GROUPS AND SYMMETRY
CHAPTER 2
Contents
1.
2.
3.
4.
5.
6.
7.
Groups and subgroups
Group actions
Symmetry groups
Covering graphs
Symmetry in graphs
Symmetry in metric spaces
Representation of graphs and their
symmetries
1. Gropus and Subgroups
Groups
• Group G consists of elements, g1, g2, ... and
an operation ². It satisfies the following:
• A1. Replacing any two symbols in the
equation a ² b = c by group elements
uniquely determines the third one.
• A2. For any three group elements gi,gj,gh we
have: (gi ² gj) ² gk = gi ² (gj ² gk)
Usual Group Axioms
• (a ² b) ² c = a ² (b ² c)
• There exists e, such that for any a
a ² e = e ² a = a.
• For each a there exists a’ such that
a ² a’ = a’ ² a = e.
• Exercise: Show that both systems of axioms
are equivalent.
Finite and Infinite Groups
• The number |G| is called the order of group
G.
• The groups of finite order are called finite
groups. All other groups are called infinite.
Abelian Group
• If a ² b = b ² a for any a,b 2 G the group G is
called Abelian or commutative.
Residues mod n: Zn.
•
•
•
•
•
•
Two views:
Zn = {0,1,..,n-1}.
Define ~ on Z:
x ~ y $ x = y + cn.
Zn = Z/~.
(Zn,+) an abelian group, called the cyclic
group of order n. Here + is taken mod n!!!
Example (Z6, +).
+ 0 1 2 3 4 5
0 0 1 2 3 4 5
1 1
2
3
4
5
0
2 2
3
4
5
0
1
3 3
4
5
0
1
2
4 4
5
0
1
2
3
5 5
0
1
2
3
4
Example
(Z6, £) is not a group.
£ 0 1 2 3 4 5
0 0 0 0 0 0 0
1 0
1
2
3
4
5
2 0
2
4
0
2
4
3 0
3
0
3
0
3
4 0
4
2
0
4
2
5 0
5
4
3
2
4
Example
(Z6\{0}, £) is not a group.
For p prime, (Zp\{0}, £) forms a group.
£ 1
1 1
2
2
3
3
4
4
5
5
2 2
4
0
2
4
3 3
0
3
0
3
4 4
2
0
4
2
5 5
4
3
2
4
R - real numbers
• (R,+) is a group
• The group is Abelian and infinite. Its unit is
0, the inverse of a is –a.
• (R \ {0},.) is a group
• The group is infinite, Abelian, its unit is 1,
the inverse of a is 1/a.
• Let R+ := {x 2 R| x > 0}. (R+,.) is also a
group.
Subgroup
• H µ G is a subgroup if H is a group for the
same group operation.
• There are subsets, closed for the group
operation, that are not subgroups. For
instance, (N,+) is not a subgroup of (Z,+).
Cosets
• G – group
• H – subgroup
• The set aH := {a ² x| x 2 H} is called (left)
coset of H in G.
• G = H t aH t bH t ...
Index
• Let H µ G be a subgroup of G.
• [G:H] := # cosets of H is called the index of
H in G. For finite groups [G:H] = |G|/|H|.
Q – rational numbers
• (Q,+) is a group.
• Rational numbers form a group under
addition.
• (Q \ {0},.) is a group.
Z - integers
• (Z,+) is a group.
• (Z \ {0},.) is not a group.
Complex numbers C.
a = a + bi 2 C.
a* = a – bi.
b = c + di 2 C.
ab = (ac –bd) + (bc + ad)i.
b  0, a/b = [(ac + bd) + (bc – ad)i]/[c2 +
d2].
• a-1 = (a –bi)/(a2 + b2).
•
•
•
•
•
C, - Complex numbers
• (C \ {0},.) is a group
• (C,+) is a group
Quaternions H.
•
•
•
•
Quaternions form a non-commutative field.
General form:
q = x + y i + z j + w k., x,y,z,w 2 R.
i 2 = j 2 = k 2 =-1.
• q = x + y i + z j + w k.
• q’ = x’ + y’ i + z’ j + w’ k.
• q + q’ = (x + x’) + (y + y’) i + (z + z’) j + (w + w’) k.
• How to define q .q’ ?
• i.j = k, j.k = i, k.i = j, j.i = -k, k.j = -i, i.k = -j.
• q.q’ = (x + y i + z j + w k)(x’ + y’ i + z’ j + w’ k)
(Q,.) – The Quaternion Units
Q
1
-1
i
1
1
-1
i
-1
-1
1
-i
i -i j -j
i -i j -j
-i i -j j
-1 1 k -k
k -k
k -k
-k k
-j j
-i -i i 1 -1 -k k j -j
j j -j k -k -1 1 i -i
-j -j j -k k 1 -1 -i i
k k -k j
-k k -k j
-j -i
-j -i
i -1 1
i 1 -1
Conjugation
• Given a subgroup H of a group G, then for
any g 2 G define: H’ = g-1Hg := {g-1hg|h 2
H}.
• H’ is called a conjugate of H.
• H’ is a subgroup of G.
• Conjugation is an equivalence relation on
the set of subgroups of G.
• |H’| = |H|.
Normal Subgroup
• If H · G has no nontrivial conjugates, it is
called normal.
• For a normal group the quotient G/H forms
also a group.
• G/H = {Ha|a 2 G}
• |G/H| = [G:H].
Group Homomorphisms and
Isomorphisms
• f:G1 ! G2 is a group homomorphism if for
any g,h 2 G1 we have: f(g ² h) = f(g) ±
f(h), where (G1,²) and (G2,±) are groups.
• If, in addition, f is bijection, then it is called
an isomorphism.
Exercises 1-1
• N1: There is only one way to complete the definition of
multiplication of quaternions and respect
distributivity!
• N2: Represent quaternions by complex matrices (matrix
addition and matrix multiplication)! Hint: q = [a b; -b*
a*].
Exercises 1
• N3. What is the index [R\{0}:R+\{0}] for
the multiplicative group.
• N4. Show that the multiplicative group {1,1,i,-i} µ C is isomorphic to Z4.
• N5. Determine all subgroups of Zn.
2. Group Actions
Symmetric Group Sym(n)
• As we know a permutation p is a bijective
mapping of a set A onto itself: p: A  A.
Permutations may be multiplied and form
the symmetric group Sym(A) = Sym(n) = Sn
= SA, that has n! elements, where n = |A|.
Permutation Group
• Any subgroup G · Sym(A) is called a permutation
group. If we consider an abstract group G then we
say that G acts on A.
• In general the group action is defined as a triple
(G, A, f), where G is a group, A a set and
f:G ! Sym(A) a group homomorphism.
• In general we are only interested in faithful actions,
i.e. actions in which f is an isomorphism between G
and f(G).
Automorphisms of Simple
Graphs
• Let X be a simple graph. A permutation
h:V(X) ! V(X) is called an automorphism
of graph X if for any pair of vertices x,y 2
V(X) x~y if and only if h(x)~h(y). By Aut X
we denote the group of automorphisms of
X.
• Aut X is a permutation group, since it is a
subgroup of Sym(V(X)).
Orbits and Transitive Action
• Let G be a permutation group acting on A
and x 2 A. The set [x] := {g(x)|g 2 G} is
called the orbit of x. We may also write
G[x] = [x].
• G defines a partition of A into orbits: A =
[x1] t [x2] t ... t [xk].
• G acts transitively on A if it induces a
single orbit.
Example
• Aut G(6,2) induces
two orbits on the
vertex set.
• Aut G(6,2) also
induces an action on
the edge set. There we
get three orbits.
