On 4-valent Frobenius circulant graphs

A 4-valent first-kind Frobenius circulant graph is a connected Cayley graph DLn(1, h) = Cay(Zn, H) on the additive group of integers modulo n, where each prime factor of n is congruent to 1 modulo 4 and H = {[1], [h], −[1], −[h]} with h a solution to the congruence equation x 2 + 1 ≡ 0 (mod n). In [A. Thomson and S. Zhou, Frobenius circulant graphs of valency four, J. Austral. Math. Soc. 85 (2008), 269-282] it was proved that such graphs admit 'perfect ' routing and gossiping schemes in some sense, making them attractive candidates for modelling interconnection networks. In the present paper we prove that DLn(1, h) has the smallest possible broadcasting time, namely its diameter plus two, and we explicitly give an optimal broadcasting in DLn(1, h). Using number theory we prove that it is possible to recursively construct larger 4-valent first-kind Frobenius circulants from smaller ones, and we give a methodology for such a construction. These and existing results suggest that, among all 4-valent circulant graphs, 4-valent first-kind Frobenius circulants are extremely efficient in terms of routing, gossiping, broadcasting and recursive construction.


Introduction
Let n ≥ 5 be an integer whose prime factors are all congruent to 1 modulo 4. Then the congruence equation is solvable (see e.g.[14,15]).For a solution h to this equation, let where for an integer x, [x] denotes the residue class of x modulo n.Define DL n (1, h) to be the circulant graph with vertex set Z n such that [x], [y] ∈ Z n are adjacent if and only if [x−y] ∈ H.We call DL n (1, h) a 4-valent first-kind Frobenius circulant graph [18,21] of order n.It is known [18,Theorem 2] that, for a fixed n = p e1 1 p e2 2 • • • p e l l such that p 1 , p 2 , . . ., p l ≡ 1 (mod 4), where p 1 , p 2 , . . ., p l are distinct prime factors of n and e 1 , e 2 , . . ., e l ≥ 1, there are precisely 2 l−1 pairwise non-isomorphic 4-valent first-kind Frobenius circulant graphs of order n.We remark that, if h is a solution to (1), then so is −h, and h and −h give rise to the same graph DL n (1, h).
The family of 4-valent first-kind Frobenius circulants was introduced in [18] in the context of optimal network design, and it was a subclass of a much larger class [3,16,21] of arc-transitive graphs, called the first-kind Frobenius graphs (see the next section for definition).The importance of such graphs lies in that they admit 'perfect' routing and gossiping schemes under the store-and-forward, all-port and full-duplex model (see [21] for detail).In the special case of 4-valent first-kind Frobenius circulants, this means that DL n (1, h) achieves the smallest possible edge-forwarding index and admits a shortest path routing which is optimal for the edge, arc, minimal-edge and minimal-arc forwarding indices [6] simultaneously.Moreover, under the store-and-forward, all-port and full-duplex model, DL n (1, h) has the smallest possible gossiping time and admits an optimal gossiping scheme under which messages are always transmitted along shortest paths, and at any time every arc is used exactly once for message transmission.(See [18,Theorem 3] for detail.)Because of these 4-valent first-kind Frobenius circulants are strong candidates for modelling interconnection networks.Such graphs are also useful in coding theory, and they were studied independently in [12] from a coding-theoretic point of view by using the language of Gaussian integers.Combining [12,Theorem 4] and the discussion in [17], it follows that the family of 4-valent first-kind Frobenius circulants is precisely the family of Gaussian graphs [12,Definition 3] of odd orders (see Lemma 5 and Remark 6).
The purpose of this paper is to study broadcasting in and recursive construction of 4-valent first-kind Frobenius circulants.We prove that such a graph achieves the smallest possible broadcasting time, namely its diameter plus two (Theorem 4).With the help of number theory we prove that it is possible to recursively construct larger 4-valent first-kind Frobenius circulants from smaller ones, and we give a methodology for such a construction (Section 4).These results make 4-valent first-kind Frobenius circulants even more attractive for modelling interconnection networks, besides their applications in coding theory.There is a long history in studying 4-valent circulants (also called double-loop networks) as models for networks; see e.g.[1,8,9] for surveys.The results in this paper and [18] suggests that, among all 4-valent circulants, 4-valent first-kind Frobenius circulants are exceedingly efficient in terms of routing, gossiping, broadcasting and recursive construction.
The reader is referred to [2] for group-theoretic terminology used in this paper.

