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Procedia Computer Science 74 (2015) 2 – 9
International Conference on Graph Theory and Information Security
Degree/diameter problem for mixed graphs
Nacho López, Hebert Pérez-Rosés
Departament de Matemàtica, Universitat de Lleida, C/ Jaume II 69, 25001 Lleida, Spain.
Abstract
The Degree/diameter problem asks for the largest graphs given diameter and maximum degree. This problem has been extensively
studied both for directed and undirected graphs, ando also for special classes of graphs. In this work we present the state of art of
the degree/diameter problem for mixed graphs.
c 2015
2015The
TheAuthors.
Authors.
Published
by Elsevier
©
Published
by Elsevier
B.V.B.V.
This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer-review
under responsibility of the Organizing Committee of ICGTIS 2015.
Peer-review under responsibility of the Organizing Committee of ICGTIS 2015
Keywords: Degree/diameter problem, Moore bound, mixed Moore graph.
2010 MSC: 05C35
1. Introduction
A mixed graph G may contain (undirected) edges as well as directed edges (also known as arcs or darts). From this
point of view, a graph [resp. directed graph or digraph] has all its edges undirected [resp. directed]. The undirected
degree of a vertex v, denoted by d(v) is the number of edges incident to v. The out-degree [resp. in-degree] of v,
denoted by d+ (v) [resp. d− (v)], is the number of arcs emanating from [resp. to] v. If d+ (v) = d− (v) = z and d(v) = r,
for all v ∈ V, then G is said to be totally regular of degree d, where d = r + z. A trail from u to v of length l ≥ 0
is a sequence of l + 1 vertices, u0 u1 . . . ul−1 ul , such that u = u0 , v = ul , and each pair ui−1 ui is either an edge or an
arc of G. A trail whose vertices are all different is called a path. The length of a shortest path from u to v is the
distance from u to v, denoted by dist(u, v). Note that dist(u, v) may be different from dist(v, u) when shortest paths
between u and v involve arcs. The maximum distance between pairs of vertices of G is called the diameter of G.
Mixed graphs arise in many practical situations, for instance, interconnection networks, where vertices represent the
processing elements and edges represent their links [4] . The connection between vertices in these mixed graphs can be
undirected or directed, depending on whether the communication between nodes is two-way or only one-way. So, it
is natural to consider the degree/diameter problem for mixed graphs.
E-mail address: {nlopez,hebert.perez}@matematica.udl.es
1877-0509 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer-review under responsibility of the Organizing Committee of ICGTIS 2015
doi:10.1016/j.procs.2015.12.066
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Nacho López and Hebert Pérez-Rosés / Procedia Computer Science 74 (2015) 2 – 9
Degree/diameter problem for mixed graphs: given three natural numbers r, z and k, find the largest possible
number of vertices n(r, z, k) in a mixed graph with maximum undirected degree r, maximum directed outdegree
z and diameter k.
This definition can be viewed as a generalization of the ones given for undirected and directed graphs. The degree/diameter problem for directed and undirected graphs has been widely studied, and most of the main results can
be found in the survey [31] . Nevertheless, little information can be found, in relation with this extremal problem on
mixed graphs.
2. Moore bound for mixed graphs
A natural upper bound is easily derived just by counting the number of vertices of a particular distance of any given
vertex v in a [di]graph with given maximum [out-]degree and diameter, either for undirected and directed graphs. This
bound is known as the Moore bound. For instance, the order n(0, z, k) of any digraph having maximum outdegree z
and diameter k is bounded by
zk+1 −1
z−1 if z > 1
n(0, z, k) ≤ 1 + z + z2 + · · · + zk =
(1)
k + 1 if z = 1
Besides, for undirected graphs we have
n(r, 0, k) ≤ 1 + r + r(r − 1) + · · · + r(r − 1)k−1 =
−1
1 + r (r−1)
if r > 2
r−2
2k + 1
if r = 2
k
(2)
Something similar can be done for mixed graphs with small diameter. The bound in this case is known as the mixed
Moore bound and it is a generalization of the classical Moore bound, where both parameters r and z should appear.