Orbits
• Let G act on space V. On V an equivalence
relation ¼ is introduced as follows:
• x ¼ y , 9 a 2 G 3: y = a(x).
• Equivalence, indeed:
» Reflexive
» Symmetric
» Transitive
• [x] ... Equivalence class to with x belongs is called
an orbit. (Also denoted by G[x].)
Example
a
1
2
c
d
b
e
3
4
• Graph G=(V,E) has
four automorphisms.
• V(G) ={1,2,3,4}
splits into two orbits
[1] = {1,4} and [2] =
{2,3}.
• E(G) = {a,b,c,d,e} also
splits into two orbits:
[a] = {a,b,e,d} and [c]
= {c}.
Stabilizers and Orbits
• Let G be a permutation group acting on A
and let x 2 A. By G(x) we denote the orbit
of x.
• G(x) = {y 2 A| 9 g 2 G 3: g(x) = y}
• Let Gx µ G be the set of group elements,
fixing x. Gx is called the stabilizer of x and
forms a subgroup of G.
Orbit-Stabilizer Theorem
• Theorem: |G(x)||Gx| = |G|.
• Corollary: If G acts transitively on A then
|A| is the index of any stabilizer Gx in G.
Burnside’s Lemma
• Let G be a group acting on A.
• For g 2 G let fix(g) denote the number of
fixed points of permutation g.
• Let N be the number of orbits of G on A.
• Then:
Regular Actions
• The transitive action of G on A is called
regular, if |G| = |A|, or equivalently, if each
stabilizier is trivial.
• An important and interesting question can
be asked for any transtive action of G on A.
• Does G have a subgroup H acting regularly
on A?
Semiregular Action
• Definition: Grup G acts on V
semiregulary,
• If there exists a 2 G 3: a = ( ...) ( ...) ...( ...)
composed of cycles of the same size r; |V| = r s.
• For each x 2 V we have: |[x]| = r.
Primitive Groups
• A transitive action of G on X is called
imprimitive, if X can be partitioned into k
(1 < k < |X|) sets:
X = X1 t X2 ,t ... t
Xk (called blocks of imprimitivity) and
each g 2 G induces a set-wise permutation
of the Xi’s.
• If a group is not imprimitive, it is called
primitive.
Example
• For a prism graph Pn, Aut Pn is imprimitive
if and only if n  4.
• There are n blocks of imprimitivity of size
2, each corresponding to two endpoints of a
side edge.
Permutation Matrices
• Each permutation p 2 Sym(n) gives rise to
a permutation matrix P(p) = [pij] with pij = 1
if j = p(i) and pij = 0 otherwise.
• Example: p1 = [2,3,4,5,1] and P(p1) is
shown below:
0
1 0 0 0
0
0 1 0 0
0
0 0 1 0
0
0 0 0 1
1
0 0 0 0
Matrix Representation
• A permutation group G can be represented
by permutation matrices. There is an
isomorphism p a P(p). And p s
corresponds to P(p)P(s). Since each
permutation matrix is orthogonal, we have
P(p-1) = Pt(p).
Alternating Group Alt(n)
• A transposition t is a permutation
interchanging a single pair of elements.
• Permutation p is even if it can be written as
a product of an even number of
transpositions (otherwise it is odd.)
• Even permutations from Sym(n) form the
alternating group Alt(n), a subgroup of
index 2.
Homewrok 2
• H1. Let X be any of the three graphs below.
• Determine the (abstract) group of automorphisms
Aut X.
• Action of Aut X on V(X).
• Action of Aut X on E(X).
X1
X2
X3
3. Symmetry Groups
Iso(M)
• Isometries of a metric space (M,d) onto
itslef form a group of isometries that we
denote by Iso(M).
Sim1(M)
• Similarities of type I of a metric space
(M,d) onto itslef form a group of
similarities that we denote by Sim1(M).
Sim2(M)
• Type II similarities of a metric space (M,d)
onto itslef form a group of similarities that
we denote by Sim2(M).
• In any metric space the groups are related:
• Iso(M) · Sim2(M) · Sim1(M).
Symmetry
• Let X µ M be a set in a metric space (M,d). An
isometry s 2 Iso(M) that fixes X set-wise: s(X) =
X, is called a (metric) symmetry of X.
• All symmetries of X form a group that we denote
by IsoM(X) or just I(X). It is called the symmetry
group of X.
• Note: this idea can be generalized to other groups
and to other structures!
Free Group F(S)
• Let S be a finite non-empty set. Form two copies
of it, call the first S+, and the second S-. Take all
words (S+ t S-)* over the alphabet S+ t S-.
Introduce an equivalence relation @ in such a way
that two words u @ v if and only if one can be
obtained from the other one by a finite series of
deletion or insertion of adjacent a+a- or a-a+.
• Let F(S) = (S+ t S-)* / @ . Then F(S) is a group,
called the free group generated by S.
• We also denote F(S) = <S | >.
Finitely Presented Groups
• Let S and <S | > be as before. Let R = {R1, R2, ...,
Rk} ½ (S+ t S-)* be a set of relators.
• The expression <S | R> is called a group
presentation. It defines a quotient group of <S | >.
• Two group elements from F(S) are equivalent if
one can be obtained from the other by successive
insertion or deletion of relators or their inverses.
• Since both sets S and R are finite, the group is
finitely presented.
Generators
• Let G be a group and X ½ G. Assume that
X = X-1 and 1  X. Then X is called the set
of generators. Let <X> denote the smallest
subgroup of G that contains X. We say that
X generates <X>.
Cayley’s Theorem
• Theorem. Every group G is isomorphic to
some permutation group.
• Proof. For g 2 G define its right action on
G by x a xg. The mapping from G to
Sym(G) defind by g a (x a xg) is an
isomorphism to its image.
Cyclic Group Cyc(n)
• Let G = <a| an>. Hence G =
{1,a,a2,..,an-1}. By Calyey’s Theorem
we may represent a as the cyclic
permutation (2,3,...,n,1) that generates
the group Cyc(n) · Sym(n).
• Note that Cyc(n) is isomorphic to
(Zn,+).
• Cyc(n) may also be considered as a
symmetry group of some polygons.
Cyc(8) is the symmetry group of the
polygon on the left.
Dihedral Group Dih(n)
t
2
1
3
6
4
5
s
• Dihedral group Dih(n) of
order 2n is isomoprihc to
the symmetry group of a
regular n-gon.
• For instance, for n=6 we
can generate it by two
permutations: s =
(2,3,4,5,6,1) and t =
(1,2)(3,6)(4,5). Dih(n) has
the following presentation:
• <s,t|sn=t2=stst=1>
Symmetry of Platnoic Solids
• There are five Platonic
solids: Tetrahedron T,
Octahedron O, Hexaedron
H, Dodecahedron D and
Icosahedron I.
Tetrahedron
•
•
•
•
•
A tetrahedron has
v = 4 vertices,
e = 6 edges and
f = 4 faces.
Determine its
symmetry group.
Octahedron
•
•
•
•
•
An octahedron has
v = 6 vertices,
e = 12 edges and
f = 8 faces.
Determine its
symmetry group.
Hexahedron
•
•
•
•
•
A cube has
v = 8 vertices
e = 12 edges and
f = 6 faces.
Determine its
symmetry group.
Dodecahedron
•
•
•
•
•
A dodecahedron has
v = 20 vertices,
e = 30 edges and
f = 12 faces.
Determine its
symmetry group.
Icosahedron
•
•
•
•
•
An icosahedron has
v = 12 vertices,
e = 30 edges and
f = 20 faces.
Determine its
symmetry group.
