorphism of F (d, k) must send the unique cycles of lengths 1 and 2 (loop and digon) to themselves. This means that vertex 00 . . . 0 must be fixed, and the vertex set {0101, . . . , 1010 . . .} must be an orbit. But the only way to preserve the adjacencies between these two vertices and 00 . . . 0 is to fix them, which implies that all the other vertices have to be also fixed, and the automorphism is the identity. Finally, the second statement of (v) is justified by the mapping . . x 2 x 1 , which is an isomorphism between F (2, k) and its converse F (2, k).
In contrast with F (2, k), the Fibonacci digraph F (d, k) with d > 2 is not isomorphic to its converse. By using the line digraph approach of Section 4 again, this is a simple consequence of the fact that, for d > 2, T d ∼ = T d . However, the same approach allows to show that most of the properties of F (d, k) related with d-Fibonacci numbers, are shared by its converse F (d, k).
To illustrate case (ii), in Figure 1 each Fibonacci digraph F (2, k) with k ≤ 4 is shown with thick lines as a subdigraph of its corresponding de Bruijn digraph B(d, k). In particular, note that F (d, 1) has d vertices, which coincides with the order d k of the de Bruijn digraph B(d, k) when k = 1. In contrast, the number of vertices of F (d, k) is much smaller when k increases, as the following result shows.
n k j and, from the conditions on the digits x i , we get or, in matrix form, Then, applying recursively (2.2), n k+1 = n 1 R k = jR k . Now, it is readily checked that the characteristic polynomial of the above recurrence we can expand the determinant relative to the first line to get or, multiplying both terms by the vector n 1 = j, Hence, j=0 n k j = n k j = jR k−1 j for k = 1, . . . , d, so that we first compute the vectors u k−1 = R k−1 j for k = 0, 1, . . . , k + 1 to get: 1, 1, (d) . . ., 1, 1) ,  Table 1: The vectors n k , for k = 0, . . . , 7, with entries n k j being the numbers of vertices Notice that, for each k = 1, . . . , d, the sum of all entries of u k−1 equals the first entry u k 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 the same recurrence as the d-step Fibonacci numbers in (1.2). For example, for d = 5, Table  1 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 k j + · · · + n k−4 j , for k ≥ 5, by looking at each j-th column of the formed array.

Fibonacci digraphs
Although a similar (although more involved) study for general d can be done, we concentrate here in the case d = 2, where we simply refer to Fibonacci digraphs F (k). The reason is that, from Proposition 2.   Figure 1. Indeed, such binary sequences also correspond to the vertices of the (undirected) Fibonacci graphs that are induced subgraphs of the k-cubes. So, two vertices are adjacent when their labels differ exactly 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. More precisely, we prove that F (k) is semi-pancyclic. Proof. A p(> 1)-periodic vertex of F (k) has 1's in the positions i(≤ k), i + p, i + 2p, . . . Then, by cyclically shifting at the left the corresponding sequence, but keeping the periodicity, such a vertex gives rise to a cycle of length p. For instance in F (7), vertex 0001000 gives the 4-cycle 0001000 → 0010001 → 0100010 → 1000100 → 0001000.
The other cycles (of lengths k + 1, k + 2, . . . , ) go either through the vertex 0 = 00 . . . 0 or the vertex 1 = 00 . . . 01. In both cases, if we look at the successive sequences of the cycle as the rows of an array, the entries 1 form a number q = 1, 2, . . . , k/2 of anti-diagonals, as shown in Table 2 for k = 7 and q = 1, 2, 3. We label the corresponding cycles with the prefixes [0, q] and [1, q], respectively. Then, summarizing, we have the following cases: • The vertex 0 gives a cycle of length 1 (a loop).
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 iterated 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 2d − 1 arcs. Moreover, it is a strongly connected digraph with diameter D = d − 1. As examples, see Figure 3.
The adjacency matrix A of T d , indexed by the vertices 0, 1, . . . , d − 1, has first row j, the all-1 vector, and i-th row the unit vector e i+1 , for i = 1, 2, . . . , d − 1 (recall that the arithmetic is modulo d). Then, A coincides with the recurrence matrix R in (2.2) and, hence, the entries of the powers of A satisfy the recurrence for k ≥ d. In the following result, we show that the d-Fibonacci digraphs can also be defined as iterated line digraphs of T d .
Proof. We know that the vertices of L k−1 T d correspond to the walks of length k − 1 in T d . But, according to Definition 2.1, such walks are in correspondence with the sequences of length k defining the vertices of F (d, k). Moreover, the adjacencies in L k−1 T d are the same as in F (d, k).
As a consequence of the last proposition and the proof of Proposition 2.3, we have the following result.
(iii) Fo any given d, k ≥ 0, the total number of closed l-walks C l (d, k) in F (d, k) satisfies the 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 − 1.
Proof. (i) Since the diameter of T d is D = d − 1 , the result follows from Proposition 4.1 and the results in Fiol, Yebra, and Alegre [7].
(iii) From (ii), the nonzero eigenvalues of F (d, k) and T (d) coincide. Then, for k ≥ 1, the total numbers of closed walks of length l, with l ≥ 1, in F (d, k) and in T d coincide because tr A(k) l = tr A l . But A coincides with the recurrence matrix R in (2.2), so that, from (4.1) and l ≥ d, and C l (d, k) = tr A k for l = 0, . . . , d − 1.
For instance, in the case of Fibonacci digraphs (d = 2), (4.2) becomes the version of 1.3 for the number of closed walks in F (2, k). Namely, initiated with C 0 (2, k) = 2 and C 1 (2, k) = 1. Compare (4.3) with the Binet's formula In fact, from (4.1) and the fact that every closed walk of length l in T d gives a closed walk of the same length in F (d, k), and vice versa, we can prove that, for any given j, d ∈ [0, d − 1], the numbers 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 holds for the total number of walks in F (d, k), which go from the vertices of type x 1 x 2 . . . j to the vertices of type x 1 x 2 . . . j for any given j, j ∈ [0, d − 1]. In the case of d = 2, this is a consequence of the following known formula for the powers of the adjacency matrix of T 2 , as a particular case of (2.3), The fact that (A k ) 00 = F k corresponds to the number of closed walks of length k rooted at vertex 0 in the digraph T 2 is cited in On-line Encyclopedia of Integer Sequences A000045 [11].