For instance the maximum number of vertices for a mixed graph of diameter k = 2 with maximum undirected degree
r and maximum out-degree z is n(r, z, 2) ≤ 1 + (r + z) + z(r + z) + r(r + z − 1) (see Fig. 1).
v
···
r−1
z
r−1
r
z
z
···
r
z
r
z
Fig. 1. Counting the number of vertices of this tree gives the Moore bound for a mixed graph of diameter two, maximum undirected degree r and
maximum out-degree z.
A generalization of this upper bound for any diameter k was first stated [32] as
1 + (r + z) + z(r + z) + r(r + z − 1) + · · · + z(r + z)k−1 + r(r + z − 1)k−1
(3)
Nevertheless, this bound is larger than the number of vertices that appears in the tree that counts the number of vertices
at a particular distance of any given vertex in a mixed graph with the prescribed restrictions. For instance, 20 is the
maximum number of vertices in a mixed graph with r = z = 1 having diameter k = 4, according to Eq. (3). But a
simpe drawing shows that this number is in fact 19. The counting problem becomes more difficult than it seems at
first sight, but recently Buset et al. [7] solved it, and the mixed Moore bound has been adjusted to
M(r, z, k) = A
uk+1 − 1
uk+1
1 −1
+B 2
u1 − 1
u2 − 1
(4)
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Nacho López and Hebert Pérez-Rosés / Procedia Computer Science 74 (2015) 2 – 9
where
v = (z + r)2 + 2(z√− r) + 1
z+r−1− v
u1 =
2
√
z+r−1+ v
u2 =
2
√
v − (z + r + 1)
A =
√
√ 2 v
v + (z + r + 1)
B =
√
2 v
In the context of the degree/diameter problem, either for undirected or directed graphs, there exist Moore-like upper
bounds (which in general are smaller than the Moore bound for general graphs). These bounds exist for bipartite
graphs [16] , planar graphs [15,38] , vertex-transitive graphs [28,37] , Cayley graphs [28,37,39] , etc. For example, the number
of
vertices of an undirected abelian Cayley graph with degree r = 2t and diameter k is bounded above by ti=0 2i ti ki .
On the other hand, the Moore bound for directed Cayley graphs is much simpler: k+z
z .
Moore-like upper bounds for mixed graphs have not been so well studied. As far as we know, there exists only one
result of this type. The following theorem [25] gives an upper bound for the number of vertices of an abelian mixed
Cayley graph, as a function of its diameter:
Theorem 1. Let Γ be an abelian group, and let Σ be a generating set of Γ containing r1 involutions and r2 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 = r1 + 2r2 , and directed out-degree z. The number of
vertices of Cay(Γ, Σ) is bounded above by
k r2 + z + i r 1 + r 2
,
k−i
i
i=0
(5)
where k represents the diameter of Cay(Γ, Σ).
The first interesting case concerning this bound is for diameter k = 2, where one can simplify Eq. (5) as:
1 + r2 + z +
1 2
d +d ,
2
(6)
where d = r + z. Note that,
lim
d→∞
1 + r2 + z +
1
2
d2 + d
M(r, z, 2)
= lim
d→∞
1 + r2 + z +
1
2
d2 + d
1 + z + d2
=
1
2
Hence, asymptotically with the total degree d, the bound for this restricted version of the degree/diameter problem
gives a maximum number of vertices which is half the maximum number of vertices for the general case.
Problem 1. Derive Moore-like upper bounds for different classes of mixed graphs
3. Mixed graphs approaching the mixed Moore bound
3.1. Mixed Moore graphs
Mixed graphs of order M(r, z, k) with maximum undirected degree r, maximum directed outdegree z and diameter
k are called mixed Moore graphs. Such extremal mixed graphs are totally regular of degree d = r + z and they have
the property that for any ordered pair (u, v) of vertices there is a unique trail from u ot v of length at most the (finite)
diameter k.
