Connectivity of some Algebraically Defined Digraphs

Let $p$ be a prime, $e$ a positive integer, $q = p^e$, and let $\mathbb{F}_q$ denote the finite field of $q$ elements. Let $f_i : \mathbb{F}_q^2\to\mathbb{F}_q$ be arbitrary functions, where $1\le i\le l$, $i$ and $l$ are integers. The digraph $D = D(q;\bf{f})$, where ${\bf f}=(f_1,\dotso,f_l) : \mathbb{F}_q^2\to\mathbb{F}_q^l$, is defined as follows. The vertex set of $D$ is $\mathbb{F}_q^{l+1}$. There is an arc from a vertex ${\bf x} = (x_1,\dotso,x_{l+1})$ to a vertex ${\bf y} = (y_1,\dotso,y_{l+1})$ if $ x_i + y_i = f_{i-1}(x_1,y_1) $ for all $i$, $2\le i \le l+1$. In this paper we study the strong connectivity of $D$ and completely describe its strong components. The digraphs $D$ are directed analogues of some algebraically defined graphs, which have been studied extensively and have many applications.


Introduction and Results
In this paper, by a directed graph (or simply digraph) D we mean a pair (V, A), where V = V (D) is the set of vertices and A = A(D) ⊆ V × V is the set of arcs. The order of D is the number of its vertices. For an arc (u, v), the first vertex u is called its tail and the second vertex v is called its head; we denote such an arc by u → v. For an integer k 2, a walk W from x 1 to x k in D is an alternating sequence W = x 1 a 1 x 2 a 2 x 3 . . . x k−1 a k−1 x k of vertices x i ∈ V and arcs a j ∈ A such that the tail of a i is x i and the head of a i is x i+1 for every i, 1 i k − 1. Whenever the labels of the arcs of a walk are not important, we use the notation x 1 → x 2 → · · · → x k for the walk. In a digraph D, a vertex y is reachable from a vertex x if D has a walk from x to y. In particular, a vertex is reachable from itself. A digraph D is strongly connected (or, just strong) if, for every pair x, y of distinct vertices in D, y is reachable from x and x is reachable from y. A strong component of a digraph D is a maximal induced subdigraph of D that is strong. For all digraph terms not defined in this paper, see Bang-Jensen and Gutin [1].
Let p be a prime, e a positive integer, and q = p e . Let F q denote the finite field of q elements, and F * q = F q \{0}. We write F n q to denote the Cartesian product of n copies of F q . Let f i : F 2 q → F q be arbitrary functions, where 1 i l, i and l are positive integers. The digraph D = D(q; f 1 , . . . , f l ), or just D(q; f), where f = (f 1 , . . . , f l ) : F 2 q → F l q , is defined as follows. (Throughout all of the paper the bold font is used to distinguish elements of F j q , j 2, from those of F q , and we simplify the notation f((x, y)) and f ((x, y)) to f(x, y) and f (x, y), respectively.) The vertex set of D is F l+1 q . There is an arc from a vertex x = (x 1 , . . . , x l+1 ) to a vertex y = (y 1 , . . . , y l+1 ) if and only if We call the functions f i , 1 i l, the defining functions of D(q; f).
If l = 1 and f(x, y) = f 1 (x, y) = x m y n , 1 m, n q − 1, we call D a monomial digraph, and denote it by D(q; m, n).
We note that F q and F l q can be viewed as vector spaces over F p of dimensions e and el, respectively. For X ⊆ F l q , by X we denote the span of X over F p , which is the set of all finite linear combinations of elements of X with coefficients from F p . For any vector subspace W of F l q , dim(W ) denotes the dimension of W over F p . If X ⊆ F l q , let v + X = {v + x : x ∈ X}. Finally, let Im(f) = {(f 1 (x, y), . . . , f l (x, y)) : (x, y) ∈ F 2 q } denote the image of function f.
