Optimal Networks
Mirka Miller
School of Information Technology and Mathematical Sciences
University of Ballarat
m.miller@ballarat.edu.au
CIAO
Centre for Informatics and
Applied Optimization
1
Outline of the talk
 Interconnection
networks
 Degree/diameter problem – directed and
undirected
 Recent new results concerning graphs close to
Moore bound
 Three extremal problems
 Open problems
2
Interconnection networks
Examples:
Communications
 Transportation
 Computer
 Social

Networks can be modeled as graphs. The
structure (topology) of the graph is useful
when designing algorithms (for
communication, broadcasting etc.) and for
analyzing the performance of a network.
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Small-world networks
4
WWW (2000)
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Network parameters
Topology
Nodes and edges and their arrangement.
Diameter
Maximum distance between any pair of nodes.
Connectivity
Number of neighbours of a given node:
d degree
Clustering
Are neighbours of a node also neighbours among
themselves?
6
Most “real life” networks
small-world
small diameter
Milgram 1967
clustered
Watts & Strogatz 1998
scale-free
degrees follow a power-law
Barabási & Albert 1999
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Network topology
Structured
 high clustering
 large diameter
 regular
|V| = 1000 d = 10
D = 100
C = 0.67
Small-world
Random
 high clustering
 small diameter
 almost regular
 low clustering
 small diameter
|V|=1000 d = 8-13
D = 14
C = 0.63
|V|=1000 d = 5-18
D=5
C = 0.01
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Definitions
We study graphs with respect to 3 parameters: diameter,
degree and order.
 Diameter
 The
longest distance between any two
vertices in the graph.
 Degree
 The number of edges attached to a vertex.
 Order
 The number of vertices in the graph.
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Degree/diameter problem
Determine the largest number of
vertices n of a graph G for given
maximum degree d and diameter at most
k.


Survey by Miller and Siran “Moore bound and beyond: A
survey of the degree/diameter problem”
Research supported by a grant from Australian Research
Council (ARC) CI’s: McKay and Miller
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Degree/diameter problem
Directed version:
Determine the largest number of vertices
n of a digraph G for given maximum
out-degree d and diameter at most k.
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Degree/diameter problem
Approaches to attack the problem
Increase the lower bound: by construction
o
o
o
o
Voltage assignments
Line digraphs
Computer search
Etc.
Decrease the upper bound: by proving a graph
with given parameters cannot exist
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Upper bound
A natural upper bound on the
number of vertices n of digraph G
with given maximum out-degree d
and diameter at most k is:
n  Md,k = 1+d +d 2 + … + d k.
v
This bound is called the Moore bound.
A digraph attaining this bound is called
a Moore digraph.
13
Moore digraphs
Plesnik & Znam ’74, Bridges & Toueg ’80 :
Moore digraphs exist only for trivial
cases, namely for d =1 (cycles of k+1
vertices) or k =1 (complete digraphs
on d+1 vertices).
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Moore digraphs
Outline of the PROOF that (d,k)-digraphs do not exist
when
d > 1 and k > 1 :
I+A+…+Ak=J where A is adjacency matrix of G,
J is unit matrix, I is identity matrix
J has eigenvalues n (once) and 0 (n-1 times)
A has eigenvalues d (once) and n-1 roots of the characteristic
equation 1+ x+x2+…xk = 0
x = 0 and roots of xk+1 -1 =0
Since tr(Aj)=0 for 1  jk, we obtain –d=-dk and so
d=1 or k=1 are the only solutions.
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Diregularity of AMDs
Is Md,k–1 attainable for all d>1 and k>1?
Notation. (d,k)-digraph is a digraph of maximum out-degree d,
diameter k and order n = Md,k – 1. (We also call such a digraph
almost Moore digraph).
Miller, Gimbert, Siran & Slamin, ‘00:
The (d,k)-digraphs are diregular of degree d.
Note that: to show the regularity of out-degree is easy (by a
counting argument). However, to show the regularity of in-degree
is not easy.
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Repeat of a vertex
Let G be a (d,k)-digraph.
 For every vertex x of G there exists a vertex y,



called the repeat of x, such that there are two walks
of lengths  k from x to y.