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Nacho López and Hebert Pérez-Rosés / Procedia Computer Science 74 (2015) 2 – 9
Moore graphs can be seen as a particular case of mixed Moore graphs when they contain only edges. These
(undirected) graphs are well studied (see [31] for a complete survey). In the particular case of diameter k = 2, Hoffman
and Singleton [20] proved that unique Moore graphs exist for k = 2 and r = 2, 3, 7, whereas the case r = 57 remains
as the most important open problem in this area. They also showed that for diameter k = 3 and degree r > 2 Moore
graphs do not exist. The enumeration of Moore graphs of diameter k > 3 was concluded by Damerell [12] , using
the theory of distance-regularity to prove nonexistence unless r = 2 (an independent proof was given by Bannai and
Ito [2] ). On the other hand, mixed Moore graphs are known as Moore digraphs when they contain only arcs. It is well
known that Moore digraphs only exist in the trivial cases, z = 1 or k = 1, which correspond to the directed cycle of
order k + 1 and the complete digraph of order z + 1, respectively (see [35,5] ).
Mixed Moore graphs containing both edges and arcs are called proper mixed Moore graphs. Nguyen, Miller and
Gimbert [33] proved the non-existence of proper mixed Moore graphs of diameter k ≥ 3. Nevertheless, some mixed
Moore graphs of diameter k = 2 are known to exist. Let A be the adjacency matrix of a mixed Moore graph (of
diameter 2), undirected degree r and directed degree z. Due to the uniqueness of trails of length at most 2 between
any pair of vertices, A satisfies
I + A + A2 = J + rI
(7)
where I and J denote the identity and the all-ones matrix, respectively. Indeed, each entry of I + A + A2 is 1, due
to the uniqueness of the trails, except in the main diagonal (which corresponds to closed trails of length ≤ 2), where
the fact that each vertex is incident to r edges gives exactly r closed trails of length 2 at any vertex. Computing the
characteristic polynomial of A, Bosák [4] gives the following necessary condition for the existence of a mixed Moore
graph of diameter 2:
Theorem 2. Let G be a (proper) mixed Moore graph of diameter 2. Then, G is totally regular with directed degree
z ≥ 1 and undirected degree r ≥ 1. Moreover, there must exist a positive odd integer c such that
r=
1 2
(c + 3) and c|(4z − 3)(4z + 5).
4
(8)
As a consequence, these extremal graphs may only exist for some (infinitely many) values of order n (see Table 1).
Nevertheless, the existence of these graphs has been proved only for a ‘few’ cases. The only known infinite family is
the Kautz [33] digraphs Ka(1 + z, 2), for every z ≥ 2 and r = 1. Gimbert [19] proved that Kautz digraphs are the unique
mixed Moore graphs with those parameters.
Table 1. Pairs (r, z) satisfying Eq. (8) and such that M(r, z, 2) ≤ 200.
M(r, z, 2)
6
12
18
20
30
40
42
54
56
72
84
88
90
108
110
132
150
154
156
180
182
r
1
1
3
1
1
3
1
3
1
1
7
3
1
3
1
1
7
3
1
3
1
z
1
2
1
3
4
3
5
4
6
7
2
6
8
7
9
10
5
9
11
10
12
d
2
3
4
4
5
6
6
7
7
8
9
9
9
10
10
11
12
12
12
13
13
Existence
Ka(2, 2)
Ka(3, 2)
Bosák graph
Ka(4, 2)
Ka(5, 2)
No
Ka(6, 2)
No
Ka(7, 2)
Ka(8, 2)
No
Unknown
Ka(9, 2)
Jørgensen graphs
Ka(10, 2)
Ka(11, 2)
Unknown
Unknown
Ka(12, 2)
Unknown
Ka(13, 2)
Uniqueness
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Unknown
Yes
No
Yes
Yes
Unknown
Unknown
Yes
Unknown
Yes
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Nacho López and Hebert Pérez-Rosés / Procedia Computer Science 74 (2015) 2 – 9
An interesting proper mixed Moore graph of order 18 was given by Bosák [4] (and its uniqueness was proved
later [33] ). The Bosák graph is a Cayley mixed graph either from S 3 × Z3 or from (Z3 × Z3 ) Z2 (see [24] ). In the
same paper it is shown that hypothetical mixed Moore graphs on 40 and 54 vertices cannot be Cayley graphs. This
result was extended by Jørgensen [21] , who proved that the 40-vertex case can not be vertex-transitive. Jørgensen also
found two non-isomorphic proper mixed Moore graphs of order 108 of Cayley type. Recently, López et al. [23] have
proved the non existence of mixed Moore graphs of orders 40, 54 and 84, by reducing the existence of their adjacency
matrix to the Boolean satisfiability problem. Besides, there are many pairs (r, z) satisfying Eq. (8), for which the
existence of a proper mixed Moore graph remains as an open problem (see Table 1). Structural properties of mixed
Moore graphs of directed degree z = 1 were studied independently in [27] and [13] . In particular it is shown that their
existence is related to the existence of some strongly regular graphs. The underlying undirected graph of any mixed
Moore graph of directed degree z = 1 (obtained by removing the directed part, that is, the arcs) is a distance-regular
graph of diameter 4.
