Strongly connected multivariate digraphs

Generalizing the idea of viewing a digraph as a model of a linear map, we suggest a multi-variable analogue of a digraph, called a hydra, as a model of a multi-linear map. Walks in digraphs correspond to usual matrix multiplication while walks in hydras correspond to the tensor multiplication introduced by Robert Grone in 1987. By viewing matrix multiplication as a special case of this tensor multiplication, many concepts on strongly connected digraphs are generalized to corresponding ones for hydras, including strong connectedness, periods and primitiveness, etc. We explore the structure of all possible periods of strongly connected hydras, which turns out to be related to the existence of certain kind of combinatorial designs. We also provide estimates of largest primitive exponents and largest diameters of relevant hydras. Much existing research on tensors are based on some other definitions of multiplications of tensors and so our work here supplies new perspectives for understanding irreducible and primitive nonnegative tensors.


Hydras and weighted hydras
We reserve the notation N for the set of positive integers.For every n ∈ N, we write [n] for the set of the smallest n positive integers.For a set K and a positive integer t, we understand K t as K [t] , namely the set of all maps from [t] to K. Thus, an element x ∈ K t can be specified by its values x(i) for i ∈ [t].We will also often directly write x i for x(i), as K t can be understood as the set of all length-t words on K.The binary semifield, denoted by T, is the ring consisting of two elements, 0 and 1, in which the arithmetic operation is the same with the binary field, excepting that 1 + 1 now equals to 1, not to 0 as in the binary field.Note that each element f ∈ T K can be identified with its support, i.e., {k ∈ K : f (k) = 1}, and then, T K is often naturally identified with the power set of K, which is more commonly denoted by 2 K .It is clear that the addition/multiplication operation in T K corresponds to the taking union/intersection operation in 2 K .
Let K be a set.A digraph Γ on K consists of a pair (K, E) where E ⊆ K × K = K 2 .Usually, we call K the vertex set of Γ and E the arc set of Γ, and denote them by V(Γ) and A(Γ).More generally, given a ring R, we can consider a weight function w from A(Γ) to R and get an R-weighted digraph.For every (k, ) ∈ (K × K) \ A(Γ), we can think that w assigns weight 0 ∈ R to (k, ) and so, in this way, a general weighted digraph on K with weights/variables from R is simply a map w Γ from K × K to R. A digraph Γ given as a pair (V(Γ), A(Γ)) can be viewed as a T-weighted digraph, where the associated weight function w Γ defined on V(Γ) × V(Γ) sends (k, ) to 1 ∈ T if and only if (k, ) ∈ A(Γ).
An R-weighted digraph Γ on K can be represented as a map f from K to R K where, for every k ∈ K, f (k) ∈ R K sends each ∈ K to w Γ ((k, )) ∈ R.This map f from K to R K and the weight function w Γ from K × K to R surely determines each other and so we will write Γ f for this digraph Γ corresponding to f .In the case that R = T, This observation allows us to identify an R-weighted digraph on K with a single variable map from K to R K , and especially, to identify a digraph with a single variable map from K to 2 K .Then, is there a multivariate counterpart for weighted digraphs?Yes, for any positive integer t, we could simply call any map from K t to R K a t-variable digraph on K with variables from a ring R. But, does it lead to any interesting mathematics?A Markov chain is a sequence of possible events in which the probability of every event relies linearly on the probability of the states attained in the previous event.More generally, for any positive integer t, an order-t Markov chain is a sequence of possible events in which the probability of each event depends multi-linearly on the probability of the states attained in the previous t events.It is widely believed that everything about higher-order Markov chains can be encoded as something about Markov chains and so it does not make sense to study higher-order Markov chains separately [LM95, Example 1.5.10,Proposition 1.5.12].We suggested multivariate graph theory in [WXZ16] as a framework to understand multi-linear phenomena, including higher-order Markov chains.arc set A(PS f ) = {A → M f (A) : A ∈ 2 K }; see Figure 1 for an example.The structure of PS f tells us the dynamical behaviour of M f .For any positive integer n, let Z n denote the cyclic group of integers modulo n and let C n denote the n-cycle, which is the digraph with V(C n ) = Z n and A(C n ) = {i → i + 1 : i ∈ Z n }.When K is finite, a traveller driven by M f in PS f will eventually run around a limit cycle repeatedly after staying for a while, so-called transient time, in the transient part, which is an in-tree attached to the limit cycle.Let K be a set and t a positive integer.As advised earlier in this note, let us define a t-variable digraph f on K to be a map from K t to 2 K .The map f induces a Boolean t-linear map from (2 K ) t to (2 K ) t , called the Markov operator associated to f and denoted by M f , such that M f (A 1 , . . ., A t ) = A 2 , . . ., A t , ∪ (k 1 ,...,kt)∈A 1 ×•••×At f (k 1 , . . .k t ) for all (A 1 , . . ., A t ) ∈ (2 K ) t .The phase space of f , denoted by PS f , is the digraph which has vertex set V(PS f ) = (2 K ) t and arc set A(PS f ) = {A → M f (A) : A ∈ (2 K ) t }.For an order-t Markov chain, we can establish its nonparametric model with a t-variable digraph in an obvious way [WXZ16, Eq. (3.3)] and the phase space of this t-variable digraph displays the evolution of the supports of the random events.Surely, when K is finite, to understand PS f is also to understand those limit cycles, transient time and transient parts, as in the case of t = 1.
