Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
SIAM J. DISCRETE MATH.
c
2007 Society for Industrial and Applied Mathematics
Vol. 21, No. 3, pp. 763–784
CAYLEY DIGRAPHS OF FINITE ABELIAN GROUPS AND
MONOMIAL IDEALS
∗
DOMINGO G
´
OMEZ
†
, JAIME GUTIERREZ
†
, AND
´
ALVAR IBEAS
†
Abstract. In the study of double-loop computer networks, the diagrams known as L-shapes arise
as a graphical representation of an optimal routing for every graph’s node. The description of these
diagrams provides an efficient method for computing the diameter and the average minimum distance
of the corresponding graphs. We extend these diagrams to multiloop computer networks. For each
Cayley digraph with a finite abelian group as vertex set, we define a monomial ideal and consider
its representations via its minimal system of generators or its irredundant irreducible decomposition.
From this last piece of information, we can compute the graph’s diameter and average minimum
distance. That monomial ideal is the initial ideal of a certain lattice with respect to a graded
monomial ordering. This result permits the use of Gr¨obner bases for computing the ideal and finding
an optimal routing. Finally, we present a family of Cayley digraphs parametrized by their diameter d,
all of them associated to irreducible monomial ideals.
Key words. monomial ideals, Cayley digraph, Gr¨obner bases, multiloop networks
AMS subject classifications. 13P10, 05C25, 68M10
DOI. 10.1137/050646056
1. Introduction. Let Γ be a group and S ⊆ Γ a subset. The Cayley digraph
associated to (Γ,S) is a directed graph whose vertex set is Γ and whose edge set is
{(g, h) ∈ Γ
2
| g
−1
h ∈ S}. Every Cayley digraph is vertex-symmetric and its degree
equals the number of elements in S. These graphs are connected if and only if the
set S generates the group. We are dealing with digraphs associated to finite abelian
groups, but we are mainly interested in those associated to cyclic groups. Let N be
a positive integer and Z
N
the integers modulo N. For any subset S = {j
1
,...,j
r
} of
this abelian group we denote by C
N
(S)=C
N
(j
1
,...,j
r
) the corresponding Cayley
digraph (see Figure 1.1), which is called the circulant digraph or multiloop computer
network of jumps j
1
,...,j
r
. It is connected if and only if gcd(j
1
,...,j
r
,N)=1.
If S is a subset of Z
N
such that for every element in S its inverse also lies in S,
then C
N
(S) is an undirected graph called a circulant graph or distributed multiloop
computer network.
Multiloop networks were first proposed in [32] for organizing multimodule memory
services and have a vast number of applications in telecommunication networking,
VLSI design, and distributed computation. Their properties, such as diameter and
reliability, have been the focus of much research in computer network design; see, for
instance, [5, 7, 12, 13, 19, 21, 25, 33].
The single-loop network or ring network is mathematically trivial. Digraphs with
r =2ordouble-loop networks and their corresponding undirected graphs (distributed
double-loop networks, with degree four) have been extensively studied; see the sur-
veys [3, 20] and the references therein. When C
N
(j
1
,j
2
) is connected, one can define a
minimum distance diagram (MDD) as an array with vertex 0 in cell (0, 0) and vertex c
∗
Received by the editors November 25, 2005; accepted for publication (in revised form) March
22, 2007; published electronically September 26, 2007. This research was partially supported by the
Research Project MTM2004-07086 of the Spanish Ministry of Science. An extended abstract of part
of this work appeared in [14].
http://www.siam.org/journals/sidma/21-3/64605.html
†
University of Cantabria, E–39071 Santander, Spain (domingo.gomez@unican.es, jaime.gutierrez
@unican.es, alvar.ibeas@unican.es).
763
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Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
764 DOMINGO G
´
OMEZ, JAIME GUTIERREZ, AND
´
ALVAR IBEAS
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Fig. 1.1. C
18
(3, 8).
0 5
14
10
19
28
15
24
9
20
29
23
25
1
4
30
6
18
2
11
32
7
16
13
12
21
27
17
26
8
22
31 3
Fig. 1.2. MDD of C
33
(5, 14).
in cell (x, y)(x is the column index and y the row index), for a particular choice satis-
fying j
1
x + j
2
y ≡ c mod N, and x + y minimum. One example is shown in Figure 1.2.
The classical work of Wong and Coppersmith [32] presents an algorithm for con-
structing an MDD of C
N
(j
1
,j
2
)inO(N
2
) steps and shows it has an “L” shape.
Several characterizations and applications of this idea for describing circulants with
desirable properties appear in [1, 8, 9, 13]. However, they do not focus on higher
degree digraphs.
Two notable parameters in a graph are the diameter d and the average minimum
distance
¯
d. The former represents the worst delay in the communication between two
nodes, and the latter represents the average delay. Given an L-shape, it is easy to
compute d and
¯
d.