Problem 2. Prove or dismiss the existence of mixed Moore graphs of parameters (r, z) satisfying Eq. (8).
3.2. Almost mixed Moore graphs
Mixed graphs of diameter k with maximum undirected degree r, maximum directed out-degree z and order just
one less than the Moore bound M(r, z, k) are called almost mixed Moore graphs. It is known that either almost Moore
graphs and almost Moore digraphs (also known as (r, k)-graphs of defect 1 and (z, k)-digraphs of defect 1, respectively)
are regular [30] . For the case of undirected graphs, Erdös, Fajtlowitcz and Hoffman [14] prove that, apart from the cycle
graph of length 4, almost Moore graphs of diameter 2 do not exist. This was generalized by Banai and Ito [2] to all
diameters. Fiol et al. [17] prove the existence of almost Moore digraphs of diameter 2 for every degree z and Gimbert [19]
completes its classification by enumerating all of them. Conde et al. [10,11] prove that almost Moore digraphs do not
exist for diameters 3 and 4. Seeking to generalize the aforementioned results, Gimbert formulated the cyclotomic
conjecture [18,9] , which states a relationship between the existence of almost Moore digraphs of any diameter k ≥ 3
and the irreducibility of certain cyclotomic polynomials. The non existence of the almost Moore digraphs for k > 2
could also be derived from the conjecture [8] based on the structure of the out-neighbours of a k-type vertex. On the
other hand, focusing on small out-degrees instead of diameter, Miller and Fris [29] proved that there are no almost
Moore digraphs of maximum out-degree z = 2, for any k ≥ 3. Moreover, Baskoro et al. [3] showed that there are no
almost Moore digraphs of maximum out-degree z = 3 and any diameter k ≥ 3.
a0
c0
a1
a4
c1
c4
c2
a2
c3
a3
Fig. 2. The unique mixed Moore graph with undirected degree r = 2 and directed degree z = 1.
Little is known about the general mixed case. Dealing with the parity of M(r, z, 2) it is easy to see [22] that there is
no almost mixed Moore graph of diameter 2 with undirected odd degree r. This can be generalized [1] to any diameter
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Nacho López and Hebert Pérez-Rosés / Procedia Computer Science 74 (2015) 2 – 9
when M(r, z, k) is odd. López and Miret [22] establish the following necessary condition for the existence of a totally
regular almost mixed Moore graph of diameter 2:
Theorem 3. Let G be a totally regular almost mixed Moore graph of diameter 2, undirected even degree r > 2, and
directed degree z ≥ 1. Then, one and only one of the following conditions must hold:
(a) There exists an odd integer c1 ∈ Z such that c21 = 4r + 1 and c1 | (4z + 1)(4z − 7).
(b) There exists an odd integer c2 ∈ Z such that c22 = 4r − 7 and c2 | (16z2 + 40z − 23).