Skeleton of Tetrahedron – TS = K4
•
•
•
•
•
K4 has
v = 4 vertices,
e = 6 edges
f = 4 triangles.
Aut(K4) = S4.
Skeleton of Octahedron – OS =
K2,2,2
• OS has
• v = 6 vertices,
• e = 12 edges
Skeleton of Hexahedron
HS =K2 ¤ K2 ¤ K2
• HS ima
• v = 8 vertices
• e = 12 edges
Skeleton of Dodecahedron
DS = G(10,2)
• G(10,2) has
• v = 20 vertices,
• e = 30 edges
Skeleton of Icosahedron
S
I
• Is has
• v = 12 vertices,
• e = 30 edges
Platonic Solids and Symmetry
• We only considered the groups of direct
symmetries (orientation preserving
isometries).
• The full group of isometries coincides (in
this case) with the group of automorphisms
of the corresponding graphs.
• In general:
• Sym+(M) · Sym(M) · Aut(MS).
The Escher Problem
Frieze Groups
• Frieze = embroidery from Frieze, horizontal
ornamental band (architecture).
• We are interested in symmetry groups of
such bands. There are 7 frieze groups.
• We start with a rectangular stamp.
Transformations
•
•
•
•
Translation
Halfturn
Vertical Reflection
Glide-reflection
Seven Frieze Types
• Groups (notation):
•
•
•
•
•
11 (translations only)
12 (translations and halfturns)
m1 (translations and vertical reflections)
1g (translations and glidereflections)
mg (translations, halfturns, vertical reflections and
glide reflections)
• 1m (translations and horizontal reflections)
• mm (translations, halfturns,vertical reflections,
glide reflections, horizontal reflections)
11
12
m1
1g
mg
1m
mm
The Groups
• Group Elements
•
•
•
•
•
•
•
Identity I
Translation T
Halfturn R
Glidereflection G
Vertical mirror V
Horizontal mirror H.
Some relations: R2 = V2 = H2 = I, RV = VR = H,..
Subgroups
•
•
•
•
•
•
•
(1) T
(2) G
(3) T,R
(4) T,V
(5) T,H
(6) G,V
(7) all(T,G,R,V,H)
P
P,R(P)
B
A
S
A,R(A)
H
C1
C1
C1 £ D1
D1
D1
D1
D1 £ D1
Discrete Isometries
• Each metric space M determines the group of
distance preserving maps, isometries Iso(M).
• A subgroup of Iso(M) is discrete, if any isometry
in it either fixes an element of M or moves it far
enough.
• Discrete subgroups of I(R2) fall into three classes:
• 17 crystallographic groups
• 7 frieze groups
• finite groups (grups of rosettes).
Theorem of Leonardo da Vinci
• The only finite groups of isometries in the
plane are the groups of rosettes (cyclic
groups Cn and dihedral groups Dn).
The Escher problem
• There is a square stamp with asymmetric motif.
• By 90 degree rotations we obtain 4 different aspects.
• By combining 4 aspects in a square 2 x 2 block, a
translational unit is obtained that is used for plane
tiling. Such a tiling is called a pattern.
• Question: What is the number of different patterns ?
• Answer: 23.
Example
Recall Burnside’s Lemma.
• Let G be a group, acting on space S.
• For g 2 G let fix(g) denote the number of
points from S fixed by g.
• Let N denote the number of orbits of G on
S.
• Then:
Application
• Determine the group (G) and the sapce (S).
• Pattern can be translated and rotated.
• Basic observation:
• Instead of pattern consider the block (signature).
• Group operations:
• H – horizontal translation
• V – vertical translation
• R – 90 degrees rotation.
(Abstract) group G
hvr
hvr2
hvr3
h
hr
hr2
hr3
1
r
r2
r3
v
vr
vr2
vr3
hv
• h2 = v2 = r4 = 1.
• hv = vh
• hr = rv
Space S
• Space S consists of
4 4 4 4 = 256 signatures.
• Count fix(g) for g 2 G.
• For instance:
• fix(1) = 256.
• fix(r) = fix(r3) = 4.
• fix(h) = fix(v) = 16.
• By Burnside’s Lemma we
obtain N = 23.
1-dimensional Escher problem
• Rectangular asymmetric
motiff
• Only two aspects.
• 1 x n block (signature)
• Determine the number of
patterns:
• Two more variations:
• II two motiffs(mirror
images)
• III reflections are allowed.
Space S
• Space S consists of
2 2 .... 2 = 2n signatures.
• Count fix(g) for g 2 G.
• For instance:
• fix(1) = 256.
• fix(r) = fix(r3) = 4.
• fix(h) = fix(v) = 16.
• By Burnside’s Lemma we
obtain N = 23.
Solution for the basic case
1
n/k
f (n) 
(  ( k )2  g ( n ))
2n k |n
• where g(n) = 0 for odd n and for even n:
g ( n )  ( n / 2) 2
n/2
Program in Mathematica
• f[n_] := (Apply[Plus,Map[EulerPhi[#]
2^(n/#)&,Divisors[n]]] +
If[OddQ[n],0,(n/2) 2^(n/2)])/(2n)
• f[n_,m_] := (Apply[Plus,Map[EulerPhi[#]
(2 m)^(n/#)&,Divisors[n]]] +
If[OddQ[n],0,(n/2) (2 m)^(n/2)])/(2n)
• g[n_] :=
(Apply[Plus,Map[If[OddQ[#],1,2]
EulerPhi[#] 4^(n/#)&,Divisors[n]]] +
If[OddQ[n],0,(n) 4^(n/2)])/(4n)
Results for a tape
•
n
•
•
•
•
•
•
•
•
•
•
1
2
3
4
5
6
7
8
9
10
I
II
1
2
2
6
2
12
4
39
4
104
9
366
10 1172
22 4179
30 14572
62 52740
III
1
4
6
23
52
194
586
2131
7286
26524
17 CRYSTALLOGRAPHIC GROUPS
6-števna os?
4-števna os?
zrcaljenje?
3-števna os?
p6mm
p6
zrcaljenje?
p4mm
zrcaljenje?
p4
Zrcala v 4 smereh?
3-osi na zrcalih?
p4gm
p3
2-števna os?
zrcaljenje?
p31m
p3m1
zrcaljenje?
Rombska mreža?
glide?
glide?
Drugo zrcalo?
Rombska mreža?
c2mm
p2mm
p2mg
c2gg
p2
cm
pm
pg
p1
p1
• p1 = <a,b|ab=ba>
p2
• p2 = <a,b,c| b2=c2=(ab)2=(ac)2=1>
pm
• pm = <a,b,c| b2=c2=1, ab=ba, ac=ca>
pg
• pg = <a,b|ab=ba-1>
cm
• cm = <a,b,c| b2=c2=1, ab=ca>
p2mm
• p2mm = <a,b,c,d| a2=b2= c2=d2= 1,
(ab)2=(ad)2=(bc)2=(cd)2=1>
p2mg
• p2mg = <a,b,c| b2 = c2 = 1, (ab)2=(ac)2=1>
p2gg
• p2gg = <a,b| (ab)2=1>
Exercises 3-1
b
d
q
• N1. Explain how the
Frieze groups can be
described by the four
letters (aspects of the
pattern):
p
• b, p, q, d.
Exercises 3-2
• N2. Determine the Cayley graph of each of
the Frieze groups.
• N3. Determine the crystallographic groups
that may arise from the classical Escher
problem.
Homework 3-1
• H1. Determine the group of
symmetries of the prism P6.
• H2. Determine the group of
symmteries of the antiprism A6.
• H3. Determine the group of
symmetries for the pyramid P6.