For A = (A 1 , . . ., A t ) ∈ (2 K ) t and B = (B 1 , . . ., B t ) ∈ (2 K ) t , we write A B provided A i ⊆ B i for i ∈ [t].We identify the set K with K 1 and so view K t as a subset of (2 K ) t .Therefore, for each A ∈ (2 K ) t and a ∈ K t , a ∈ A is equivalent to a A. For any two t-variable digraphs f and g on K, we say that f is bigger than g provided f = g and f (a) ⊇ g(a) for all a ∈ K t .Under this partial order, the biggest t-variable digraph on K is the digraph f t,K satisfying f t,K (a) = K for all a ∈ K t .
We often directly call the set K t = V(Γ f ) the vertex set of f , denoted by V(f ), and call A(Γ f ) the arc set of f , denoted by A(f ).The De Bruijn form of f t,K is just the famous dimension-t De Bruijn digraph on the alphabet K [dB46, DW05,Goo46], which we denote by B(t, K), and the De Bruijn form of any digraph of several variables must be a spanning subgraph of a De Bruijn digraph.Note that B(t, K) has vertex set V(f t,K ) = K t and arc set A(f t,K ) = {(k 1 , . . ., k t ) → (k 2 , . . ., k t+1 ) : k 1 , . . ., k t+1 ∈ K}, the latter corresponding to K t+1 in a natural way.By mapping a multivariate digraph to its De Bruijn form and mapping a De Bruijn form to its arc set, we get a bijection from the set of t-variable digraphs on K to the set of spanning subgraphs of B(t, K) and then to the set of subsets of K t+1 .This suggests that multivariate digraphs are generalizations of both hypergraphs [Ber89, Bol86,Wan08] and digraphs [BJG09, BM08,BR91].For this reason, we propose to call a t-variable digraph a t-head hydra, or simply a t-hydra.
When we mention a digraph below, we mean a 1-hydra; we speak of a hydra for any thydra for some t ∈ N .We should mention that a natural generalization of both digraphs and hypergraphs, called directed hypergraphs, have been well studied in the literature [All14,GLPN93].For a t-hydra f , one can view Γ f as the local dynamical mechanism and PS f as the global evolving picture.The question is to see how to link the local with the global.In light of the higher-order Markov chain model, one can also think of Γ f as the particle version of f and PS f as the wave version of f .Both versions encode full information about f in some way but the transformation between different representations may involve nontrivial mathematics.Note that the study of PS f is nothing but the study of the tlinear map M f .The aim of this paper is to develop some hopefully new perspective for getting a better appreciation of multilinear phenomena.
Let K be a set, t a positive integer and R a ring.A t-fold tensor over R K is a map f from K t to R. Following the convention in graph algebra [Rae05], we define the adjacency tensor of a t-hydra f on K, denoted by A(f ), to be the (t + 1)-fold tensor over T K , also called a Boolean t+1 K × • • • × K-array, whose (i 1 , . . ., i t+1 )-entry, where (i 1 , . . ., i t+1 ) ∈ K t+1 , is given by When t = 1, A(f ) coincides with M f and it is more commonly known as the adjacency matrix of f.The set of all Boolean t+1 K × • • • × K-arrays are often named as order-(t + 1) dimension-K Boolean tensors (in coordinate forms).Note that every order-(t + 1) tensor is the adjacency tensor of a corresponding t-hydra.Therefore, hydras and tensors are the electronic journal of combinatorics 24(1) (2017), #P1.47 different representations of the the same objects and we often do not distinguish between a hydra f and its associated tensor A(f ) in the rest of the paper.For any two t-hydras f and g on K, we can surely do the composition of M f and M g .But, unlike the digraph case, we should warn the reader that there may not exist another t-hydra h such that M h = M f M g .In an obvious way, we can introduce R-weighted hydras for any ring R. All the concepts discussed above for hydras, which are simply T-weighted hydras or Boolean hydras, can be extended to general weighted cases.Especially, t-fold tensors over R K can be viewed the same as R-weighted (t + 1)-hydras via the general concept of adjacency tensors.Note that tensors and its practical applications [Lan12,Stu16] have been a very active field of research in last decade.Especially, some definitions of tensor multiplication/composition are available from [Gro87,Sha13,Wil73,Yam65]. By resorting to the higher-order Markov chain background, the following definition of tensor multiplication looks to be natural and it can be checked to be isomorphic with the one [Gro87, Eq. ( 2)] posed by Grone in 1987.For any two (t + 1)-fold tensors B and C over R K , we define the product of B and C to be the (t + 1)-fold tensor over R K , which is denoted by B C and sends (k (1) Surely, when K is an infinite set, to make Eq.(1) well-defined, we need to impose some local finiteness assumption or some convergence assumption.For t = 1, Eq. (1) becomes the rule for the usual matrix multiplication: The idea of Eq. ( 1) is that the weight for a transition from 1 • • • t at initial t slots to to t+2 at time t + 2 via B C should be the sum of all weights as given by the following transition , where t+1 runs through all possible states of the system at time t + 1.That is, matrix multiplication corresponds to walks in weighted complete digraphs while tensor multiplication corresponds to walks in general weighted De Bruijn digraphs.With the multiplication given in Eq. (1), the set of (t + 1)-fold tensors over R K form the tensor algebra A(t + 1, R, K).The map f that sends (k 1 , . . ., k t+1 ) ∈ K t+1 to δ k 1 ,k 2 is the left identity for the tensor multiplication in A(t + 1, R, K).Like the Lie bracket product in Lie algebras, the tensor product defined in Eq. (1) may not be associative and so A(t + 1, R, K) is generally not an associative algebra.We should not forget that dynamics is the study of change and change takes place within time [Fur14,Preface].Since our multiplication is abstracted from the evolution of a dynamical system, it is natural that we should do the composition of maps according to the flow of time, that is, multiplication should be done from right to left, and so it does not really make sense to require the associativity.To understand the structure of A(t + 1, R, K) as an nonassociative algebra [Sch95] may be an interesting direction.If the adjacency matrix A(f ) of a digraph f is symmetric, the digraph f is called a symmetric digraph or simply a graph.A large part of graph theory is about graphs and so, for the purpose of extending that part to multivariable case, let us define symmetric hydras.Let f be a t-hydra on a set K. The reversal of f, denoted by In Figure 2, we depict a 2-hydra on {a, b} and its reversal.We call a t-hydra symmetric provided f = ← − f .Though the local correspondence between a hydra and its reversal is quite straightforward, the relationship between their phase spaces seems not so trivial to tell in case the hydra has more than one variables.
