New Moore-Like Bounds and Some Optimal Families of Abelian Cayley Mixed Graphs

Mixed graphs can be seen as digraphs that have both arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case where such graphs are Cayley graphs of abelian groups. Such groups can be constructed using a generalization to Zn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}^n$$\end{document} of the concept of congruence in Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}$$\end{document}. Here we use this approach to present some families of mixed graphs, which, for every fixed value of the degree, have an asymptotically large number of vertices as the diameter increases. In some cases, the results obtained are shown to be optimal.


Introduction
The choice of the interconnection network for a multicomputer or any complex system is one of the crucial problems the designer has to face. In fact, the network topology largely affects the performance of the system and it has an important contribution to its overall cost. As such topologies are modeled by either graphs, digraphs, or mixed graphs, this has led to the following optimization problems: (a) Find graphs, digraphs or mixed graphs, of given diameter and maximum out-degree that have a large number of vertices. (b) Find graphs, digraphs or mixed graphs, of given number of vertices and maximum out-degree that have small diameter.
The research of C. Dalfó has also received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie Grant agreement no. 734922.
For a more detailed description of these problems, their possible applications, the usual notation, and the theoretical background, see the comprehensive survey of Miller andŠiráň [14]. For more specific results concerning mixed graphs, which are the topic of this paper, see, for example, Nguyen and Miller [18], and Nguyen, Miller, and Gimbert [19]. Mixed graphs, with their undirected and directed connections, are a better representation of a complex network (as, for example, the Internet) than the one given by both graphs or digraphs. As two examples of the applications of mixed graphs, we have (a) Mixed graphs modeling job shop scheduling problems, in which a collection of tasks is to be performed. In this case, undirected edges represent a constraint for two tasks to be incompatible (they cannot be performed simultaneously). Directed edges may be used to model precedence constraints, in which one task must be performed before another. (b) Mixed graphs are also used as models for Bayesian inference. The directed edges of these graphs are used to indicate a causal connection between two events, in which the outcome of the first event influences the probability of the second event. Undirected edges, instead, indicate a non-causal correlation between two events. In this first section, we give some preliminaries, together with some details on Abelian Cayley graphs from congruences in Z n . In Sect. 2, we provide a combinatorial way to obtain the Moore bound for mixed Abelian Cayley graphs. Taking into account some symmetries, we rewrite in a more condensed way the Moore bound for some cases. Finally, in Sect. 3, we obtain some Moore Abelian Cayley mixed graphs when they exist, and some infinite families of dense graphs when they do not exist.

Preliminaries
The degree/diameter or (d, k) problem asks for constructing the largest possible graph (in terms of the number of vertices), for a given maximum degree and a given diameter. In the degree/diameter problem for mixed graphs we have three parameters: a maximum undirected degree r, a maximum directed outdegree z, and diameter k. A natural upper bound for the maximum number of vertices M (r, z, k) for a graph under such degrees and diameter restrictions is (see Buset et al. [2]): where with d = r + z and v = (d − 1) 2 + 4z, Besides this general bound given above, researchers are also interested in some particular versions of the problem, namely when the graphs are restricted to a certain class, such as the class of bipartite graphs (which was studied by the authors [3]), planar graphs (see Fellows et al. [6], and Tischenko [22]), maximal planar bipartite graphs (see Dalfó et al. [4]), vertex-transitive graphs (see Machbeth et al. [12], andŠiagiová and Vetrík [20]), Cayley graphs ( [12,20] and Vetrík [23]), Cayley graphs of Abelian groups (Dougherty and Faber [5]), or circulant graphs (Wong and Coppersmith [24], and Monakhova [15]). In this paper, we are concerned with mixed Abelian Cayley graphs.
For most of these graph classes there exist Moore-like upper bounds, which in general are smaller than the Moore bound for general graphs, although some of them are quite close to the Moore bound. For example, the Moore-like upper bound for bipartite mixed graphs is (when r > 0): where u 1 , u 2 , A, and B are given by (2) and (3) (see Dalfó et al. [3]). The upper bound for mixed Abelian Cayley graphs was given by López et al. [10]: Let Γ be an Abelian group, and let Σ be a generating set of Γ containing r 1 involutions and r 2 pairs of generators and their inverses, and z additional generators, whose inverses are not in Σ. Thus, the Cayley graph Cay(Γ, Σ) is a mixed graph with undirected degree r, where r = r 1 + 2r 2 , and directed out-degree z. An upper bound for the number of vertices of Cay(Γ, Σ), as a function of the diameter k, is Some interesting (proper) cases, that are mentioned later, are r 1 = 0, r 2 = 1, z = 1 and r 1 = 1, r 2 = 0, z = 2, for which (5) gives the same Moore bound (k + 1) 2 .
Circulant graphs are Cayley graphs over Z n , and they have been studied for the degree/diameter problem for both the directed and the undirected case. As in the general case, the definition of circulant graphs can be extended to allow both edges and arcs. Let Σ be a generating set of Z n containing r 1 involutions and r 2 pairs of generators, together with their inverses, and z additional generators, whose inverses are not in Σ. The (mixed) circulant graph Circ(n; Σ) has vertex set V = Z n , and each vertex i is connected to i + a (mod n) vertices, for all a ∈ Σ. Thus, Circ(n; Σ) has undirected degree r, where r = r 1 + 2r 2 , and directed degree z. In fact, r 1 ≤ 1 since Z n has either one involution (for n even), or none (for n odd).