On the other hand, let d
r
(N):=min{d(C
N
(j
1
,...,j
r
) | j
1
,...,j
r
∈ Z
N
}. An im-
portant problem is to determine this value and find a specific C
N
(j
1
,...,j
r
) attaining
this minimum. The network C
N
(j
1
,...,j
r
) is said to be optimal if its diameter equals
d
r
(N). In some cases, it is difficult to obtain optimal networks; however, one can find
general simple functions serving as upper and lower bounds for d
r
(N); see [3]. The
paper [32] shows d
2
(N) ≤
√
3N − 2 and presents a family of circulant digraphs with
diameter 2
√
N − 2.
In this article we present monomial ideals as a natural tool for studying the
MDDs of arbitrary Cayley digraphs, provided that the vertex group is finite and
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CAYLEY DIGRAPHS AND MONOMIAL IDEALS 765
abelian. Given a graded monomial ordering and a Cayley digraph (Γ,S), we build a
monomial ideal in the polynomial ring K[X
1
,...,X
r
], where K is an arbitrary field
and r =#S. We obtain some properties of this monomial ideal: in particular, a
certain generalization of the two-dimensional L-shape is shown. On the other side, it
is the initial ideal of a certain lattice. This result permits the use of Gr¨obner bases
for computing the ideal and finding an optimal routing for each pair of nodes. Given
the representation of the monomial ideal via its irreducible decomposition, we provide
formulae to compute d and
¯
d. We also show a family of circulant digraphs of degree
two which coincides with the family obtained in paper [32]. Finally, we present a
new and attractive family of circulant digraphs of arbitrary degree parametrized by
the diameter d, with average minimum distance d/2, and whose associated monomial
ideals are irreducible.
The paper is divided into nine sections. In section 2 we collect several known
facts about monomial ideals, presenting examples and fixing notation for later use.
Section 3 presents the key idea of associating monomial ideals to digraphs in order to
obtain an MDD, and it also provides an algorithm to construct an MDD for Cayley
digraphs with a finite abelian group as vertex set. Section 4 is devoted to present-
ing the relation between MDDs and the ideal of a lattice. In section 5 we present
an algorithm to compute a shortest path between two vertices by means of Gr¨obner
bases. Section 6 presents an algorithm specifically tailored for degree three circulants.
It computes the minimal system of generators in O(s log N) arithmetic operations,
where s is the number of generators and N is the number of nodes. Section 7 is dedi-
cated to providing formulae to find the diameter and the average minimum distance.
Then section 8 presents a family of multiloop computer networks with an arbitrary
number of jumps, parametrized by the diameter d, and all of them associated to ir-
reducible monomial ideals. We conclude with a short summary and a discussion of
open questions.
2. Monomial ideals. Monomial ideals form an important link between com-
mutative algebra and combinatorics. Here we review several basic related results and
definitions concerning monomial ideals; see, for instance, [2, 30].
Let K be an arbitrary field and K[X
1
,...,X
r
] the polynomial ring in the variables
X
1
,...,X
r
. Throughout the paper, we very often identify monomials of K[X
1
,...,X
r
]
with vectors of N
r
and use the following notation:
x
a
= X
a
1
1
···X
a
r
r
←→ a =(a
1
,...,a
r
),
x
a
|x
b
⇐⇒ a =(a
1
,...,a
r
) ≤ b =(b
1
,...,b
r
)
def
⇐⇒ a
i
≤ b
i
∀i =1,...,r,
a =(a
1
,...,a
r
) b =(b
1
,...,b
r
)
def
⇐⇒ (b
i
> 0 ⇒ a
i
<b
i
) ,
e
i
:= (0,...,
i
1
,...,0), m
a
:= (X
a
i
i
| a
i
> 0), 1 := (1,...,1).
The definition of suits the characterization in (2.2), and when it is employed
(in expressions like a b), we usually have 1 ≤ b.
A monomial ideal is an ideal generated by monomials, i.e., I ⊂ K[X
1
,...,X
r
]is
a monomial ideal if there is a subset A ⊆ N
r
such that
I =(x
a
| a ∈ A)=(A).
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766 DOMINGO G
´
OMEZ, JAIME GUTIERREZ, AND
´
ALVAR IBEAS
y
3
x
2
y
3
x
2
y
2
x
4
y
2
x
4
Fig. 2.1. Staircase diagram and Buchberger’s graph.
There are two standard ways of describing a nontrivial monomial ideal:
• Via the (unique) minimal system of monomial generators I =(x
a
1
,...,x
a
s
),
we have
(2.1) x
u
∈ I ⇐⇒ ∃ i ∈{1,...,s}|a
i
≤ u.