Recall that a digraph f on a set K is strongly connected if for all a, b ∈ K there exists a nonnegative integer Here is an easy way to generalize this concept for hydras.Let f be a t-hydra on a set K. For a, b ∈ K t , we define RI f (a, b) to be the set a)} and call it the set of reachable indices of f from a to b.An element from (2 K ) t which has ∅ as one of its t components is called a vacant element.It is clear that for any vacant element a in (2 K ) t , N \ RI f (a, ∅ t ) is a finite subset, that is, M N f (a) = ∅ t when N is large enough.We say that f is strongly connected if RI f (a, b) = ∅ for all a, b ∈ K t .When f is strongly connected and |K| 2, both ∅ t and K t give rise to length-1 limit cycles in PS f , i.e., they are fixed points of M f .We call f primitive if there exists an integer N > 0 such that for all a ∈ K t , M N f (a) = K t .Equivalently, f is primitive if and only if we can find For a primitive t-hydra f on K, the primitive exponent of f , which we denote by g(f ), is the minimum positive integer N such that M N f (a) = K t for all a ∈ K t .If g is a primitive t-hydra on K and if f is bigger than g, we can derive that f is primitive and g(f ) g(g).
Note that f t,K is primitive with In terms of the concept of reachable indices, many more basic concepts for digraphs can be naturally extended to hydras, say distance, diameter, radius, girth, etc.For example, let us define the distance from a to b in f to be for every a, b ∈ K t , and the diameter of f to be Dia(f ) = max a,b∈K t Dist f (a, b).So far, we have introduced some basic concepts in multivariate graph theory via the phase space, namely by examining the action of the Markov operator.It is surely possible to give these definitions via the concept of tensor multiplication as presented in Eq. (1).The current approach focuses on the combinatorial core of many situations.But, to start from Eq. (1) will allow us go beyond binary semifield to max-times semiring [Ser09] or to real/complex numbers [CPZ08,Lim05], and even go from homogeneous to nonhomogeneous [WZ15].
In graph theory, more precisely, in one-variable graph theory, there are very rich results about strongly connected digraphs, say the cyclicity theorem [BCOQ92, Theorem Then, should we expect some new mathematics in multivariate graph theory?We will first display in § 2 some possibly counter-intuitive examples to acquaint the reader with some concepts in multivariate graph theory and to show the difference caused by several variables.Our main contribution in this paper will be an analysis of the structure of strongly connected hydras, including some results related to the aforementioned cyclicity theorem and Wielandt's theorem.We summarize our main observations on strongly connected hydras in § 3 and then, in § 4, § 5 and § 6 we develop some technical apparatus to verify our claims in § 3.

Surprises from several variables
Let f be a hydra on a finite set K. (2) Note that in [WXZ16, Example 3.5] we construct a primitive 3-hydra f on [2] whose De Bruijn form is not strongly connected and 6 = g(f  Example 1.In Figure 3 we depict the De Bruijn form of a primitive 2-hydra Let us recall the classical wheels-within-wheels theorem of Knuth on strongly connected digraphs [Knu74, Lemma 1]. Theorem 2 (Knuth, 1974).Every strongly connected digraph f (possibly infinite) is either a single vertex with no arcs, or it can be represented as in Figure 4 for some n ∈ N.Here Γ 1 , . . ., Γ n are strongly connected digraphs; x i and y i are (possibly equal) vertices of Γ i ; and e i is an arc from y i to x i+1 .The original digraph f consists of the vertices and arcs of Γ 1 , . . ., Γ n plus the arcs e 1 , . . ., e n .
In fact, if σ is any given cycle of f , there exists such a representation in which each of the e i is contained in σ.
The next example is indeed reporting our failure in extending the wheels-within-wheels theorem to hydras.
Example 3. Let f be the 2-hydra on [4] whose De Bruijn form Γ f is shown in Figure 5.The hydra f is strongly connected but its De Bruijn form Γ f has two weakly connected components.There is no way to partition V(f ) = [4] 2 into V 1 , . . ., V n for some n ∈ N such that the following hold.(a) For every i ∈ Z n and all x, y ∈ V i , we have For a strongly connected digraph f , its reversal must also be strongly connected and we can establish a natural one-to-one length-preserving correspondence between the limit cycles of PS ← − f and those of PS f .However, when we enter the world of several variables graph theory, the arrow of time plays some magic, as Example 4 below illustrates.Example 4. In Figure 6, we demonstrate a primitive 2-hydra f on K = {a, b} whose reversal is not strongly connected.Note that PS f contains two limit cycles while PS ← − f has three limit cycles.