Preliminaries
In this section we collect a few results on 4-valent first-kind Frobenius circulants that will be used in later sections.
Given a group X with identity element 1 and a subset S ⊆ X \ {1} such that s −1 ∈ S for every s ∈ S, the Cayley graph Cay(X, S) is defined to have vertex set X such that x, y ∈ X are adjacent if and only if xy −1 ∈ S.
A Frobenius group is a transitive permutation group with the property that there are nonidentity elements fixing one point but only the identity element can fix two points.It is well known [2] that a finite Frobenius group is a semidirect product K H, where K is a nilpotent normal subgroup, and we may think of K H as acting on K in such a way that K acts on K by right multiplication and H acts on K by conjugation.A first-kind K H-Frobenius graph is defined [3,21] as a Cayley graph Cay(K, a H ) on K, for some a ∈ K such that a H = K, where a H is the H-orbit containing a and either H is of even order or a is an involution.There is another class of graphs, called the second-kind Frobenius graphs [3], associated with finite Frobenius groups.The reader is referred to [4] for gossiping and routing properties of second-kind Frobenius graphs. Let n and Z * n acts on Z n by the usual multiplication: a group of automorphisms of G such that any arc of G can be permuted to any other arc of G by an element of X, where an arc is an ordered pair of adjacent vertices.
Lemma 1 ( [18]) Let n ≥ 5 be an integer all of whose prime factors are congruent to 1 modulo 4. Let h be a solution to (1) and H be as given in (2).
In fact, by (1) The last two statements in the lemma follow from [18, Theorem 2].We may represent DL n (1, h) by a plane tessellation of squares [9,19,20] such that the square with coordinates (x, y) represents the vertex [x + yh] of DL n (1, h) (see Figure 1).Thus the squares adjacent to For 0 ≤ i ≤ r, define [18] x The image of A under the action of H is given by Equivalently, AH intersects with the four quadrants at A, A[h], −A, −A[h], respectively, and H permutes these four parts cyclically (see Figure 1), where } is an algebraic expression [18] of the minimum distance diagram [9,19,20] of DL n (1, h) as shown by the following lemma.It is well known (e.g.[15,Corollary 6.8.2]) that the Diophantine equation Lemma 3 ([12, Theorem 6], see also [17]) Let n ≥ 5 be an integer all of whose prime factors are congruent to 1 modulo 4. Let h be a solution to (1).Let 0 < a < b be the unique primitive solution of (3) such that ah ≡ b (mod n).Then r = (a + b − 1)/2 (4) and, for 0 ≤ i ≤ r, x i = max{r − i, r − a}. ( In particular, the diameter of DL n (1, h) is given by

Broadcasting time
A common process in communication networks is to disseminate a message from a specific source vertex to all other vertices in such a way that in each time step any vertex who has received the message already can retransmit it to at most one of its neighbours.This process is called broadcasting, and the minimum number of time steps required is denoted by b(G, u) if G is the network and u is the source vertex.The broadcasting time [7] of G, denoted by b(G), is defined to be the maximum among b(G, u) for u running over V (G).In general, it is difficult to determine b(G).See [7, Section 5.2] for a survey.
Theorem 4 Let n ≥ 5 be an integer all of whose prime factors are congruent to 1 modulo 4. Let h be a solution to (1) and d the diameter of DL n (1, h) (as given in Lemmas 2 and 3).Then Moreover, we can explicitly give an optimal broadcasting in DL n (1, h).
Without loss of generality we may assume that [0] has a message to be broadcasted in G.In the following we prove for each since no vertex is allowed to send the message to two of its neighbours at the same time.
Let r be as in (4).We successively send the message to the vertices in the negative x-direction at times 1, 2, . . ., r, in the x-direction at times 2, 3, . . ., r + 1, in the y-direction at times 3, 4, . . ., r + 2, and in 49 4 50 3 51 2 52 1 0 0 1 2 2 3 3 4 4 5   19 5 20 5 21 4 22 3 23 3 24 4 25   For 0 ≤ j ≤ r, define L([j In this way we define a broadcasting on all vertices of G other than those Consider the remaining vertices above.Such a vertex is d and hence is the only exceptional vertex to be considered.In this case, if x r = 1, then since [r] receives the message at time r Thus, if a < r, then [r − x r h] with x r + r = d receives the message at time d + 1 or d + 2. This together with ( 7)-( 13) gives a broadcasting in G using d + 2 time steps.See Figure 2 for this broadcasting in DL 53 (1,23).It remains to deal with the case a = r.In this case, x i = r − i for 0 ≤ i ≤ r by ( 5), d = r by Lemma 2 (b), and so This implies received the message at time d + 1 (see (7)).So we have finished defining a broadcasting in G using d + 2 time steps when a = r.See Figure 3 for an illustration of this broadcasting.
In summary, we have proved b(G, [0]) ≤ d + 2 up to now.
Part 2: We now prove that d + 2 is a lower bound for b(G, [0]).This is true when d = 1 since in this case G is the complete graph . Suppose to the contrary that there exists a broadcasting using d + 1 time steps.One of the neighbours should receive the message at time 1, and assume that the other three neighbours receive the message at times t 1 < t 2 < t 3 respectively, where t 1 ≥ 2. Note that receives the message at time t 2 or t 3 , then the last vertex [uh i ] of P i receives the message at time d + 2 or later, which is a contradiction.Similarly, if t 1 ≥ 3, then no [v i ] receives the message at time t 1 .Hence each [v i ] receives the message at time and one of [uh i ] and [uh j ] receives the message at time d + 2 or later, a contradiction.Hence at most one of receives the message at time 2 and the remaining three or four vertices receive the message at time 1.Suppose without loss of generality that An optimal broadcasting for any source vertex [w] can be obtained from the optimal broadcasting L in Part 1 for source vertex [0] by translation.Define ) is as in (6); that is, the time when the message originated from [w] is sent from [v u + w] to [u + w] is the same as the time when the message originated from In the special case of DL n d (1, 2d + 1) the broadcasting described in Part 1 of the proof above is similar to the one given in [11].Nevertheless, this is by no means the only optimal broadcasting in DL n d (1, 2d + 1).In general, optimal broadcastings of DL n (1, h) other than the one given in the proof of Theorem 4 can be found by using similar approaches.