If r(x) = x then vertex x is called a selfrepeat.
The function r : V(G)  V(G) is an automorphism
on V(G); namely, (x,y)  E(G) iff (r (x), r (y)) 
E(G).
Let the order of vertex v be the smallest positive
integer (v) so that r (v)(v)= v.
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Diameter 2
Fiol, Allegre, and Yebra ’83:
(d,2)-digraphs exist for any d  2.
Example:
The line digraph of L(Kd+1) of the complete
digraph Kd+1.
But Gimbert ’01 showed that this line
digraph is the only (d,2)-digraph if d  3.
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Diameter 2, degree 2
There are exactly three (2,2)-digraphs.
1
2
1
3
4
4
5
3
2
1
4
5
2
5
6
3
6
(1)(2)(3)(4)(5)(6)
(123)(456)
All selfrepeats
All order 3
6
(12)(3456)
Two order 2; four order 4
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Degree 2 and degree 3
Miller and Fris ’92:
There are no (2,k)-digraphs for any k  3.
Baskoro, Miller, Siran & Sutton (in press):
There are no (3,k)-digraphs for any k  3.
The remaining cases are still open:
Do there exist any (d,k)-digraphs, d  4, k  3?
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Existence of (d,k)-digraphs
A (d,k)-digraph, d  4, k  3 (if it exists) may
contain a selfrepeat or no selfrepeat.
Further study may focus on the existence of:
 (d,k)-digraphs with selfrepeats
 (d,k)-digraphs with no selfrepeats
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Structure of the orders of
vertices
A (d,k)-digraph contains either k
selfrepeats or none.
[Baskoro, Miller, Plesnik, ’98]
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The orders of vertices
We can determine the structure of the
orders of vertices in a (d,k)-digraph with
selfrepeats, d  2, k  2.
[Baskoro, Cholily, Miller, ’04]
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The orders of vertices
Example. k=2, d=6. Label vertices 0,1,2,…,41. Suppose
the digraph G contains a selfrepeat and that the outneighbourhood of a selfrepeat consists of vertices of
orders 2 and 3 (as well as a selfrepeat). Then (up to
isomorphism) the permutation cycles of repeats of G are
(0) (1) (2,3)(4,5,6) (7,8)(9,10,11)(12,18)(13,19)(14,20)
(15,21,16,22,17,23)(24,30,36)(25,31,37)(26,32,38)(27,33,39)
(28,34,40)(29,35,41)
Two cycles of length 1, five cycles of length 2,
eight cycles of length 3, one cycle of length 6.
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Open problems
Are there any (d,k)-digraphs, d  4, k  3, with
selfrepeats?
Are there any (d,k)-digraphs, d  4, k  3, without
selfrepeats?
For d = 3, k  3, are there any digraphs of order
M3,k – 2?
For d = 2, k  3, are there any digraphs of order
Md,k – 3?
Are almost almost Moore digraphs diregular?
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Degree/diameter problem
Undirected version:
Determine the largest number of vertices n
of a graph G for given maximum degree d
and diameter at most k.
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Upper bound
A natural upper bound on the number of
vertices n of a graph G of given
maximum degree d and diameter at most
k is:
n  Md,k=1+d+d(d-1)+…+d(d-1)k-1
v
 This bound is called the Moore bound.
 A graph attaining this bound is called a Moore
graph.
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Moore graphs
k =1: Moore graphs are complete graphs on
d+1 vertices.
Hoffman and Singleton, ’60:
k =2: Moore graphs exist for d =2
(pentagon) or d =3 (Petersen graph) or
d =7 (Hoffman-Singleton graph) or d
=57?
k =3: Moore graph exists for d =2 (7-gon).
Damerell; Bannai and Ito, ’73:
k >3: Moore graph are (2k+1)-gons.
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Almost Moore graphs
Is Md,k–1 attainable for all d>2 and k>1?
Erdos, Fajtlowicz and Hoffman, ’80:
k =2: almost Moore graph exists only
for d =2 (4-cycle).