Question 5.For any hydra f , let w(f ) be the number of weakly connected components of Γ f .If f is a strongly connected t-hydra, is there any good upper bound estimate for w(f )? Surely, w(f ) equals 1 when t = 1.
As with most non-numerical properties, the study of nonnegative tensors is equivalent to the study of corresponding Boolean tensors; see our brief discussion in [WXZ16,p. 404].Let us then formulate some discussions in the literatures on nonnegative tensors in the language of Boolean tensors below.Following [CPZ08,Lim05], we call an order-t dimension-K Boolean tensor A irreducible if for all nonempty subsets I K we can find i 1 ∈ I and i 2 , . . .i t ∈ K \ I such that A i 1 ,i 2 ,...,it = 1.A hydra is irreducible whenever so is its adjacency tensor.It is well-known that a digraph is strongly connected if and the electronic journal of combinatorics 24(1) (2017), #P1.47The 2-hydra f on K = {a, b} is primitive while its reversal is not strongly connected.We do not include those vacant vertices when displaying the phase spaces.
The symbols aa and aK appeared in the two phase spaces should be understood as {a} × {a} and {a} × K, respectively, and so on.
only if it is irreducible.It is easy to show one direction of it for general hydras.But the equivalence itself cannot be generalized to hydras.
Proposition 6.Let f be a t-hydra on a set K. If f is strongly connected, then the adjacency tensor A(f ) is irreducible.
Proof.Assume to the contrary that there is a set I such that ∅ I K and that A(f ) i 1 ,...,i t+1 = 0 for every i 1 ∈ I and i 2 , . . ., i t+1 ∈ K \ I. Pick any a ∈ (K \ I) t and b ∈ I t .It is clear that RI f (a, b) = ∅ and so f is not strongly connected, yielding a contradiction.
Let A be an order-(t + 1) dimension-K Boolean tensor and let f be the corresponding hydra.Let C A be the map from 2 we display a primitive hydra which is not CPZ-primitive.The next example, Example 8, shows that being CPZ-primitive may not imply being strongly connected.In Figure 8, we briefly demonstrate the relationship among primitive, CPZ-primitive, strongly connected and irreducible hydras.
We depict Γ f and PS f in Figure 7.We can check that A(f ) is CPZ-primitive but not strongly connected.
Let A be a t-hydra on a set K. Let us define A 1 to be the digraph on K with A n+1 inductively by setting A n+1 i,j . ., i t ∈ K}.For every j ∈ K, A is said to be j-primitive [YHY14, Definition 2.14] provided there exists n ∈ N such that A n i,j = 1 holds for all i ∈ K.One can check that A n i,j > 0 if and only if i ∈ C n A ({j}) for every i, j ∈ K. Therefore, A is CPZ-primitive if and only if A is j-primitive for all j ∈ K.For t = 1, it is well-known that A is primitive if and only if it is CPZ-primitive and if and only if it is irreducible and j-primitive for any j ∈ K. Yuan et al. posed the conjecture [YHY15, Conjecture 4.5] that this observation for digraphs holds for general hydras, namely a hydra A on K is CPZ-primitive if and only if A is irreducible and there exists some j ∈ K such that A is j-primitive.The following example shows that the situation is more intricate than expected.
We can check that A is irreducible.For n ∈ N, we can also check that This means that A is neither 2-primitive nor 3-primitive, and hence not CPZ-primitive, but 1-primitive.Note that this refutes [YHY15, Conjecture 4.5].Also observe that A is not strongly connected as

Main results
Let f be a t-hydra on K.It is natural to consider the asymptotic equivalence for f , denoted by ∼ f , which is the binary relation on K t such that, for all a, b ∈ K t , a ∼ f b if and only if there exists a positive integer m such that M m f (a) = M m f (b).It is clear that asymptotic equivalence is an equivalence relation.Let C(f ) be the set consisting of all equivalence classes of ∼ f .We construct the digraph f * with vertex set C(f ) and arc set Let per(f ) denote the greatest common divisor of a∈K t RI f (a, a) and call it the period of f .Having in mind the partial order relation of divisibility on the nonnegative integers, we adopt the convention that the greatest common divisor of the set {0} is 0. We call a digraph on one vertex without any arc a 0-cycle.Note that both 0-cycle and 1-cycle are strongly connected, but among them only 1-cycle is primitive.If f is strongly connected and per(f ) = 0, we can deduce that V(f ) is a singleton set and A(f ) = ∅ and hence Γ f is a 0-cycle.The next result means that the cyclicity theorem for digraphs indeed holds for all hydras, namely every strongly connected hydra with a finite diameter looks like a cycle when viewed from a distance.
the electronic journal of combinatorics 24(1) (2017), #P1.47 Theorem 10.Let f be a strongly connected hydra with a finite diameter.Then f * is a per(f )-cycle.Moreover, if per(f ) > 0, then there exists m ∈ N such that M n f (b) is an equivalence class of ∼ f for every b ∈ V(f ) and n m.
Here is an immediate consequence of Theorem 10.It is noteworthy that a counterpart of it for the CPZ-primitiveness has been established by Cui et al. [CLN15,Theorem 4] in the language of directed hypergraph.
Corollary 11.Let f be a hydra with a finite diameter.Then, f is primitive if and only if f is strongly connected and per(f ) = 1.