In this paper we study strong connectivity of D(q; f). We mention that by Lagrange's interpolation (see, for example, Lidl, Niederreiter [12]), each f i can be uniquely represented by a bivariate polynomial of degree at most q − 1 in each of the variables. We therefore also call functions f i defining polynomials.
In order to state our results, we need the following notation. For every f : As g(0) = h(0) = 0, one can view the coordinate function g i of g (respectively, h i of h), i = 1, . . . , l, as the sum of all terms of the polynomial f i containing only indeterminate the electronic journal of combinatorics 22 (2015), #P00 x (respectively, y), and having zero constant term. We, however, wish to emphasise that in the definition off (x, y), g is evaluated at y, and h at x. Also, we will often write a vector (v 1 The main result of this paper is the following theorem, which gives necessary and sufficient conditions for the strong connectivity of D(q; f) and provides a description of its strong components in terms of Im(f 0 ) over F p .
Then the following statements hold.
(i) If q is odd, then the digraphs D and D 0 are isomorphic. Furthermore, the vertex set of the strong component of D 0 containing a vertex (u, v) is The vertex set of the strong component of D containing a vertex (u, v) is (ii) If q is odd, then D ∼ = D 0 has (p el−d + 1)/2 strong components. One of them is of order p e+d . All other (p el−d − 1)/2 strong components are isomorphic, and each is of order 2p e+d .
If q is even, then the number of strong components in D is 2 el−d , provided f(0, 0) ∈ W 0 , and it is 2 el−d−1 otherwise. In each case, all strong components are isomorphic, and are of orders 2 e+d and 2 e+d+1 , respectively.
We note here that for q even the digraphs D and D 0 are generally not isomorphic. We apply this theorem to monomial digraphs D(q; m, n). For these digraphs we can restate the connectivity results more explicitly.
the electronic journal of combinatorics 22 (2015), #P00 Theorem 2. Let D = D(q; m, n) and let d = (q − 1, m, n) be the greatest common divisor of q − 1, m and n. For each positive divisor e i of e, let q i := (q − 1)/(p e i − 1), and let q s be the largest of the q i that divides d. Then the following statements hold.
(i) The vertex set of the strong component of D containing a vertex (u, v) is In particular, D is strong if and only if q s = 1 or, equivalently, e s = e.
(ii) If q is odd, then D has (p e−es + 1)/2 strong components. One of them is of order p e+es . All other (p e−es − 1)/2 strong components are all isomorphic and each is of order 2p e+es .
If q is even, then D has 2 e−es strong components, all isomorphic, and each is of order 2 e+es .
Our proof of Theorem 1 is presented in Section 2, and the proof of Theorem 2 is in Section 3. In Section 4 we suggest two areas for further investigation.
2 Connectivity of D(q; f ) Theorem 1 and our proof below were inspired by the ideas from [15], where the components of similarly defined bipartite simple graphs were described.
We now prove Theorem 1.
Proof. Let q be odd. We first show that is clearly a bijection. We check that φ preserves adjacency. Assume that ((x 1 , x 2 ), (y 1 , y 2 )) is an arc in D, that is, and so (φ((x 1 , x 2 )), φ((y 1 , y 2 ))) is an arc in D 0 . As the above steps are reversible, φ preserves non-adjacency as well. Thus, We now obtain the description (1) of the strong components of D 0 , and then explain how the description (2) of the strong components of D follows from (1).
Note that as f 0 (0, 0) = 0, we have the electronic journal of combinatorics 22 (2015), #P00 In order to do this, we write an arbitrary y ∈ h(a)−g(u)+W 0 as for some a 1 , . . . , a d ∈ F p , and consider the following directed walk in D 0 : Traveling through vertices whose first coordinates are 0, x 1 , y 1 , 0, 0, and 0 again (steps 6-11) as many times as needed, one can reach vertex (0, v − g(u) + a 1α1 ). Continuing a similar walk through vertices whose first coordinates are 0, x i , y i , 0, 0, and 0, 2 i d, as many times as needed, one can reach vertex (0, v − g(u) + (a 1α1 + . . . + a iαi )), and so on, until the vertex (0, −v + g(u) − (a 1α1 + · · · + a dαd )) is reached. The vertex (a, v + y) will be its out-neighbor. Here we indicate just some of the vertices along this path: Hence, (a, v + y) is reachable from (u, v) for any a ∈ F q and any y ∈ h(a) − g(u) + W 0 , as claimed. A slight modification of this argument shows that (a, −v + y) is reachable from (u, v) for any y ∈ h(a) + g(u) + W 0 .