• H4. Determine the group of
symmetries of the double
pyramid B6.
• H5. Generalize to other values
of n.
• H6. Repeat the problems for the
corresponding skeleta.
Homework 3-2
• Consider the Escher problem
with the motiff on the left.
• H7. Determine the abstract
group and its Cayley graph.
• H8. What is the number of
different patterns?
• H9. What is the number of
different patterns if reflections
are allowed?
• H10. What is the number of
different patterns in the original
Escher problem if reflections
are allowed?
4. Covering Graphs
Covering Graphs
• Motivation:
• Suppose you are taken to two different
labyrinths. Is it possible to tell they are
distinct just by walking around?
• Let us call the first graph maze X, and the
second one Y.
Question
• Is it possible to
distinguish between
the two mazes?
• Answer: Yes, we can.
In the upper maze
there are two
adjacent trivalent
vertices. This is not
the case in the lower
maze.
Local Isomorphism
• On the other hand we
cannot distinguish
(locally) between the
upper and lower
graph.
• To each walk upstairs
we can associate a
walk downstairs.
One More Example
• C4 over C3 is no good
(Why not?). However,
C6 over C3 is Ok.
Fibers and Sheets.
• We say that C6 is a two-sheeted
cover over C3. Red vertices are
in the same fiber. Similarly, the
dotted lines belong to the same
fiber.
• The graph mapping f: C6  C3
is called covering projection.
• The pre-image of a vertex f-1(v)
(or an edge f-1(e)) is called a
fiber.
• The cardinality of a fiber is
constant. k =|f-1(v)| is called the
number of sheets.
One More Example
• The cube graph Q3 is a
two fold cover over
the complete graph K4.
• The vertex fibers are
composed of pairs of
antipodal vertices.
Covers over Pregraphs
• K4 can be understood
as a four-fold cover
over a pregraph on one
vertex (one loop and
one half-edge).
Voltage Graphs
• X = (V,S,i,r) – connected (pre)graph.
• (G,A) – permutation group G acting on
space A.
• g:S  G – voltage assignment.
• Condition: for each s 2 S we have
g[s]g[r(s)] = id.
Voltage Graph Determines a
Covering Graph
• Each voltage graph (X,G,A,g) determines a covering graph
Y with covering projection
f: Y  X as follows:
• Covering graph Y = (V(Y),S(Y),i,r)
•
•
•
•
V(Y) := V(X) x A
S(Y) := S(X) x A
i: S(Y)  V(Y): i(s,a) := (i(s),a).
r: S(Y)  S(Y): r(s,a) := (r(s), g[s](a)).
• Covering projection f
• f: V(Y)  V(X): f(x,a) := x.
• f: S(Y)  S(X): f(s,a) := s.
• Sometimes we denote the covering graph Y by Cov(X;g).
(Rhetorical) Questions
• “Different” voltage graphs may give rise to the
“same” cover. What does the “same” mean in this
context and how do we obtain all “different”
voltage graphs?
• A voltage graph is determined in essence by the
abstract group. What is the role of the permutation
group?
• How do we ensure that if X is connected that Y is
connected, too?
Kronecker Cover
• Let X be a graph. The
canonical double cover
or Kronecker cover,
KC(X), is a twofold cover
that is defined by a
voltage graph that has
nontrivial voltage from Z2
on each of its edges. It can
also be described as the
tensor product
KC(X) = X £ K2.
Regular Covers
• Let Y be a cover over X. We are interested in
fiber preserving elements of Aut Y (covering
transformations).
• Let Aut(Y,X) · Aut Y be the group of covering
transformations.
• The cover Y is regular, if Aut(Y,X) acts
transitively on each fiber.
• Regular covers are denoted by voltage graphs,
where the permutation group (G, A) acts regularly
on itself by left or right translations: (G, G).
Dipole qn
• A dipole qn has two
vertices joined by n
parallel edges. We may
call one vertex black, the
other white. On the left
we see q5.
• Each dipole is bipartite,
that is why each cover
over qn is bipartite, too. q3
is cubic and sometimes
called the theta graph q.
Cyclic cover over a dipole – Haar
graph H(n).
0 3 5
Z6
• H(37) is determined by the
number 37, actually by its binary
representation (1 0 0 1 0 1).
• k = 6 is the length of the
sequence, hence we choose the
group Z6.
• (0 1 2 3 4 5) – positions of “1”.
• Positions of “1”s: 0, 3 and 5.
{0,3,5} are the voltages on q. The
corresponding covering graph is
H(37).
Cages as Covering Graphs
• A g-cage is a cubic graph of girth g that has
the least number of vertices.
• Small cages can be readily described as
covering graphs.
1-Cage
• Usually we consider only
simple graphs as cages.
For our purposes it makes
sense to define also a 1cage. We define it to be
the pregraph on the left.
• A 1-cage is the unique
smallest cubic pregraph.
2-Cage
• The only 2-cage is the
q graph.
• We may view the 2cage as the Kronecker
cover over 1-cage.
1
1
Z2
3
K4, the 3-cage
2
1
0
2
1
Z4
• K4 is a Z4 covering over
the 1-cage.
• In general, we obtain a Z2n
covering over the 1-cage
by assigning voltage 1 to
the loop and voltage n to
the half-edge.
• Exercise: What is the
covering graph for the
cover described above?
K3,3, the 4-cage
5
4
3
2
1
0
3
1
Z6
• K3,3 is a Z6 covering over
the 1-cage.
• It can also be seen as a Z3
covering over the 2-cage
q.
• Exercise: Express K3,3 as a
covering graph over q.
Dtermine a natural
number n, such that K3,3 is
a Haar graph H(n).
The Handcuff Graph G(1,1)
• By changing the
voltage on the loop of
the 1-cage we obtain a
double cover G(1,1),
the smallest
generalized Petersen
graph, known as the
Handcuff graph.
1
0
Z2
I graphs I(n,i,j) and Generalized
Petersen graphs G(n,k)
0
i
Zn
j
• Cyclic covers over the
handcuff graph are called
I-graphs. Each I-graph can
be described by three
parameters n,i, and j, with
i · j. In case i = 1 we call
I(n,i,k) = G(n,k), the
generalized Petersen
graph.
• In particular, I(5,1,2) is the
5-cage.
The 6-cage
b
a
D7
• The 6-cage is the
Heawood graph on 14
vertices. It is a 7-fold
cyclic cover over the q
graph. But it is also a
dihedral cover over the 1cage.
• Let the presentaion of Dn
be given as follows:
Dn = <a,b|an,b2, ab=ba-1>
• Then the Heawood graph
is the covering described
on the left.
(3,1)-trees
• A (3,1)-tree is a tree
whose vertices have
valence 3 or 1 only.
• On the left we see the
smallest (3,1)-trees I,Y
and H.
(3,1)-cubic graphs
• A (3,1)-cubic graph is
obtained from a (3,1)tree by adding a loop
at each vertex of
valence 1.
• On the left we see the
smallest (3,1)-cubic
graphs
I(1,1,1),Y(1,1,1,1) and
H(1,1,1,1,1).
Coverings over (3,1)-cubic graphs
j
i
Zn
j
i
k
i
j
k
l
• By putting 0 on the tree
edges and appropriate
voltages on the loops of
(3,1)-cubic graphs we
obtain their Zn coverings.
• For the graphs on the left
we obtain the I-graphs, Ygraphs and H-graphs:
I(n,i,j),Y(n,i,j,k) and
H(n,i,j,k,l).
Covers Determined by Graphs
• We know already that there exists a cover,
namely the Kronecker cover, that depends
only on X itself and the voltage assignment
plays a minor role.