Abelian Cayley Graphs from Congruences in Z n
Let M be an n × n non-singular integral matrix, and Z n the additive group of n-vectors with integral components. The set Z n M , whose elements are linear combinations (with integral coefficients) of the rows of M is said to be the lattice generated by M . By the Smith normal form theorem, M is equivalent to the diagonal matrix S(M ) = S = diag(s 1 , . . . , s n ), where s 1 , . . . , s n are the invariant factors of M , which satisfy s i |s i+1 for i = 1, . . . , n − 1. That is, there exist unimodular matrices U and V , such that S = UMV . The canonical form S is unique, but the unimodular matrices U and V certainly not. However, this fact does not affect the results below. For more details, see Newman [17]. The concept of congruence in Z has the following natural generalization to Z n (see Fiol [7]). Let u, v ∈ Z n . We say that u is congruent with The Abelian quotient group Z n /Z n M is referred to the group of integral vectors modulo M . In particular, when M = diag(m 1 , . . . , m n ), the group Z n /Z n M is the direct product of the cyclic groups Z mi , for i = 1, . . . , n. Let us consider again the Smith normal form of M , S = diag(s 1 , . . . , s n ) = UMV . Then (6) holds if and only if uV ≡ vV (mod S) or, equivalently, where V i denotes the ith column of V . Moreover, if r is the smallest integer such that s n−r = 1; hence, s 1 = s 2 = · · · = s n−r = 1 (if there is no such a r, let r = n), then the first n − r equations in (7) are irrelevant, and we only need to consider the other ones. This allows us to write where V stands for the n × r matrix obtained from V by taking off the first n−r columns, and S = diag(s n−r+1 , s n−r+2 , . . . , s n ). So, the (linear) mapping φ from the vectors modulo M to the vectors modulo S given by φ(u) = uV is a group isomorphism, and we can write Notice that since for any Abelian group Γ there exists an integral matrix M ∈ Z n * n such that Γ ∼ = Z n /Z n M , the equality in (9) is just the fundamental theorem of finite Abelian groups (see also Proposition 1.1(b)). The next proposition contains more consequences of the above results. For instance, (b) follows from the fact that s 1 s 2 · · · s n = d n = m and s i |s i+1 , for i = 1, 2, . . . , n − 1. Clearly, a multidimensional circulant (digraph, graph, or mixed graph) is a Cayley graph of the Abelian group Γ = Z n /Z n M . In our context, if Γ is an Abelian group with generating set Σ containing r 1 + 2r 2 + z generators (with the same notation as before), then there exists an integer n × n matrix M with size n = r 1 + r 2 + z such that

Proposition 1.1. (a) The number of equivalence classes modulo
. . , e r1+r2+z }, and the e i 's stand for the unitary coordinate vectors. For example, the two following Cayley mixed graphs Indeed, Z 3 /Z 3 M is a cyclic group of order |detM | = 24 and, according to (7), the generators ±e 1 , e 2 , and e 3 of Z 3 /Z 3 M give rise to the generators ±2, 3, and −12 = 12 (mod24) of Z 24 ; see the last column of V .