Example 12. Let Z be the set of all integers.Let f be the digraph with V(f ) = Z and Let t ∈ N and let K be a set.We use P(t, K) to stand for the set of periods of all those strongly connected t-hydras f on K with A(f ) = ∅ and Dia(f ) < ∞.For any k ∈ N, we write P(t, k) for P(t, [k]).Let P(t) stand for ∪ k∈N P(t, k).Considering all cycles, we see that P(1) = N.
The periods of strongly connected hydras turn out to be related to a problem on combinatorial design, namely the construction of cyclic decompositions.We elucidate this problem in the sequel.For a map Φ defined on a cyclic group Z p , we often write Φ i for Φ(i) for any i ∈ Z p .Let t be a positive integer, K a set, and X ⊆ K t .A cyclic decomposition of (X, K, t) with period p ∈ N, also called a cyclic decomposition of X relative to K with period p, is a map Φ : for all {i, j} ∈ Zp 2 .We use the notation per(Φ) for the period p of Φ. Theorem 13.Let K be a set and t be a positive integer.There exists a cyclic decomposition Φ of (K t , K, t) with per(Φ) = p if and only if p ∈ P(t, K).Theorem 14.Let t be a positive integer and let K be any infinite set.Then P(t) = P(t, K) ⊇ • • • P(t, 4) P(t, 3) P(t, 2) P(t, 1) = {1}.the electronic journal of combinatorics 24(1) (2017), #P1.47 We can conclude that 9 / ∈ P(3), for which we do not include a proof here due to the length constraint.We mention that this implies α(3) 10.Note that Theorem 22 shows α(2) = 8 > 1 = α(1).
The minimum possible m that satisfies the requirement in Theorem 10 is called the transient time of f .When f is primitive, the transient time is just the primitive exponent g(f ).A good estimate of transient time for a strongly connected hydra may not be easy.We only have a look at the primitive exponent in this paper.For any t, k ∈ N, define γ(t, k) to be the maximum possible value of the primitive exponent of a primitive t-hydra on [k].
Let f and g be two t-hydras on a set K. We say that f is weakly isomorphic to g if there exists a bijection φ from V(f ) to V(g), called a weak isomorphism, such that for all x, y ∈ V(f ), xy ∈ A(f ) if and only if φ(x)φ(y) ∈ A(g).We say that f is isomorphic to g if there exists a permutation τ on K such that (τ, . . ., τ t ) gives rise to a weak isomorphism from f to g.It is not obvious if there will be any relationship between PS f and PS g provided f and g are weakly isomorphic.But if f and g are isomorphic, their dynamical behaviours can be said to be really of no difference.
Let r k be the minimum number of multiplications to multiply two k by k complex matrices.By coincidence, for all k ∈ N? Note that we [WXZ16, Conjecture 3.6] have made the conjecture that when k approaches the infinity.If both Eq.(5) and Eq. ( 6) are correct, we will have the astounding result that, asymptotically, it is as easy to multiply matrices as it is to add them.
In [CW17], we will show that γ(t, k) t(k − 1)(k t − 1) + 1 for all k, t ∈ N. Unlike the case of t = 1, to determine γ(t, k) for general t and k looks to be quite challenging.In contrast, we mention that for the concept of CPZ-primitiveness, it turns out that the maximum primitive exponents of Boolean tensors are not related to the order of the tensors [YHY14, Theorem 1.2], namely the number of variables does not make any difference then.
For any t, k ∈ N, let D t,k be the maximum possible diameter of a strongly connected t-hydra f on a set of size k.
Example 33.Making use of computer search, we can determine that D 2,2 = 4 and D 2,3 = 15.Up to isomorphism, there are four 2-hydras on [2] whose diameters achieve D 2,2 = 4 and twenty-six 2-hydras on [3] whose diameters achieve D 2,3 = 15.It is surprising that all these 2-hydras are primitive while the reversal of none of them is strongly connected.Indeed, those four 2-hydras on [2] all have primitive exponent 4 and the primitive exponents of those twenty-six 2-hydras on [3] range among 15, 16, 17 and 18.We display three such extremal hydras in Figure 11.The hydra g 4 in Figure 10 shows the fact that D 2,4 28.Example 33 is about our knowledge of D 2,k for k 4. We now provide an estimate of D 2,k for k 5.
Theorem 34.For any integer k greater than 4, it holds the electronic journal of combinatorics 24(1) (2017), #P1.47 Our proof of Theorem 31 and Theorem 34 relies on the construction of a family of 2-hydras introduced at the beginning of § 6.The experience of going through that proof leads us to the following conjecture.
In order to unfold deep secrets hidden in seemingly messy high dimensional data, it is often crucial to get a "correct" definition of a cycle in a hypergraph [Wan08,Wan11].For hydras, we also face the problem of deciding what is a real cycle or a real hole of a multilinear map.A cycle in the usual digraph setting could be characterized in many ways.For example, the n-cycle C n could be characterized as the strongly connected digraph on n vertices with maximum possible diameter and then with minimum number of arcs.We leave it as an open problem to find a good definition of a cycle hydra, which hopefully can bring forth many new concepts for hydra theory, say zeta functions [Ter11] or homology theory [Big97], besides those followed from the definition of reachable indices.

Generalized cycles
This section is mainly devoted to a proof of Theorem 10, which closely follows the usual proof of the corresponding result for digraphs.With the help of Theorem 10, we will deduce Theorem 13, a result which becomes trivial for digraphs.
Lemma 36.Take a set K, an integer t ∈ N, and a, b Lemma 37. Let f be a strongly connected t-hydra on a set K.
(a) For every a ∈ K t , the greatest common divisor of RI f (a, a) equals to per(f ).