the electronic journal of combinatorics 22 (2015), #P00 Let us now explain that every vertex of D 0 reachable from (u, v) is in the set {(a, ±v ∓ g(u) + h(a) + W 0 ) : a ∈ F q }.
We will need the following identities on F q and F 2 q , respectively, which can be checked easily using the definition off: The identities immediately imply that for every t, x, y ∈ F q , Consider a path with k arcs, where k > 0 and even, from (u, v) to (a, v + y): Using the definition of an arc in D 0 , and setting f 0 ( Hence, y ∈ −g(x 0 ) + h(x k ) + W 0 . Similarly, for any path with k arcs, where k is odd and at least 1, we obtain y ∈ g(x 0 ) + h(x k ) + W 0 . The digraph D 0 is strong if and only if W 0 = Im(f 0 ) = F l q or, equivalently, d = el. Hence part (i) of the theorem is proven for D 0 and q odd.
Let (u, v) be an arbitrary vertex of a strong component of D. The image of this vertex under the isomorphism φ, defined in (5), is (u, v − 1 2 f(0, 0)), which belongs to the strong component of D 0 whose description is given by (1) with v replaced by v − 1 2 f(0, 0). Applying the inverse of φ to each vertex of this component of D 0 immediately yields the description of the component of D given by (2). This establishes the validity of part (i) of Theorem 1 for q odd. For q even we first apply an argument similar to the one we used above for establishing components of D 0 for q odd. As p = 2, the argument becomes much shorter, and we obtain (3). Then we note that if is a path in D, then where δ = 1 if k is odd, and δ = 0 if k is even.
For (ii), we first recall that any two cosets of W 0 in F kl p are disjoint or coincide. It is clear that for q odd, the cosets (1) coincide if and only if v ∈ g(u) + W 0 . The vertex set of this strong component is {(a, h(a) + W 0 ) : a ∈ F q }, which shows that this is the unique component of such type. As |W 0 | = p d , the component contains q · p d = p e+d vertices. In all other cases the cosets are disjoint, and their union is of order 2qp d = 2p e+d . Therefore the number of strong components of D 0 , which is isomorphic to D, is For q even, our count follows the same ideas as for q odd, and the formulas giving the number of strongly connected components and the order of each component follow from (3).
For the isomorphism of strong components of the same order, let q be odd, and let D 1 and D 2 be two distinct strong components of D 0 each of order 2p e+d . Then there exist Consider a map ψ : V (D 1 ) → V (D 2 ) defined by (a, ±v 1 + h(a) ∓ g(u 1 ) + y) → (a, ±v 2 + h(a) ∓ g(u 2 ) + y), for any a ∈ F q and any y ∈ W 0 . Clearly, ψ is a bijection. Consider an arc (α, β) in D 1 . If α = (a, v 1 + h(a) − g(u 1 ) + y), then β = (b, −v 1 − h(a) + g(u 1 ) − y + f 0 (a, b)) for some b ∈ F q . Let us check that (ψ(α), ψ(β)) is an arc in D 2 . In order to find an expression for the second coordinate of ψ(β), we first rewrite the second coordinate of β as −v 1 + h(a) + g(u 1 ) + y ′ , where y ′ ∈ W 0 . In order to do this, we use the definition of f 0 and the obvious equality where y ′ = (g(b) − h(b)) − y +f 0 (a, b) ∈ W 0 . Now it is clear that ψ(α) = (a, v 2 + h(a) − g(u 2 ) + y) and ψ(β) = (b, −v 2 + h(b) + g(u 2 ) + y ′ ) are the tail and the head of an arc in D 2 . Hence ψ is an isomorphism of digraphs D 1 and D 2 . An argument for the isomorphism of all strong components for q even is absolutely similar. This ends the proof of the theorem.