Expansion and Contraction of Abelian Cayley Graphs
The following basic results are simple consequences of the close relationship between the Cartesian product of Abelian Cayley graphs and the direct products of Abelian groups (see, for instance, [7,8]).

A New Approach to the Moore Bound for Mixed Abelian Cayley Graphs
In [11], López, Pérez-Rosés, and Pujolàs derived the Moore bound (4) using recurrences and generating functions. In this section, we obtain another expression for M AC (r 1 , r 2 , z, k) in a more direct way from combinatorial reasoning. In Proof. A vertex u at distance at most k from 0 can be represented by the situation of k balls (representing the presence/absence of the edges/arcs in the shortest path from 0 to u) placed in 1 + r 1 + r 2 + z boxes (representing the presence/absence of the generators) with the following conditions: • One box contains the number of (white) balls of the non-existing edges/arcs. Joining all together, we obtain the result.

Algebraic Symmetries
As commented in Introduction, the numbers M AC (r 1 , r 2 , z, k) have some symmetries. For instance, as it is well known, the so-called Delannoy numbers F t,k , where we have used (10). Also, we have already mentioned that In fact, this is a particular case of the following result.

Lemma 2.2.
For any integer ν such that −r 2 ≤ ν ≤ min{r 1 , z}, the Moore bounds for the mixed Abelian Cayley graphs satisfy Proof. We only need to prove it for ν = 1. Remembering the proof of Proposition 2.1, notice that there is an equivalence between (i) and (ii) as follows: (i) The balls of a box corresponding to a generator b ∈ Σ (with −b ∈ Σ) together with the (0 or 1) balls of the box representing an involution ι. (ii) The ('white' or 'black') balls in a box representing the pair of generators ±b ∈ Σ. So, in our counting process, each generator pair of type {ι, b} can be replaced by a generator of type a, without changing the result. In fact, notice that a more direct proof is obtained using the expression (5) for M AC (r 1 , r 2 , z, k) since it is invariant under the changes r 1 → r 1 −ν, r 2 → r 2 +ν, and z → z − ν.
By combining (11) for the extreme values of ν with (10), we get the following result.
Notice that, by changing the summation variable j to k − i in (12), we get (5), so proving that the expressions given for M AC (r 1 , r 2 , z, k) in (5) and (10) are equivalent.

Existence of Mixed Moore Abelian Cayley Graphs and Some Dense Families
Graphs (or digraphs or mixed graphs) attaining the Moore bound are called Moore graphs (or Moore digraphs or mixed Moore graphs). The same applies for particular versions of the problem and, hence, mixed Abelian Cayley graphs attaining the Moore bound are called mixed Moore Abelian Cayley graphs. Moore graphs are very rare and, in general, they only exist for a few values of the degree and/or diameter. In this section, we deal with the existence problem of these extremal graphs and, if they do not exist, we give some infinite families of dense graphs.