(b) For any a, b ∈ K t , the elements of RI f (a, b) are congruent modulo per(f ).
Proof.(a) Let a and b be two vertices of f .Let p a and p b denote the greatest common divisors of RI f (a, a) and RI f (b, b), respectively.It suffices to show p a = p b .Let s ∈ RI f (a, a).Due to the symmetry between a and b, we need only verify that p b is a divisor of s.Since f is strongly connected, we can take t 1 ∈ RI f (b, a) and t 2 ∈ RI f (a, b).By Lemma 36, t 1 + t 2 and s + t 1 + t 2 are contained in RI f (b, b) and therefore p b divides s, as required.
(b) Take t 1 , t 2 ∈ RI f (a, b).Pick s ∈ RI f (b, a) and hence, by Lemma 36, s+t 1 , s+t 2 ∈ RI f (a, a).According to (a), per(f ) is a divisor of both s + t 1 and s + t 2 .This shows that t 1 − t 2 = (s + t 1 ) − (s + t 2 ) is a multiple of per(f ), finishing the proof.
For any hydra f with period p, any a ∈ V(f ) and i ∈ N, use C i,a (f ) as a shorthand for f ) and so it is often natural to view the parameter i of C i,a (f ) as an element of Z p .
the electronic journal of combinatorics 24(1) (2017), #P1.47 Lemma 38.Let f be a strongly connected t-hydra on a set K with period p > 0.
(c) This is a consequence of (a) and (b).
The next result, Lemma 39, is in the folklore [KS76, Theorem 1.4.1][RA05, Theorem 1.0.1].The number N (X) appeared in it is the so-called Frobenius number in the Postage Stamp Problem.Its estimate may be useful in bounding the transient time of a strongly connected hydra.
Lemma 39.Let X be a nonempty subset of positive integers closed under integer addition and let gcd(X) = p.Then there exists a smallest nonnegative integer N (X) such that, for all integers x > N (X), x ∈ X holds if and only if p | x.
Lemma 40.Let f be a strongly connected t-hydra on set K with Dia(f ) < ∞ and per(f ) = p ∈ N .Let a ∈ K t .There exists a number m ∈ N such that M f (b) = C j,a (f ) whenever b ∈ C i,a (f ), > m and + i ≡ j (mod p).
Proof.Applying Lemma 39 for X = RI f (a, a), we know the existence of a positive integer R such that pr ∈ RI f (a, a) whenever r R. Lemma 36 then implies that x + pr ∈ RI f (a, b) for any b ∈ V(f ), x ∈ RI f (a, b) and r R. By Lemma 38(b), we can take m = Dia(f ) + Rp, completing the proof.
Proof of Theorem 10.The result is trivial when per(f ) = 0.When per(f ) ∈ N, it follows from Lemma 38 and Lemma 40.
Proof of Theorem 13.Let Φ be a cyclic decomposition of (K t , K, t) with per(Φ) = p ∈ N.For each i ∈ Z p , we let C i be the nonempty set the electronic journal of combinatorics 24(1) (2017), #P1.47 It is clear that f is a strongly connected t-hydra with Dia(f ) p + t and with C i , i ∈ Z p , being all the ∼ f equivalence classes.It then follows from Theorem 10 that p = per(f ) ∈ P(t, K).
Conversely, assume there exists a strongly connected t-hydra f on K with period p ∈ N and finite diameter.Observe that, for any b ∈ K t and n ∈ N, M n f (b) must be of the form L 1 × • • • × L t , where L i are nonempty subsets of K for i ∈ [t].Therefore, according to Theorem 10, there exists a cyclic decomposition Φ of (K t , K, t) with period p ∈ N, for which Φ i × • • • × Φ i+t−1 is an equivalence class of ∼ f for every i ∈ Z p .

Periods
For any set K, t ∈ N and X ⊆ K t , we call a cyclic decomposition Φ of (X, K, t) a discrete cyclic decomposition if per(Φ) = |X|.
Lemma 41.Let K be a set and and t be a positive integer.Then (K t , K, t) admits a discrete cyclic decomposition if and only if |K| ∈ N.
Lemma 42.Let t be a positive integer.
Proof.For any map Φ from Z p to 2 K \ {∅}, any k ∈ K and any set X disjoint from K, we put Φ k,X to be the map from with period p ∈ N, we can check that, for any k ∈ K 1 , Φ k,K 2 \K 1 is a cyclic decomposition of (K t 2 , K 2 , t) with period p. Claim (a) now follows from Theorem 13.If Φ is a strong cyclic decomposition of ( the electronic journal of combinatorics 24(1) (2017), #P1.47 Lemma 43.Let , k, t ∈ N and assume k.Then t ∈ P(t, k).If 1 < , then we also have t ∈ P * (t, k).
Proof.This follows immediately from Lemma 41 and Lemma 42.
Take p ∈ P(t, K).By Theorem 13, there exists a cyclic decomposition Φ of (K t , K, t) with period p.For any x ∈ K t , let ξ(x) denote the unique element i ∈ Z p such that and so, the definition of a cyclic decomposition ensures ξ(x) = ξ(y), which means that ξ(x) is the only element i in Z p such that (c 1 , . . ., c t ) ∈ Ψ i × • • • × Ψ i+t−1 .We now conclude that Ψ is a cyclic decomposition of (C t , C, t) with period p, and so, by Theorem 13, p ∈ P(t, |C|) ⊆ P(t), as was to be shown.
For a map Φ defined on Z p , we let σ p (Φ) be the map on Z p such that for all i ∈ Z p .If Φ is a map defined on [p], we naturally identify it as a map on Z p and in this way we can also talk about the map σ p (Φ).