• Now we will present some covers
possessing a similar property.
Coverings and Trees
• Let X be a connected graph and let Cov(X) denote
all connected covers over X:
• Cov(X) = {(Y,f)| Y connected and f: Y ! X,
covering projection}. For each connected X we
have (X,id) 2 Cov(X).
• Proposition: For a connected X we have Cov(X)
= {(X,id)} if and only if X is a tree.
• This fact holds both for finite and locally finite
trees.
Universal cover
•
•
•
•
•
•
•
Let X, Y and Z be connected graphs and let f: Y ! X and :Z ! Y be covering
projections.
Let us consider the class Cov(X) of all coverings over X. We may introduce a
partial order in Cov(X). (Y,f) < (Z,) if there exists a covering projection
(Z,) 2 Cov(Y) so that  =  f.
Proposition: Any connected finite or locally finite graph X can be covered by
some tree T; f: T ! X.
Proposition: Any connected finite or locally finite graph X can be covered by
at most one tree T.
Proposition: Let f: T ! X be a covering projection from a tree to a connected
graph X. Then for each covering : Y ! X there exists a covering q: T ! Y such
that f = q .
Corollary: For each connected X the poset Cov(X) has a maximal element
(T,f) where T is a tree.
The maximal element (T,f) 2 Cov(X) is called the universal covering of X.
Construction of Universal Cover
• There is a simple construction of the universal covering projection.
• Let X be a connected graph and let T µ X be a spanning tree.
Furthermore, let S = E(X) \ E(T) be the set of edges not in tree T.
• Consider S to be the set of generators for a free group F(S) and let
F(S) be the voltage group.
• Let us assing voltages on E(X) as follows:
• If e 2 E(T) the voltage on e is the identity.
• If e 2 S the voltage is the corresponding generator (or its inverse)
• Note: The construction does not depend on the choice of edge
directions.
• Proposition: The described construction yields the universal cover.
Examples
• Example: The
universal cover over
any regular k-valent
graph is a regular
infinte tree T(1,k).
Valence Partition and Valence
Refinement
• Let G be a graph and let B = {B1, ..., Bk} be a partition of
its vertex set V(G) for which there are constants rij, 1 · i,j ·
k such that for each v 2 Bi there are rij edges linking v to
the vertices in Bj. Let R = [rij] be the corresponding k £ k
matrix. B is called valence partition and R is called
valence refinement. If k is minimal, then B is called
minimal valence partition and R is called minimal
valence refinement.
• Two refinements R and R’ are considered the same if one
can be transformed into the other by simultaneous
permutation of rows and columns.
• A refinement is uniform, if each row is constant.
Construction
•
•
•
•
•
•
•
•
•
•
•
•
Given graphs G and G’ with a common refinement.
Let mij denote the number of arcs in G of type i ! j.
Let ni denote the number of vertices in G of type i.
Let bij = lcm(mij)/mij. (If mij = 0 , let bij undefined).
Let ai = lcm(mij)/ni.
Note that bij and ai depend only on the common matrix R and are the same for
both graphs G and G’.
Let l(e) or l(e’) be a linear order given to all type i ! j arcs with a common
initial vertex i(e) (or i(e’)).
Let V(H) = {(i,v,v’,p)|v and v’ of type i, p 2 Zai}
Let S(H) = {(i,j,e,e’,q)|e and e’ of type i ! j, q 2 Zbij}
r(i,j,e,e’,q) := (j,i,r(e),r(e’),q)
i(i,j,e,e’,q) := (i,i(e),i(e’),q rij + l(e)-l(e’)}
H is a common cover of G and G’.
Computing Minimal Valence
Refinement
• Let r[u,B] denote the number of edges linking u to the vertices in B.
• Algorithm [F.T.Leighton, Finite Common Coverings of Graphs,
JCT(B) 33 1982, 231-238.]
• Step 1. Place two vertices in the same block if and only if they have
the same valence.
• Step 2. While there exist two blocks B and B’ and two distinct vertices
u,v in B with r[u,B’]  r[v,B’] repeat the following:
• Partition the block B into subblocks in such a way that two vertices u,b of B
remain in the same block if and only if r[u,B’] = r[v,B’] for each B’ of the
previous partition.
• Step 3. From the minimal valence partition B compute the minimal
vertex refinement R.
• Note: We may maintain R during the run of the algorithm as a matrix
whose elements are sets of numbers.
Comon Cover
•
1.
2.
3.
4.
5.
•
Theorem. Given any two finite graphs G and H, the
following statements are equivalent:
G and H have the same universal cover,
G and H have a common finite cover,
G and H have a common cover,
G and H have the same minimal valence refinement.
G and H have the same some valence refinement.
Homework. Find the result in the literature and
construct a finite common cover of G(5,2) and G(6,2).
Petersen graph
• An unusual drawing of
the Petersen graph.
Petersen graph G(5,2) and graph X.
Kronecker Cover - Revisited
• The Kronecker cover KC(G) is an example
of a cover determined by the graph itself.
• Exercise. Show that G(5,2) and X have the
same Kronecker cover.
THE covering graph
• Let G be a graph with the vertex set V. By
THE(G) we denote the following covering graph.
• To each edge e = uv we assign the transposition te
= (u,v) 2 Sym(V). The resulting covering graph
has two components, one being isomorphic to G.
The other component is called THE covering
graph.
Examples
• On the left we see
THE(K2,2,2).
• The construction
resembles truncation.
• Each vertex is truncated
and an inverse figure is
placed in the space
provided for it.
• Theorem: If G is planar,
then THE(G) is planar.
The fundamental group of a graph.
• Let G be a connected graph rooted at r 2 V(G) and let 
denote the collection of closed walks rooted at r.
• Let a and b be two closed walks rooted at r. The
composition a b is also a closed walk rooted at r.
• We may also define a-1 as the inverse walk.
• Finally, we need equality (equivalence).
• a1 a2 ~ a1 e e-1 a2.
p(G,r) := /~ is a group, called the fundamental group
of G (first homotopy group).
• Fact: p(G,r) is a free group on m-n+1 generators.
•
The first Homology group of a graph
• Let G be a connected graph and T one of its spanning trees.
Each edge h 2 G\T not in T defines a unique cycle C(h) µ
E(T) [ h.
• The charactersitic vector h 2 {0,1}m, h(e) = 1, if e 2 C(h)
and h(e) = 0, represents C(h). The set of all charactersitic
vectors spans a m-n+1 dimensional Z-module in Zm. This
can be also viewed as a free abelian group isomorphic to
Zm-n+1.
• This group is called the first homology group H1(G,Z).
We may replace Z by Zk and obtain the first Zk homology
group Zkm-n+1.
Pseudohomological Covers
• Idea: Let G be a graph, T one of its spanning trees
and H = {h1,h2,...,hm-n+1} = E(G)\E(T). Let G(H)
be a group with m-n+1 interchangeable generators
H. The pseudohomological G-cover HOM(G,G,T)
is determined by a voltage graph with g(e) = id, for
e 2 E(T) and g(h) = h, for h 2 E(G)\E(T).
• Main Question. Is HOM(G,G,T) independent of
the choice of T and the selection of the generators
or their inverses? If the answer is yes, the covering
is called homological cover.
Pseudohomological 2-cover
• Let G be a graph and T a spanning tree of
G. The pseudohomological 2-cover
HOM(G,Z2,T) is determined by a voltage
graph with g(e) = 0, for e 2 E(T) and g(e) =
1, for e  E(T).
• Theorem. If G is connected then
HOM(G,Z2,T) is connected if and only if G
is not a tree.