The One-Involution Case
We begin with the case when r = 1, that is, when our graphs contain a 1-factor. This means that the corresponding generating set of the group must contain exactly one involution. If, in addition, we have just one arc generator, that is z = 1, then the Moore bound (5) becomes 2k + 1. This bound is unattainable since a graph containing a 1-factor must have even order. Nevertheless, it is easy to characterize those mixed Abelian Cayley graphs with maximum order 2k in this case: let Γ be an Abelian group and Σ = {ι, b} a generating set of G , where ι is an involution of Γ (the edge generator) and b is an element whose inverse is not in Σ (the arc generator). The set of vertices at distance l from 0 is G l (e) = {ι + (l − 1)b, lb}, for 1 ≤ l ≤ k − 1. Since the order of G is 2k and the diameter is k, then G i (e) ∩ G j (e) = ∅ for i = j, and moreover, G k (e) = {ι+(k−1)b}. Now, due to the regularity of G, we have two possibilities according to the arc emanating from (k − 1)b, that is, the value of kb: (a) kb = ι. Then ι + (k − 1)b = 2ι = 0, and the mixed graph has been completed. In this case, b has order 2k, that is, b is a generator of Γ. Thus, G contains a Hamiltonian directed cycle (generated by b), and it is trivial to see that G is isomorphic to a circulant graph of order 2k with generators 1 and k. (b) kb = 0. Then, the arc emanating from ι(k − 1)b goes to ι, and the mixed graph has been completed. This graph contains two disjoint directed cycles of order k, both joined by a matching. Then, G is isomorphic to So, a mixed Abelian Cayley graph with r = z = 1 and maximum order for diameter k is isomorphic either to Circ(2k; {1, k}) or Cay(Z 2 × Z k ; {(1, 0), (0, 1)}).
For z > 1, the problem seems to be much more difficult, but we have a complete answer for the case z = 2. Theorem 3.1. Depending on the value of the diameter k ≥ 2, the maximum order of a mixed Abelian Cayley graph with r 1 = 1, r 2 = 0, and z = 2 is given in Table 2, together with some graphs attaining the bound.
Proof. Let G be an Abelian Cayley graph with the above parameters and maximum order N ≤ (k+1) 2 . Then, by Lemma 1.2(ii), the graph G = G/K 2 is an Abelian Cayley digraph with z = 2 generators and diameter k ∈ {k − 1, k}.
For the values k ≤ 4, Table 2 shows the optimal values of N found by computer search.
Otherwise, if k > 4, we prove that G has diameter k = k−1. Indeed, note first that, since G is a circulant digraph with two 'steps' (generators), it admits a representation as an L-shaped tile that tessellates the plane (see Fiol, Yebra, Alegre, and Valero [9]). Moreover, if such a tile L has dimensions , h, x, y as shown in Fig. 1 to take two equal L s, say L 1 and L 2 , as shown in Fig. 2 (where L 2 is shaded). Now, since the diameter of G is k, we have two possibilities: (i) The set of the two equal Ls, L 1 = L 2 , corresponds to a diagram of minimum distances (from 0) with D(L 1 ) = D(L 2 ) = k − 1. Notice that, in this case, L 1 and L 2 correspond to the sets of vertices, G 1 (0) and G 2 (0), at minimum distance from 0, whose respective shortest paths do, or do not, contain vertex ι (the involution), and we have |G 1 (0)| = |G 2 (0)| (Fig. 3). (ii) Otherwise, as happens in our example, the diagram of minimum distances is a single L with D(L) = k, as shown in Fig. 4c. This is because, apart from the case (i), the only tile that tessellates the plane is the above L 1 "extended" to L using L 2 . In this case, |G 1 (0)| ≥ |G 2 (0)|. In [9], it was proved (by considering a continuous version of the problem) that the order N of a circulant digraph with two generators satisfies the bound N ≤ (k +2) 2