We illustrate Theorem 1 by the following example.
Example 3. Let p 3 be prime, q = p 2 , and F q ∼ = F p (ξ), where ξ is a primitive element in F q . Let us define f : F 2 q → F q by the following table: As 1 and ξ are values of f , Im(f ) = F 2 q . Nevertheless, D(q; f ) is not strong as we show below.

Connectivity of D(q, m, n)
The goal of this section is to prove Theorem 2.
For any t 2 and integers a 1 , . . . , a t , not all zero, let (a 1 , . . . , a t ) (respectively [a 1 , . . . , a t ]) denote the greatest common divisor (respectively, the least common multiple) of these numbers. Moreover, for an integer a, let a = (q − 1, a). Let < ξ >= F * q , i.e., ξ is a generator of the cyclic group F * q . (Note the difference between < · > and · in our notation.) Suppose A k = {x k : x ∈ F * q }, k 1. It is well known (and easy to show) that A k =< ξ k > and |A k | = (q − 1)/k.
We recall that for each positive divisor e i of e, q i = (q − 1)/(p e i − 1).

Lemma 4.
Let q s be the largest of the q i dividing k. Then F p es is the smallest subfield of F q in which A k is contained. Moreover, A k = F p es .
Proof. By definition of k, q s divides k, so k = tq s for some integer t. Thus for any x ∈ F q , as x (p e −1)/(p es −1) is the norm of x over F p es and hence is in F p es . Suppose now that A k ⊆ F p e i , where e i < e s . Since A k is a subgroup of F * p e i , we have that |A k | divides |F * p e i |, that is, (q − 1)/k divides p e i − 1. Then k = r · (q − 1)/(p e i − 1) = rq i for some integer r. Hence, q i divides k, and a contradiction is obtained as q i > q s . This proves that A k is a subfield of F p es not contained in any smaller subfield of F q . Thus A k = F p es .
Let A m,n = {x m y n : x, y ∈ F * q }, m, n 1. Then, obviously, A m,n is a subgroup of F * q , and A m,n = A m A n -the product of subgroups A m and A n .
Proof. As A m and A n are subgroups of F * q , we have It is well known (and easy to show) that if x is a generator of a cyclic group, then for any integers a and b, < x a > ∩ < We wish to show that |A m,n | = |A d |, and since in a cyclic group any two subgroups of equal order are equal, that would imply A m,n = A d .
Since d = (q − 1, d) = d and |A d | = (q − 1)/d, we have |A m,n | = |A d | and so A m,n = A d .
We are ready to prove Theorem 2.
Proof. For D = D(q; m, n), we have Im(f 0 ) = Im(f ) = Im(x m y n ) = A m,n = A d = F p es , where the last two equalities are due to Lemma 5 and Lemma 4.
Part (i) follows immediately from applying Theorem 1 with W = F p es , g = h = 0. Also, D is strong if and only if F p es = F q , that is, if and only if e s = e, which is equivalent to q s = 1.
The other statements of Theorem 2 follow directly from the corresponding parts of Theorem 1.

Open problems
We would like to conclude this paper with two suggestions for further investigation. Problem 1. Suppose the digraphs D(q; f) and D(q; m, n) are strong. What are their diameters? Problem 2. Study the connectivity of graphs D(F; f), where f : F 2 → F l , and F is a finite extension of the field Q of rational numbers.

Acknowledgement
The authors are thankful to the anonymous referees whose thoughtful comments improved the paper; to Jason Williford for pointing to a mistake in the original version of Theorem 1; and to William Kinnersley for carefully reading the paper and pointing to a number of small errors.