Lemma 44.Let K be a set and t be a positive integer.Let X ⊆ K t .If Φ is a cyclic decomposition of (X, K, t) with period p, then so is σ p (Φ).
the electronic journal of combinatorics 24(1) (2017), #P1.47 For any p ∈ N, when we are considering the cyclic group Z p of p elements, we often think of it as the set of all residue classes modulo p.Thus, we may use [m, n] p for the residue classes to modulus p as represented by m, m + 1, . . ., n, and view it as a subset of Z p ; we sometimes drop the subscript p from the notation [m, n] p if it is clear from the context.Let Φ be a map on Z p and Ψ a map on Z q .We view them as words indexed by Z p and Z q and so it is natural to use ΦΨ to denote the map defined on Z p+q , called the concatenation of Φ and Ψ, such that, for i ∈ Z p+q , it holds Lemma 45.Take a set K, t ∈ N and X, Y ⊆ K t .Assume that X ∩ Y = ∅.Let Φ be a cyclic decomposition of (X, K, t) with period p and let Ψ be a cyclic decomposition of (Y, K, t) with period q.
By checking some other obvious conditions, we conclude that ΦΨ is a cyclic decomposition of (X ∪ Y, K, t).
For any t ∈ N and any set K, we define d K,t to be the map on For any two subsets X and Y of K t , we write d K,t (X, Y ) for min{d K,t (x, y) Lemma 46.Let K be a set and t a positive integer.Let X, Y ⊆ K t and let Φ and Ψ be discrete cyclic decompositions of (X, K, t) and (Y, K, t), respectively.If d K,t (X, Y ) = 1, then there is a discrete cyclic decomposition of X ∪ Y relative to K.
Proof.Suppose d K,t (X, Y ) = 1.Then, in light of Lemma 44, we can assume Φ i = Ψ i for i ∈ [t − 1].Applying Lemma 45, we see that ΦΨ is a cyclic decomposition of (X ∪ Y, K, t), as desired.
Lemma 47.Let c and t be two positive integers.Put X = [c + 1] t \ [c] t .If t > 1, then there exists a discrete cyclic decomposition ∆ of (X, [c + 1], t) such that Proof.If t = 2, we let and the resulting ∆ as a map on Z 2c+1 is what we want.We proceed to deal with the case of t > 2.
We view the elements of [c + 1] t as maps on Z t and this allows us consider the action of σ t on the elements of X -recall the definition of σ as given in Eq. ( 7).For each x ∈ [c + 1] t , let x refer to the orbit of x under the action of σ t .It is clear that σ t gives rise to a permutation of X, namely X is the disjoint union of several orbits of σ t .
For any x ∈ X, let n x := |{x i : and define Λ x to be the map from Z ox to 2 [c+1] that sends i ∈ Z ox to {x i }.For our subsequent proof, the important thing is that For any i ∈ [t], let Ω i := {x ∈ X : n x = i} and Π i := {x ∈ X : n x i}, which are both closed under the action of σ t .Let Ω i = {x : n x = i, x ∈ X} and is adjacent to some vertex in Ω i ⊆ Π i .By successive applications of Lemma 46 and fact (10), we derive the existence of a discrete cyclic decomposition Q of ( Combining Lemma 46 and fact (10), we can start from Q to get a discrete cyclic decomposition Φ of (X \ y, [c + 1], t).
Lemma 49.For every t ∈ N, the set P * (t) contains a complete residue system modulo t.
Proof.For i = 0, . . ., t − 1, we shall show inductively the existence of p i ∈ P * (t) such that p i ≡ i (mod t).
When i = 0, by Lemma 41 there exists a discrete cyclic decomposition of [t] t relative to [t].Since t | t t , we can set p 0 = t t ∈ P * (t).
Assume that i t − 1 is a positive integer and we have known the existence of the required p i−1 .We are going to prove the existence of p i .Let us assume that p i−1 ∈ P * (t, d) for some d ∈ N, namely there exists a strong cyclic decomposition Φ of ([d] t , [d], t) with period p i−1 .Let c be a positive multiple of t satisfying c d.By Lemma 42(b), we can find a strong cyclic decomposition Ψ of ([c] t , [c], t) with period p i−1 .By Lemma 44 and Lemma 47, we can find a strong cyclic decomposition ∆ of [c + 1] t \ [c] t relative to [c + 1] with period (c + 1) t − c t .We then employ Lemma 45 to deduce that Ψ∆ is a strong cyclic decomposition of ([c + 1] t , [c + 1], t) with period p i−1 + (c + 1) t − c t .Noting that (c + 1) t − c t ≡ 1 (mod t), we can take p i = p i−1 + (c + 1) t − c t , finishing the proof.
Proof.Let r = 3 t − 2 t .By Lemma 49, P * (t) contains a complete residue system modulo t, say p 1 , . . ., p t .Take i ∈ [t].According to Lemma 42(b), there is s i ∈ N such that For any with period p i .By Lemma 48, for any there exists a strong cyclic decomposition Ψ p,k i of ([ with period p + p i .Putting together Eq. ( 12) and Eq. ( 13), we see that p can take all values of the form r + t, ∈ N. As p 1 , . . ., p t form a complete residue system modulo t, the freedom in choosing i from [t] says that every large enough integer can be expressed as p+p i , the period of Φ p i ,k i Ψ p,k i for suitable i, k i and p.This demonstrates that Proof of Theorem 15.It follows from Theorem 13 that P * (t) ⊆ P(t).Thus, the claim is direct from Lemma 50.