Recursive construction
An important issue in network design is whether it is possible to 'expand' an existing network to larger ones of similar structures.And if this is possible, how can we construct efficiently such larger networks from smaller ones?For instance, an attractive feature of hypercubes is that we can easily expand a given hypercube to larger ones of higher dimensions.In this section we prove that 4-valent Frobenius circulants share this property.We will give algorithms for constructing larger 4-valent Frobenius circulants from smaller ones by using number theory.To this end we will use an equivalent definition of a 4-valent Frobenius circulant in terms of Gaussian integers.
A Gaussian integer is a complex number a + bi with both a and b in Z.(In this section we reserve i for the imaginary unit of complex numbers.)The set Z[i] of all Gaussian integers is a ring under the usual addition and multiplication of complex numbers, the ring of Gaussian integers.Its units are 1, −1, i and −i, and for α ∈ Z[i] and a unit ε we call εα an associate of α.It is well known that Z[i] is an Euclidean domain with the norm function defined by N (a + bi) = a 2 + b 2 for 0 = a + bi ∈ Z[i].In other words, for any α, β ∈ Z[i] with β = 0, there exists γ, δ ∈ Z[i] such that α = γβ + δ and either δ = 0 or N (δ) < N (β).Hence Z[i] is a principal ideal domain and so a unique factorization domain.It is easy to see that N (αβ) = N (α)N (β) for any nonzero α, β ∈ Z[i].All these results and definitions about Gaussian integers can be found in, for example, [10].
Given Theorem 2].In the case when gcd(a, b) = 1, the Gaussian graph G α generated by α is defined [12] to have vertex set Note that, if α is an associate of 1 + i (that is, N (α) = 2), then the cardinality of H α is 2 and so G α is 2-valent.In general, G α is a 4-valent graph as long as gcd(a, b) = 1 and α is not an associate of 1 + i.
One can verify that G α ∼ = G εα for any unit ε of Z[i], and εα defines an isomorphism between the two graphs.Since εα = a + bi, −a − bi, −b + ai, b − ai when ε runs over the four units of Z[i], in studying Gaussian graphs we may assume without loss of generality that both a and b are positive integers.Gaussian graphs above were introduced in [12] with motivation from coding theory.It turns out that in the case when N (α) = a 2 + b 2 is odd, they are exactly the family of 4-valent Frobenius circulants.This was first noticed by Alison Thomson (personal communication).It is implied in the following lemma in which N (α) can be odd or even.