Table 2 provides examples of values for which an almost mixed Moore graph could exist (but no example is known
so far). The case r = 2 is excluded from Th. 3 and is completely open, except for the case z = 1, where a mixed graph
of 10 vertices does exist (see Fig. 2). Recently, Buset [6] proves the uniqueness of this mixed graph, but in general
there are many open questions related to these extremal graphs.
Question 1. Are there almost mixed Moore graphs of diameter 2, undirected degree r = 2 and directed degree z ≥ 2?
Question 2. Are almost mixed Moore graphs of diameter 2 totally regular?
Question 3. Does an almost mixed Moore graph of diameter 2 with parameters (r, z), satisfying the necessary conditions given in Th. 3, exist?
Table 2. The first even values for the undirected degree r and their corresponding values for parameters c1 , c2 and z as they appear in Th. 3.
r
4
6
8
10
12
14
16
18
20
22
c1
5
7
9
-
c2
3
5
7
9
z
1, 4, 7, 10, . . .
1, 3, 6, 8, . . .
5, 7, 12, 14, . . .
2, 4, 11, 13, . . .
-
n
26, 68, 128, 206, . . .
50, 84, 150, 204, . . .
294, 368, 588, 690, . . .
486, 580, 972, 1102, . . .
-
Existence
Unknown
Unknown
non-existent
non-existent
Unknown
non-existent
non-existent
non-existent
Unknown
non-existent
3.3. Large mixed graphs
The construction of large graphs with small diameter in the context of the degree/diameter problem has received
a lot of attention in the literature, and we refer the reader to the surveys [31,34] to get an idea about the techniques
and results. Most of the latest largest graphs have been found with the aid of computer search and clever techniques
to reduce the search space. Optimal graphs, that is, graphs with the largest possible number of vertices given the
(undirected) degree r and the diameter k, have been obtained in some cases. The situation is very different for the
directed case: The best constructions are Kautz digraphs, and the Alegre digraph, together with their line digraphs.
These constructions are known to be optimal for diameter k = 2 and any degree z ≥ 1, and for diameter k = 3 and
degree z = 2. Tables of the orders of the largest known [di]graphs of the degree/diameter problem can be seen on the
Combinatoricswiki website1 , including those of the restricted versions of the problem.
Not much work has been done in this direction for the mixed case. Araujo et al. [1] construct a family of mixed
2
graphs of diameter 2 with parameters r = q + 2t, z = q−1
2 − 2t and 2q vertices, where q ≥ 3 is an odd prime power
and t belongs to a set of integers depending on q. Such a construction is based on biaffine planes, and it yields optimal
mixed graphs for some cases. In particular, for q = 5 and t = 0 this construction gives an optimal mixed graph of
1
http://combinatoricswiki.org/wiki/Main_Page
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Nacho López and Hebert Pérez-Rosés / Procedia Computer Science 74 (2015) 2 – 9
order 50 with parameters r = 5 and z = 2. Recently, another optimal mixed graph with the same set of parameters has
been found [26] by a different approach. Besides, Šiagiová [36] approaches the mixed Moore bound for diameter k = 2
using Cayley mixed graphs, by extending a known construction for the undirected case.
Regarding restricted versions of the problem, very little has been done. For mixed abelian Cayley graphs. Eq. (5)
becomes
k 1 1+k−i
= 1 + 2k
i
k−i
i=0
when r = 1 and z = 1. This bound is unattainable since the subgraph induced by the (undirected) edges of any mixed
graph with r = 1 is a disjoint union of complete graphs of order 2, hence has even order. So the upper bound becomes
2k. Now, the (mixed) circulant graph with order 2k and generators 1 and k has diameter k, so this case is done. For
r = 1 and z > 1, López and Pérez-Rosés [25] approach the Moore bound
k 1 z+k−i
i=0
which is assimptotically close to
i
k−i
2k + z k + z
=
k+z k
2k − 2 + 2z z
2kz
, by constructing circulant graphs with order
.
z!
2z − 1
Problem 3. Find families of mixed graphs with large order.
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