• Via the (unique) irredundant decomposition by irreducible monomial ideals
I = m
b
1
∩···∩m
b
n
,wehave
(2.2) x
u
∈ I ⇐⇒ ∃ i ∈{1,...,n}|u b
i
.
The so-called staircase diagram is a useful graphical representation of monomial
ideals.
Example 2.1. The monomial ideal I
1
:= (x
4
,x
2
y
2
,y
3
)=(x
2
,y
3
) ∩ (x
4
,y
2
)is
represented on the left in Figure 2.1.
There is an algorithm for finding the irredundant irreducible decomposition of a
monomial ideal based on Alexander duality; see [27]. An irreducible component m
a
can be associated to lcm(X
a
1
1
,...,X
a
r
r
)=x
a
. On the other hand, if K[X
1
,...,X
r
]/I
is an artinian ring, then the monomial x
a
associated to the irreducible component m
a
must coincide with the least common multiple of a subset of the minimal generators
of I. In the above Example 2.1 we have
x
2
y
3
= lcm(x
2
y
2
,y
3
),x
4
y
2
= lcm(x
4
,x
2
y
2
).
The diagram on the right in Figure 2.1 is called Buchberger’s graph of the mono-
mial ideal I
1
; see [28]. At any stage in Buchberger’s algorithm for computing Gr¨obner
bases, one considers the S-pairs among the current polynomials and removes those
which are redundant; the minimal S-pairs define a graph on the generators of any
monomial ideal.
Theorem 2.2. Let I be a nontrivial monomial ideal given by a minimal system
of generators I =(x
a
1
,...,x
a
s
) and by the irredundant irreducible decomposition
I = m
b
1
∩···∩m
b
n
. The following are equivalent:
1. K[X
1
,...,X
r
]/I is an artinian ring.
2. ∀i =1,...,r, one of the generators’ exponents is a
j
= α
i
e
i
for some α
i
∈ N.
3. ∀i =1,...,n, ∀j =1,...,r, b
i,j
> 0.
Proof. We need to prove that the number of monomials outside I is finite if and
only if either of the two last items is satisfied. We do that using the characterizations
in (2.1) and (2.2).
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CAYLEY DIGRAPHS AND MONOMIAL IDEALS 767
8,0,0 4,2,0 0,5,0
7,0,1
3,2,2
0,3,1
3,0,4
0,0,5
8,2,1
7,2,4
3,3,5
4,3,2
4,5,1
Fig. 2.2. Planar graph associated to I
2
.
If the second item is true, then the number of monomials which do not lie in the
ideal is bounded by the product
α
i
. Conversely, if that item is false, there exists
an index i ∈{1,...,r} such that X
α
i
∈ I ∀α ∈ N.
The third item is obviously equivalent to #{u ∈ N
r
| u b
i
for some i ∈
{1,...,r}}< ∞.
We conclude this section by illustrating those facts in the following example.
Example 2.3. In [28], a planar graph is associated to every monomial ideal in three
variables satisfying the conditions in Theorem 2.2. The monomial x
b
associated to
an irreducible component m
b
is identified with a connected component in the graph’s
complement and can be obtained as the least common multiple of the generators in
its boundary. In Figure 2.2 we show this construction for the ideal:
I
2
:= (x
8
,x
4
y
2
,y
5
,y
3
z,z
5
,x
3
z
4
,x
7
z,x
3
y
2
z
2
)
=(x
8
,y
2
,z) ∩ (x
7
,y
2
,z
4
) ∩ (x
4
,y
3
,z
2
) ∩ (x
4
,y
5
,z) ∩ (x
3
,y
3
,z
5
).
The description of those relations permits the simplification of some computations
on Cayley digraphs, as pointed out in section 7.
3. Minimum distance diagrams. There are different ways to relate monomial
ideals with graphs (see, for instance, [30]). In this section we propose a new approach
to studying Cayley digraphs in which we associate a graph with a monomial ideal.
The routing problem for Cayley digraphs reduces to studying paths originating at
a fixed vertex, as these graphs are vertex-symmetric. Given a graph associated to
(Γ, {s
1
,...,s
r
}), where Γ is finite and abelian, we are looking for the shortest path
from node 0
Γ
to node c ∀c ∈ Γ, i.e., a minimum distance diagram (MDD). We can
construct the routing mapping R:
(3.1)
R : N
r
−→ Γ
a → a
1
s
1
+ ···+ a
r
s
r
.
Thus, we need to find a right inverse map of R:
D :Γ−→ N
r
,
such that
R(D(c)) = c ∀c ∈ Γ and D(c)
1
= min{x
1
| x ∈ R
−1
(c)}.
In general, map D is not unique; see Figure 3.1. This happens when the set R
−1
(c)
contains two or more elements with minimum
1
-norm for some c ∈ Γ.
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