Let k, t and p be three positive integers and let Φ be a map from Z p to 2 [k] .We write ρ Φ,t for the map from Z p to 2 [k] Lemma 51.Let k, t and p be three positive integers and let Φ be a cyclic decomposition of ([k] t , [k], t) with period p.We adopt the shorthand ρ for the map ρ Φ,t .
. ., a) as a common element.This violation with Eq. (4) says that Φ cannot be a cyclic decomposition.
In both cases, Eq. (4) becomes impossible to hold, which is a desired contradiction.
Proof of Proposition 16.It is immediate from Lemma 51 (c) and Lemma 51(d).
Proof of Theorem 17.This is a result of Theorem 13 and Proposition 16.
Proof of Proposition 20.If the assertion would not hold, then, according to Theorem 13, there is a cyclic decomposition Φ of ([k] t , [k], t) with per(Φ) = k t − 2t + 1.We write p for per(Φ) and note that p 2t.It is also obvious that Φ is not discrete and hence max j∈Zp |Φ j | > 1.
Case 1.There exists j ∈ Z p such that |Φ j | 3.This implies giving a desired contradiction.
Case 2. There is a unique j ∈ Z p such that |Φ j | = 2 and |Φ j | = 1 for all j ∈ Z p \{j}.We now have again a contradiction.
It remains to consider the case that j 1 / ∈ [j 2 − t + 1, j 2 + t − 1] p .We can derive that which is a contradiction, and so the proposition follows.
Proof of Proposition 21.If k = 1, the result is trivial.Assume then k 2. By Theorem 13, our task is to show the nonexistence of a cyclic decomposition Φ of ([k] 2 , [k], 2) with period k 2 − 1.If such a Φ exists, then there exists j ∈ Z k 2 −1 such that |Φ j | 2. Hence This is impossible and hence we are done.
Let k be a positive integer and let Φ be a cyclic decomposition of ( (a) Let S be a subset of δ + (x, Φ) and let A i be a subset of Φ i for all i ∈ S.Then, i∈S A i = [k] if and only if S = δ + (x, Φ) and Φ i = A i for all i ∈ S. (b) Let S be a subset of δ − (x, Φ) and let A i be a subset of Φ i for all i ∈ S.Then, i∈S A i = [k] if and only if S = δ − (x, Φ) and Φ i = A i for all i ∈ S. Proof.(a) By Eq. (3), each element of {x} × K appears in a set of the form Φ i−1 × Φ i for exactly one i ∈ S. The result thus follows.
(b) By Eq. (3), each element of K × {x} appears in a set of the form Φ i × Φ i+1 for exactly one i ∈ S.This implies the claim, as desired.
Let Φ be a cyclic decomposition of (X, [k], t) with period p.The matrix form of Φ is the k × p matrix M Φ whose (i, j) entries, (i, j) ∈ [k] × [p], equal to 1 if i ∈ Φ j and equal to 0 otherwise.
The latter part of the above proof of Proposition 26 is devoted to proving 6 / ∈ P(3).We should mention that it is a special case of the aforementioned result (see Table 1) that 2t ∈ P(t) if and only if t ∈ {1, 2, 4} [QWZ17].
Proof.The result is trivial for k = 1.We hence assume k 2. For all q ∈ [ k 2 ], we define Note that they are all connected components of Γ h k and they all induce cycles in Γ h k .Moreover, we have For q ∈ [2, k 2 ], the subgraph of Γ f k induced by C q is a path which starts at (1, q + 1) and ends at (q, 1).For q = 1, the subgraph of Γ f k induced by C q is a 2k-cycle plus an under the assumption of Eq. (32).By the definition of the Markov operator, our task at this moment is to prove By the definition of h k and f k , we have h k (v) = f k (v) for all v ∈ Z 2 k \V k .Hence we only need to prove the fact that and If φ k (x)×φ k (x ) ∩V k = ∅, Eq. (38) and Eq.(39) will be trivially true.So, we proceed to the case that which surely implies 1 ∈ φ k (x ).
Lemma 62.Let k be an integer not less than 4. Then the 2-hydra f k is primitive with primitive exponent g(f k ) = 4k 2 − 4k + 2.
Proof of Theorem 31.This follows immediately from Lemma 62.

Figure 1 :
Figure 1: A digraph f on {a, b, c, d} and its phase space PS f .
Figure6: The 2-hydra f on K = {a, b} is primitive while its reversal is not strongly connected.We do not include those vacant vertices when displaying the phase spaces.The symbols aa and aK appeared in the two phase spaces should be understood as {a} × {a} and {a} × K, respectively, and so on.
(c) Let a, b and c be three vertices of f .If b ∼ f c, then there exists i ∈[p] such that b, c ∈ C i,a (f ).Proof.(a) Since f is a strongly connected, i∈[p] C i,a (f ) = V(f ).By Lemma 37(b), C 1,a (f ), . . ., C p,a (f ) are mutually disjoint.(b) It is obvious from the definition that M f C i,a (f ) ⊆ C i+1,a (f ).By Lemma 37(a), RI f (a, a) contains a positive multiple of p, say rp for some r ∈ N .Since p > 0, rp − 1 0 holds.Take any b ∈ C i+1,a (f ).By definition, there exists a nonnegative integer j such that b ∈ M j f (a) and j , . . ., {c + 1}, {c}, t−1 {c + 1}, . . ., {c + 1}), as wanted.
Our new example below says that Eq. (2) does not hold for general hydras f.