Example
0
1
0
0
g1
1
Z2
0
1
0
1
0
g2
• The two voltage
graphs on the left
determine different
pseudohomological Z2
covers.
• Cov(G,g2) is bipartite
and Cov(G,g1) is not.
Switching
• Let (G,g) be a voltage graph. Let : V(G) ! G be an arbitrary mapping,
called switching, that assigns voltages to vertices. Define a new
voltage assignment  as follows:
• (s) := (i(s)) g (s) (i(r(s))-1.
•  is well-defined.
• Namely (r(s)) = (i(r(s))) g(r(s)) (i(s))-1.
• Hence (r(s))-1 = (i(s)) g(r(s))-1 (i(r(s)))-1 = (i(s)) g(s) (i(r(s)))-1 =
(s).
• Clearly for any switching  the graphs Cov(G,g) and Cov(G,)
coincide.
• Given (G,g) and any spanning tree T. There exists a switching  such
that the resulting  is the identity on T.
• If, in addition, T is rooted at v, we may select (v) = id (or arbitrarily)
and this determines switching completely.
Homological Elementary Abelian
Covers
• Let G be a graph with a spanning tree T.
Let k = m-n+1 be the number of edges in
G\T. Define the voltage assignment g such
that each non-tree edge gets the voltage ei =
(0,0,..,0,1,0,...,0) 2 Zpk.
• Claim: If p is prime, then Cov(G,g) is
independent of T.
• Question: What happens in the case p is not
prime?
Tree-To-Tree Switch
• Let T and T’ be two spanning trees of G. Let H = {h1, h2, ..., hk} be the
co-tree edges of T. Let r be the root of G. For each vertex w 2 V(G)
there is a unique path P(T’,w,r) on the three T’ from w to v. Let S(w) µ
H be the collection of co-tree edges on this path. Let S(w) be the label
given to w. Hence (w) = S{ hi| hi 2 S(w)}.
• Claim: Starting with a homological voltage assignment relative to T
and applying the tree-to-tree switch , the voltages are given as
follows:
• The edges on T’ get voltage 0.
• An edge e = uv on a co-tree T’ get the voltage:
•
•
k(e) = S(u) + S(v) if e 2 T.
k(e) = S(u) + S(v) + h(e) if e  T.
• Each co-tree edge e defines a cycle C(e). The net voltage on C(e) is
equal to k(e).
• The voltages k(e), for e  T’ span the whole Z2k.
Real Homological Cover
(0,1)
(0,1) (1,1)
(1,0)
(1,0)
Z2
2
• Let G be a graph with a
given cycle basis C1, C2,
..., Ck. Direct each cycle
and assign to each edge of
Ci the voltage ei 2 Znk. The
final voltage assignmnet is
given by adding the partial
voltages.
• An example is given on
the left. The cycle basis is
determined by a spanning
tree.
Least Common Cover
• Theorem: There exist finite connected
graphs H1, H2, G1, G2 such that G1 and G2
are both double covers of H1 and H2.
• Proof. We start with graphs G = G(5,2) and
X that we know from earlier.
G+X and G + G
• Given two graphs G
and H we form G+H
by adding an edge
between them.
• On the left we see G +
X and G + G.
• The resulting graph
depends on the choice
of endpoints of the
added edge.
H1 and H2
• Define H1 and H2 as
follows:
• H1 = G + X + X and
H2 = G + G + X.
Covers of G+H.
• A double cover of G+H
can be split into two
double covers G* and H*
which are then joint by a
pair of edges. We denote
the resulting graph by G*
++ H*.
• For instance KC(G + X) =
KC(G) ++ KC(X) =
G(10,3) ++ G(10,3).
End of Proof
• Let G1 = G(10,3) ++ G(10,3) ++ G(10,3)
and G2 = G(10,3) ++ G(10,3) ++ 2X.
• G1 and G2 are distinct. They are both covers
of H1 and H2.
Exercises 4-1
• N1: Prove that each
double sheeted cover is
regular.
• N2: Find an example of a
three sheeted cover that is
not regular.
• N3: Express the graph on
the left as a 6-fold cover
over a pregraph on a
single vertex.
Exercises 4-2
• Let Znk be an elementary abelian group. Let
S be a set of generators with the following
property. Each element is a 0-1 vector. They
generate the whole group.
• N4. Show that |S| = k.
• N5. Show that there is an automorphism of the
group mapping S to the standard generating set.
Exercises 4-3
• The graph on the left is
called the Heawood graph
H. Prove:
– N6. H is bipartite.
– N7. H is a Haar graph
(Determine n, such that H =
H(n))
– N8. Express H as a cyclic
cover over q.
– N9. Show that there are no
cycles of lenght < 6 in H.
– N10. Show that H is the
smallest cubic graph with
no cycles of length < 6.
Exercises 4-4
• N11. Express the 7-cage as a covering
graph.
• N12. Express the 8-cage as a covering
graph.
Homework 4-1
• H1: Prove that the Kronecker cover is bipartite.
• H2: Prove that generalized Petersen graph G(10,2)
is a twofold cover over the Petersen graph G(5,2).
• H3: Determine the Kronecker cover over G(5,2).
• H4: Determine a Zn covering over the handcuff
graph G(1,1), that is not a generalized Petersen
graph G(n,r).
Homework 4-2
• H5. Given a connected graph G with n
vertices and e edges and with valence
sequence (d1, d2, ..., dn). Determine the
parameters for THE(G).
• H6. Determine all connected graphs G for
which girth(G)  girth(THE(G)).
5. Symmetry in Graphs
The symmetries of a graph are the elements of its automorphism
group. In particular, the symmetry of a graph does not depend on its
drawing. Sometimes it is possible to draw a graph to show all its
graph symmetries, and sometimes not.
For example, the most symmetric drawing of K_5 has only 10
symmetries, however the graph K_5 has 5! = 120 automorphisms.
Aut G revisited.
• Recall that the automorphism group Aut G
for a simple graph G can be viewed as a
subgroup of Sym(V(G)) or a subgroup of
Sym(E(G)).
Example for Aut G acting on
V(G).
a
1
2
c
d
b
e
3
4
•
•
•
•
•
•
|Aut G| = 4 .
V(G) ={1,2,3,4}
Id = (1)(2)(3)(4)
a = (1)(3)(2 4)
b = (1 3)(2)(4)
g = a b = (1 3)(2 4)
Example for Aut G acting on
E(G).
a
1
2
c
d
b
e
3
4
•
•
•
•
•
•
|Aut G| = 4.
EG ={a,b,c,d,e}
Id = (a)(b)(c)(d)(e)
a = (a d)(b c)(e)
b = (a b)(c d)(e)
g = a b = (a c)(b d)(e)
Induced Action on E(G)
• For a simple graph G the action of Aut G on
V(G) induces an action of Aut G on E(G).
• For example: since a = 1 ~ 2 and a(1) = 1,
a(2) = 4, we have a(a) = 1 ~ 4 = d.
Example for Orbits
a
1
2
c
d
b
e
3
4
• |Aut G| = 4
• V(G) ={1,2,3,4} is
partitioned into two
orbits R = {1,4} and
S={2,3}.
• E(G) = {a,b,c,d,e} has
two orbits: Z =
{a,b,e,d} and M={c}.
Cayley Table for the dihedral
group Dih(3) = D3.
1
X
X2
Y
XY X2Y
1
1
X
X2
Y
XY X2Y
X
X
X2
1
XY X2Y
X2
X2
1
X
X2Y
Y
XY
Y
Y
X2Y XY
1
X2
X
XY
XY
X2Y
X
1
X2
Y
X2
X
1
Y
X2Y X2Y XY
Y
Cayley Color Digraph
a(v)
a
b
v
LEFT
a(v)
a
• Information in Cayley
table is redundant!
ba(v) • Two possibilities:
– Left Cayley graph (will
not be used )
– Right Cayley graph.
b
v
RIGHT
ab(v)
Cayley Color Digraph for D3.