Then from (15) and (16), we see that N (ii) > N (i) only for k ≤ 4. Hence, we proved that, if k > 4, we can assume that G = G/K 2 , with G having maximum order, has diameter k = k − 1, as claimed. Now, using again the results in [9], the maximum number of vertices of G are reached by the following graphs: • If k = 3x − 2, then N = 3x 2 attained by Cay(Z 2 /Z 2 M, {e 1 , e 2 , e 3 }) Hence, by Lemma 1.2, we have the following mixed Abelian Cayley graphs with r = 1 and z = 2 with maximum order N = 2N for every diameter k = k + 1 > 4: (iii) If k = 3x + 1, then N = 6x 2 + 8x + 2 attained by Assume that M is a 3 × 3 matrix (the cases with 2 × 2 matrices are more simple). In case (i), we distinguish two cases: f x is even, say x = 2s, then the Smith normal form of M turns out to be S = diag(2, x, 3x) = diag(2, 2s, 6s) and 3, 2)}), as shown in Table 2. The reasoning of the odd case, x = 2s + 1, is similar, with the Smith normal form being now S = diag(1, x, 6x). Then in this case, the group Z 3 /Z 3 M is of rank two, namely, Z x × Z 6x . Cases (ii) and (iii) are proved analogously. For instance, when k = 3x is even, the Smith normal form of M is S = diag(1, 2, N/2) (group of rank two); whereas when k = 3x is odd, we get S = diag(1, 1, N) (cyclic group).
Notice that, using our method, circulant graphs attain the maximum order in some cases, but not in all of them. More precisely, for a diameter of the form k = 3x − 1, we must always use a group with rank 2 or 3, whereas for the other cases, k ∈ {3x, 3x + 1}, we can use a cyclic group. Moreover, we remark that in some cases there are other generators and/or groups that produce non-isomorphic mixed graphs with the same degree, diameter and order than the ones given in Table 2 (1,11), (0, 5), (1, 8)} as generators) produces an optimal mixed graph for k = 6. Another example is given in Fig. 6, where we show an alternative mixed graph with diameter 3x = 6, maximum order 32 and generators 1, 10, 16 (different from the one corresponding to 5, 2, 16, provided by Table 2).
As happens with the case z = 1, the Moore bound cannot be attained either for z ≥ 2. Now we have the following result.  (1, 0, z, k), where ι is an involution. Let us consider the Abelian Cayley mixed graph G = Cay(Γ, Σ), and suppose that the diameter of G is k. The set of vertices at distance l, 1 ≤ l ≤ k from 0, G l (0), can be split into two disjoint sets: The shortest path from b 1 to b t , for all 2 ≤ t ≤ z, must pass through a vertex v t ∈ G k (0).   Claim For any 2 ≤ t ≤ z, there exists v t ∈ G k (0), such that b 1 + v t = b t . Indeed, let v t ∈ G k (0) be the predecessor of b t in the shortest path from b 1 to b t . Then, there exists b i , for 1 ≤ i ≤ z, such that v t + b i = b t . We already know that v t = s i b i , with s i ≥ 0 and s i = k, but since the shortest path starts at b 1 , we have the extra condition that s 1 ≥ 1. (a) v t ∈ G 1 k (0). Then b 1 + s i b i = b t , for some vector (s 1 , . . . , s z ), where 0 ≤ s i ≤ k, and s i = k. That is, w t = (s 1 +1)b 1 + · · ·+(s t − 1)b t + · · ·+ s z b z = 0. Observe that w t ∈ G k (0) if s 1 < k, which is a contradiction with 0 ∈ G k (0) for k ≥ 1. Hence, s 1 = k, that is, (k + 1)b 1 = b t . Equivalently, v t = kb 1 , which means that v t does not depend on the vertex b t . (b) v ∈ G 2 k (0). As in the previous case, it is not difficult to see that v t = ι + (k − 1)b 1 ; hence, also in this case, v t does not depend on the vertex b t .
Hence, for z ≥ 4, we have at least two different vertices b and b from the set {b 2 , . . . , b z }, such that either (k +1)b 1 = b and (k +1)b 1 = b or ι+(k −1)b 1 = b and ι +(k − 1)b 1 = b , which is a contradiction. It remains to consider the cases z = 2 and z = 3. The first one can be solved as follows: using the reasoning given in (a) and (b), we have that either (k + 1)b 1 = b 2 or ι + kb 1 = b 2 . Now, apply the same argument to the shortest path from b 1 to 2b 2 ( = b 2 since k ≥ 2), showing that either (k + 1)b 1 = 2b 2 or ι + kb 1 = 2b 2 , which is a contradiction with the two cases given before. To solve the last case z = 3, it is enough to use the same argument to the shortest path from b 1 to 2b 3 , in addition to the others described before.
Note that mixed Moore Abelian Cayley graphs could exist for other values of r > 1. improving the result in (19) when z > 2. For example, for z = 3, the limit in (19) is 24/125, whereas the limit in (21) is 2/9. For finite values, the improvement is more noteworthy as z and k increase, as shown in Fig. 7 for z = 5 and k ≤ 10.