Bannai and Ito; Kurosawa and Tsujii, ’81:
k >3: almost Moore graphs exist only
for d =2 (2k-gons).
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Graphs with defect > 1
Defect 2:
d =2: (2k-1)-gons.
d >2: only 5 such graphs are known so far:
(d,k) = (3,2) (two); (4,2); (5,2); (3,3)
(unique).
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Almost almost Moore graphs
 There are no almost almost Moore graphs of
degree 3 and diameter k>3.
[Jorgensen, ’92]
 Theorem 6. For k>2, there are no almost
almost Moore graphs of degree 4.
[Miller and Simanjuntak, ’04]
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Repeat of a vertex
Define (d,k,d)-graph to be a graph of degree d,
diameter k and defect d.
Vertex y is a maximal repeat of x if y appears in
R(x) d times (x has no other repeats).
Theorem 7. For d >1, the number of
maximal repeats in a (d,2,d)-graph is
0 or 2 or 6.
[Nguyen and Miller, ’04]
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Structure of a (d,2,2)-graph
Possible repeat configurations in a (d,2,2)-graph:
u
r1(u)
u
u
r2(u)
r(u)
u
u
r1(u)
r2(u)
r1(u)
r1(u) r2(u)
r2(u)
Define n0,n1,n2a,n2b,n2c.
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Structure of a (d,2,2)-graph
Theorem 7. A (d,2,2)-graph contains
 if d is even then n0 = 3 and n2b = d2 – 4
 if d = 3 then (n0,n1,n2c) = (3,2,3)
 if d is odd then
(n0,n1,n2c,n2a,n2b) = (9,6,9,4a,d2-25-4a)
or n2b = d2 – 1.
[Nguyen and Miller, ’04]
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Open problems
 Are there any (d,2,2)-graphs for d  6?
 Are there any (d,k,2)-graphs for d  5 and
k  3?
 Are there any (3,k,3)-graphs for k  4?
 Are there any (4,k,3)-graphs for k  3?
.
.
.
 Is there a Moore graph with diameter 2 and
degree 57?
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Open problems
 We know that for directed graphs N(d,k) is
monotonic in both d and k. Let K(n,d) be the
smallest possible diameter of a digraph on n
vertices and maximum out-degree d. Is K(n,d)
monotonic in n?
For undirected graphs we know that the
corresponding K(n,d) is not monotonic in n.
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Example. K(10,3) = K(8,3) = 2
n=8
n = 10
but K(9,3) = 3!
n cannot be more
than 10
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Three optimisation problems
= max{n: G(n,d,k)}
 D(n,k) = min{d: G(n,d,k)}
 K(n,d) = min{k: G(n,d,k)}
 N(d,k)
G(n,d,k)
denotes the set of all directed
graphs of order n, degree d, and diameter k.
Question: are these three problems equivalent to each other?
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Known monotonic relationships
 d1<d2 implies N(d1,k)< N(d2,k)
 k1<k2 implies N(d,k1)< N(d,k2)
 d 1 <d 2 implies K(n,d 1 ) K(n,d 2 )
? n1<n2 implies K(n1,d)  K(n2,d)
? k1<k2 implies D(n,k1)  D(n,k2)
? n1<n2 implies D(n1,k)  D(n2,k)
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K(n,d) problem
digraph G of degree d, diameter k
and order n
line digraph construction
digraph L(G) of degree d, order
dn and diameter k+1
Vertex deletion scheme
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K(n,d) problem
Theorem [Slamin & Miller,’00]
If L(G)  G(dn,d,k) is a line digraph of a diregular digraph
GG(n,d,k-1) then there exists digraph L(G) G(dn-r,d,k’),
k’k, for every 1r (d-1)n-1
n
nd
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K(n,d) problem
The largest n such
that G(n,d,k-1) is
not empty for
given d.
G(nd,d,k)
The largest n’
such that
G(n’,d,k) is not
empty for given d.
42
K(n,d) problem
 K(n,d) is monotonic in n in some intervals of d
values
 K(n,d) problem is equivalent to N(d,k) in those
intervals
 N(d,k)  K(n,d)  D(n,k)
For directed graphs, are these three problems in
fact all equivalent?
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