A chordal ring is an augmented ring, or a circulant graph with
a chord of length 1. Formally it is defined by the pair (n,L) where
n is the number of nodes of the ring, and L is the set of chords,
.
Each chord
connects
every pair of nodes of the ring that are at distance l in the ring.
We denote by
,
,
the
chordal ring defined by
.
The dimension of
is t+1. Let us emphasize that the degree of
chordal rings is 2t in general, except whenever there is a chord of
length n/2. In this case n is even and the degree is 2t-1.
A ``natural'' labeling of the nodes for chordal rings is the cyclic
labeling: Consecutive labeling of nodes around the ring. This labeling
is not always the best possible one. For instance, [KL95] showed
that
has no shortest path 1-IRS with cyclic labeling
whereas it is of compactness 1. In fact it is possible to
characterize graphs supporting a shortest path 1-IRS according a
cyclic labeling of nodes.
By the use of the Ádám property, [Man98] showed that
is isomorphic to
if and only if
.
For instance
,
which has no shortest path 1-IRS with
cyclic labeling (cf. [KL95]), is isomorphic to
which
satisfies Theorem 26. Therefore,
.
Let
.
Since
is of diameter i
(cf. [BCH95]), it follows that 1-
,
for
every
.
(Actually,
is the chordal ring of degree 4
with the largest number of nodes for a given diameter). Moreover,
in [NO97], it is shown that 1-
.
Also, the
smallest chordal ring known to be of compactness greater than 1 is
,
and 1-
.
[NO97] showed that
there exists some chordal rings,
,
where every shortest path
IRS using a cyclic labeling requires
intervals, and also
that
has no shortest path 1-IRS of stretch factor
with a cyclic labeling. Other results about directed chordal rings
are mentioned in [KL95].
For two chords we have:
And for higher dimensions,
The latter result can be easily generalized to
when
,
and
for every i.
In [FGS94] is presented several results concerning IRS without the disjointness property of the labels assigned to the arcs, i.e., with Condition 2b of Definition 1 relaxed. See Paragraph 4.3 for a discussion of these results.
Note also that most of the results extend to circulant graphs in general, i.e., does not need to specify the chord of length 1.