X
• Right Cayley Color
Digraph
• Convention: Since 1 =
Y2 we may use the
undirected version of
the edge..
XY
Y
X2Y
1
X2
X
Y
Cayley Graph (Right)
• Let G be a group and  ½ G a set of
generators, such that:
• Symmetric:  = -1
• Does not contain identity: 1  .
• To a pair (G,) we can associate a Cayley
graph X = Cay(G,) as follows:
• V(X) = G
• g ~ h , g-1h 2 .
Basic Theorem about Cayley
graphs
• Graph X is a Cayley graph, if and only if
there exists a subgroup G · Aut X, acting
regularly on V(X)!
• Exercise: Prove that the Petersen graph is
not a Cayley graph.
Direct Product
• The Cayley graph of a direct product
corresponds to the Cartesian product of
Cayley graphs.
• Problem: Define the free product of groups
and explore the corresponding product
construction of rooted Cayley graphs.
Frucht’s Theorem
• Theorem: For each finite group G there
exists a graph X, such that G isomorphic to
Aut X.
Vertex-Transitive Graphs
• If group G acts on a space V with a single
orbit ([x] = V), we say that the action is
transitive.
• Let (G,V) denote a permutation group acting
on V. Let [x] be any of its orbits. The
restriction (G,[x]) is transitive.
Vertex Transitvity
• Graph X is vertex
transitive, if Aut X acts
transitively on V(X).
• Example: Three out of the
four graphs on the left are
vertex transitive.
• Question: Which
Generalized Petersen
graphs G(n,r) are vertex
transitive?
Vertex Transitvity and Regularity
• Proposition: Each vertex transitive graph is
regular.
• Proof: If an automorphism maps vertex u to
vertex v, then deg(u) = deg(v). Hence all
vertices of an orbit have the same valence.
A vertex transtive graph has a single vertex
orbit, therefore deg(v) is constant and the
graph is regular.
Vertex-Transitive Subgraphs
• Let G be a graph and [x] ½ V(G) an orbit of Aut
G. The induced subgraph <[x]> is vertex
transitive.
• Let H ½ G be an induced subgraph of G. Let G <
Aut H be the group of those automorphisms that
can be extended to the group of automorphisms of
G.
• Given H and given G < Aut H. Find a graph G,
such that H is induced (isometric, convex) in G.
Edge Transitive Graphs
• Graph X is edge
transitive, if Aut X
acts transitively on
E(X).
• On the left we see
antiprisms A7, A3, the
Möbius ladder M4 and
the prism P6. Which
of these graphs are
edge transitive?
Edge Transitive Graphs that are
not Vertex Transitive
• Theorem: An graph X which is edge transitive but
not vertex transitive must be bipartite.
• Lemma: If both endvertices of an edge transitive
graph belong to the same orbit, the graph is vertex
transitive.
• Lemma: An edge transitive graph has at most two
vertex orbits.
• Lemma: If an edge transitive graph has two
vertex orbits, each of them is an independent set.
Arc Transitive Graphs
• Graph X is arc
transitive, if Aut X
acts transitively on the
set of arcs S(X).
• Example: G(5,2) is arc
transitive, P3 is not.
Arc and Edge Transitivity
• Proposition: Any arc transitive graph X is
edge transitive.
• Proof: Take any edges e and f. Each of them
has two arcs e+ , e- and f+ , f-. Since X is arc
transitive, there exists an automorphism a 2
Aut X, mapping e+ to f+: a(e+ ) = f+.
Therefore it maps e- to f-: a(e- ) = f-. We
conclude that a(e) = f.
Arc and Vertex Transitivity
• Theorem: An arc transitive graph X
without isolated vertices is vertex transitive.
• Proof. Take any vertices u and v. Since they
are not isolated there are arcs e and f such
that i(e) = u and i(f) = v. Since X is arc
transitive there exists an automorphism a 2
Aut X, mapping e to f. By definition it maps
u to v.
Arc Transitive I-graphs
• The only arc transitive
I-graphs are the seven
generalized Petersen
graphs: G(4,1), G(5,2),
G(8,3), G(10,2),
G(10,3), G(12,5),
G(24,5).
Arc-transitive Y graphs
• Horton and Bouwer
showed in 1991 that
the only arc-transitive
Y graphs are
Y(7,1,2,4),
Y(14,1,3,5) (girth 8),
Y(28,1,3,9) (girth 8)
and Y(56,1,9,25)
(girth 12).
Vertex and Edge Transitivity.
• Proposition: There
exists a graph X, that
is vertex transitive, but
not edge transitive.
• Proposition: There
exists a graph X, that
is edge transitive, but
not vertex transitive.
Arc-transitive H graphs
• There are only two
arc-transitive H
graphs: H(17,1,2,4,8)
and H(34,1,9,13,15)
(girth 12).
Arc-transitive (3,1)-cubic graphs
• There is a complete characterization of arctransitive connected (3,1)-cubic graphs.
• 7 – I-graphs
• 4 – Y-graphs
• 2 – H-graphs
• Exercise: Prove that if the connectivity
condition is dropped the number of arctransitive graphs is infinite.
s-Arc-Transitive Graphs
• An s-arc in a graph X is a sequence (a0,a1,
..., as) of vertices of X such that aiai+1 is an
edge in E(X) and ai-1  ai+1.
• A graph X is s-arc-transitive if its
automorphism group acts transitively on the
set of its s-arcs and does not act transitively
on the set of its (s+1)-arcs.
1/2-Arc-Transitive Graphs
• A vertex-transitive graph X that is edgetransitive but not arc –transitive is called ½arc-transitive graph.
Vertex, Edge and Not-Arc
Transitvity
• Theorem: There exist
vertex- and edgetransitive graphs that
are not arc-transitive.
• The Holt graph on the
left is the smallest
such example. It has
27 vertices and is 4valent.
Holt graph - Revisited
• 4-valent Holt graph H
is a Z9-covering over
the graph on the left.
-4
-1
+4 +1
+2
-2
Z9
Half Arc Transitive Graph
• There are several families of ½-arc-transitive graphs (many
discovered by mathematicians in Slovenia).
• Theorem: Each ½-arc-transitive graph is 2k-regular for some
natural number k.
• Proof: Half arc transitive action on X means an action on S(X) with two
equaly sized orbits. For each s 2 S(X) the orbits [s] and [r(s)] are different.
No edge may be mapped to itself by an automorphism without fixing both of
its endvertices. This implies that giving direction to one edge implies
directions in every other edge. Aut X acts transitively on such directed edges.
• If we have at any vertex v the inequality indeg(v) > outdeg(v), the same
inequality would hold at every vertex. This contradicts the well-known fact:
• S indeg(x) = S outdeg(x).
LCF Notation for Cubic Graphs
• A cubic graph X on 2n
vertices, with a given
Hamilton cycle, can be
easily encoded by
successive lengths of the
cords along the Hamilton
cycle.
• Example: For the graph
on the left:
• LCF[3,4,2,3,4,2] =
LCF[3,-2,2,-3,-2,2]
LCF – Example
• Let us introduce simple
notation (by example):
• (a,b,c)2 = (a,b,c,a,b,c)
• (a,b)-2 = (a,b,-b,-a)2
• Example: LFC[(3,-3)4] =
LCF[(3)-4] = Q3.
Heawood Graph - LCF
• LCF[(5)-7] denotes the
Heawood graph.