Lemma 5 (a) Let
, where l is the unique solution to where [x + yl] is the residue class of x + yl modulo N (α).
The set H α defined in ( 14) is a subgroup of the group Z[i] * α of units of ring Z[i] α .One can verify that (x+yi) i s = (x+yi)i s defines an action (as a group) of H α on the additive group of Z[i] α , where x+yi, i s and (x + yi)i s are interpreted as their residue classes modulo α.Hence Z[i] α H α is well-defined and moreover it acts on Z[i] α (as a set) by where the Gaussian integers involved are interpreted as their residue classes modulo α.One can verify that Z[i] α H α preserves adjacency and non-adjacency of G α .So it can be regarded as a group of automorphisms of G α .Moreover, by [21, Lemma 2.1] and the fact that the group H α is transitive on the connection set H α of G α , we have When N (α) is odd this is known in [18] in view of Lemmas 1 and 5.In this case one can prove that To construct larger 4-valent Frobenius circulants from smaller ones, by Lemma 5 it suffices to find an approach to constructing larger Gaussian graphs of odd order from smaller ones.The following lemma serves for this purpose, and it applies to a broader family of graphs.Note that for any , where H α is as in (14).To ensure G α is a nontrivial graph with valency 4, we require that α is not a unit or an associate of 2 or 1 + i, or equivalently N (α) ≥ 5.Such generalized Gaussian graphs were studied in [13] (but with the necessary condition N (α) ≥ 5 neglected).
A graph G 1 is called a cover of a graph G 2 if there exists a surjective mapping φ : Then G αβ can be constructed from G α and is an N (β)-fold cover of G α .
, when it is viewed as a subgroup of the additive group of Now we construct G αβ from G α as follows.Consider an arbitrary pair of adjacent vertices (16) Note that this adjacency relation is defined for all pairs of adjacent vertices Hence the adjacency relation ( 16) is symmetric.Moreover, it is independent of the choice of representatives of [ξ] α and Therefore, G αβ is identical to Ĝαβ and so can be constructed from G α as in the previous paragraph.It is obvious that the quotient graph of G αβ with respect to the partition Note that αβ = (ac−bd)+(ad+bc)i gives rise to the solution (ac−bd, ad+bc) to x 2 +y 2 = np e .We claim that this is a primitive solution.Suppose otherwise.Then there exists a prime q in Z which divides both ac − bd and ad + bc.If q divides a, then it divides both bd and bc.Since q cannot divide b due to gcd(a, b) = 1, it follows that q divides both c and d, which contradicts the assumption gcd(c, d) = 1.So q is not a divisor of a.Similarly, q is not a divisor of b, c or d.Since q divides ac − bd and ad + bc, it divides c(ac−bd)+d(ad+bc) = a(c 2 +d 2 ) = ap e and a(ac−bd)+b(ad+bc) = c(a 2 +b 2 ) = cn.Since q divides neither a nor c, it follows that q divides p e and n, and hence q = p is a prime factor of n, which contradicts our assumption.Therefore, (ac−bd, ad+bc) is a primitive solution to x 2 +y 2 = np e .It is clear that there is a unique unit ε 1 of Z

Concluding remarks
In this paper we proved that 4-valent first-kind Frobenius circulants have the minimum possible broadcasting time, namely their diameter plus two, and we explicitly gave optimal broadcasting in such graphs.We developed an approach to constructing larger 4-valent first-kind Frobenius circulants from smaller ones by using number theory.Our results in this regard can be easily generalised to Gaussian graphs of even order.
As mentioned in the introduction, if n ≥ 5 has l distinct prime divisors and all of them are congruent to 1 modulo 4, then there are exactly 2 l−1 pairwise non-isomorphic 4-valent first-kind Frobenius circulants of order n [18, Theorem 2].
Question 15 What is the minimum diameter among such 2 l−1 graphs of a given order n?
As mentioned earlier, there is a one-to-one correspondence between solutions h to x 2 + 1 ≡ 0 mod n and nonnegative primitive solutions (a, b) to x 2 + y 2 = n with 0 < a < b and ah ≡ b mod n.Since

Fig. 2 :
Fig. 2: An optimal broadcasting in DL53(1, 23).Subscripts represent the times that the corresponding vertices receive the message originated from [0].The part in the first quadrant bounded by bold lines is A.
and one of [u], [uh], [−u] receives the message at time d + 3 or later.This final contradiction proves that b(G, [0]) is at least d + 2. Combining Parts 1 and 2, we obtain b(G, [0]) = d + 2 and so b(G) = d + 2. Hence the broadcasting given in Part 1 is optimal for source vertex [0].
Output: 11 We may use standard algorithms in number theory to find (a, b) in Step 1 and h j in Step 4. See for example the proofs of [15, Theorems 6.4 and 6.5].We may obtain (c, d) in Step 2 by recursively computing h(e) in Procedure 9 and then applying the algorithm implied in the proof of [15, Theorem 6.5].Theorem 12 Procedure 10 is correct, that is, DL np e (1, h 1 ) and DL np e (1, h 2 ) above are 4-valent firstkind Frobenius circulants, and moreover DL np e (1, h 1 ) ∼ = DL np e (1, h 2 ).Proof: Using the notation above, we have DL n (1, h) ∼ = G α by Lemma 5. Since p ≡ 1 (mod 4), the Diophantine equation x 2 + y 2 = p e has exactly two nonnegative primitive solutions.(This can be deduced from, say, [15, Corollaries 6.5.1 and 6.8.1].)Thus (c, d) in Step 2 exists and the other nonnegative primitive solution is (d, c).