Edge Orbits of a VertexTransitive Graph.
• Theorem: In a vertex-transitive graph X of
valence d the number of edge orbits · d.
• Proof: Let i(e) = v, hence the arc e has endpoint v.
Each vertex u has at least one arc f, with i(f) = u
and vertex-transitivity implies [f] = [e]. Around
vertex v there are at most d edge orbits passing by
automorphism from vertex to vertex. This way we
exhaust all edges and therefore their orbits.
Regular action of Aut X.
• Definition: A vertex-transitive graph X,
such that |Aut X| = |V(X)| is called a
graphical regular representation (GRR)
of its automorphism group G = Aut X.
• Remark: If Aut X acts transitively on V(X),
it does not mean that there exists a subgroup
G · Aut X, acting on V(X) regularly.
0-Symmetric Graphs
• Definition: A vertex-transitive cubic graph
X with three edge orbits is 0-symmetric.
• Theorem: The class of cubic graphs, that
are GRR coincides with the class of 0symmetric graphs.
• Proof: Use Lemma on orbits and stabilizers
and two other lemmas.
Two Lemmas
• Let X be a graph and G a group of automorphisms.
The stabilizer Gx of vertex x acts on the set of
neighbors of x: X(x).
• Lemma: In a vertex transitive graph the number w
of edge orbits equals the number of orbits of the
action of Gx on X(x).
• Lemma: The only permutation group acting
faithfully and fixing all elements of a space is
trivial.
Examples
• Each 0-symmetric
graph is a Haar graph.
• The smallest example
is H(9;S) = H(28 + 27 +
25), where S = {0, 1,
3}.
• LCF[{5,-5}9].
The Mark Watkins Graph
• The smallest 0-symmetric Haar graph H(n;{0,a,b}) with
the property that gcd(a,n) > 1, gcd(b,n) > 1,gcd(b-a,n) > 1,
and gcd(a,b) = 1, has parameters n = 30, a = 2, b = 5. It is
called the Mark Watkins graph.
Semi-Symmetric Graphs.
• Definition: A regular
graph X which is edge
transitive, but not
vertex transitive, is
called semisymmetric.
• On the left we see one
of them, the 4-valent
Folkman graph.
Direct Product of Groups - Revisited.
• A £ B – direct product of groups A and B,
defined on the cartesian product of the
elements. Group operation by components.
• Example. Z3 £ Z3 has 9 elements:
(0,2) + (1,2) = (1,1).
• Finite abelian groups $ (finite) direct
products of (finite) cyclic groups.
6. Symmetry in Metric Spaces
•
•
•
•
•
Let (M,d) be a metric space.
Iso(M) is the group of isometries.
Sim1(M) is the group of similarities of type 1.
Sim2(M) is the group of similarities of type 2.
Let B(a,r) = {x 2 M|d(a,x) · r} be the ball centered at a
with radius r.
• Let S(a,r) = {x 2 M|s(a,x) = r} the sphere centered at a
with radius r.
Isotropic Metric Spaces
• A metric space (M,d) is said to be isotropic
at point x 2 M, if all spheres S(x,r) centered
at x are homogeneous. M is said to be
isotropic, if it is isotropic at each of its
points.
Homogeneous Metric Spaces
• A metric space (M,d) is said to be
homogeneous, if all points are
indistinguishable, i. e. if Iso(M) acts
transitively on the points.
• For connected graphs the above condition is
equivalent to being vertex-transitive.
Some Results
• Claim 1. Every sphere of an isotropic space is
homogeneous.
• Let X ½ M.
• Iso(M,X) is the group of isometries fixing X set-wise.
• Iso(M;rel X) is the group of isometries fixing X pointwise.
• Iso(X) are the isometries of X.
• S(X) is the set of isometries of X that can be extended to
isometries of M.
Distance Set
• Let (M,d) be a metric space and let x 2 M.
Let D(x) = {d 2 R+| d(x,v), v 2 M}. D(x) is
called a distance set at x. M is said to have
constant distance set if D(u) = D(v) for
any pair of points u,v 2 M.
Distance-Transitive Metric
Spaces
• A metric space (M,d) is said to be distancetransitive if for any four points a,b,p,q 2 M
with d(a,b) = d(p,q) there exists an isometry
h of M, mapping a to p and b to q.
• Theorem. (M,d) is distance-transitive if and
only if it is homogeneous and isotropic.
• Note: There are isotropic non-homogeneous
metric spaces.
Distance-Transitive Graphs
• Connected graph G is also a metric space.
We may speak of isotropic graphs and
distance-transitive graphs.
• For instance Km,n is isotropic but not
distance-transitive.
Cubic Distance-Transitive
Graphs
•
Theorem: There are only
12 cubic distancetransitive graphs:
1.
2.
3.
4.
5.
6.
7.
8.
4, nonbipartite, girth = 3, K4
6, bipartite, girth = 4, K3,3
10, nonbipartite, girth = 5, G(5,2)
8, bipartite, girth = 4, Q3
14, bipartite, girth = 6, Heawood
18, bipartite, girth = 6, Pappus
28, nonbipartite, girth = 7, Coxeter
30, bipartite, grith = 8, Tutte 8-cage
Cubic Distance-Transitive
Graphs
•
Theorem: There are
only 12 cubic
distance transitive
graphs:
09. 20, nonbipartite, girth = 5, G(10,2)
10. 30, bipartite, girth = 6, G(10,3)
11. 102, nonbipartite, girth = 9, Biggs –
Smith H(17:1,2,4,8)
12. 90, bipartite, grith = 10,Foster
Example: Foster Graph
• The bipartite Foster
graph on 90 vertices is
the largest cubic
distance-transitive
graph.
• LCF[{17,-9,37},-15]
Biggs-Smith Graph
• The Biggs-Smith
graph H(17;1,2,4,8)
has 102 vertices and
girth 9.
Biggs-Smith Graph
• The Biggs-Smith
graph H(17;1,2,4,8)
has 102 vertices and
girth 9.
• Its Kronecker cover is
bipartite and has girth
12.
Odd graph On.
• Vertex set: all n-1
subsets of a 2n-1 set:
• |V(On)| = C(2n-1,n-1).
• Two sets are adjacent
if they are disjoint.
• Valence: n.
• O2 = K3
• O3 = G(5,2)
• O4 = Gewirtz graph.
Quartic Distance-Transitive
Graphs
•
Theorem: There are only
15 quartic distance
transitive graphs:
1.
2.
3.
4.
5.
K5
K4,4
L(K4)
L(K3,3)
L(G(5,2))
Quartic Distance Transitive
Graphs
1.
2.
3.
4.
5.
L(Heawood)
K2 £ K5
Heawood[3].
(4,6) cage
Gewirtz graph O4.
Quartic Distance-Transitive
Graphs
1.
2.
3.
4.
5.
L(Tutte8cage)
Q4
4-fold cover of K4,4
(4,12) cage
K2 £ O4.
Hamiltonicity
• Most vertex-transitive
graphs have Hamilton
cycles.
• There are only 4
known vertextransitive graphs
without Hamilton
cycle. [All four of
them have a Hamilton
path.]
Exercises 6-1
• N1. Find an isotropic metric space that is not
homogeneous.
• N2: Prove that G(n,k) is vertex transitive, if and
only if k2  § 1 mod n, or else n=10 and k=2.
• N3: Prove that Cn, Kn, Qn are all vertex transitive.
• N4: Which complete multipartite graphs Ka,b,
Ka,b,c, ... are vertex transitive?
• N5: Prove that the Cartesian product of vertex
transitive graphs is vertex transitive.
Exercises 6-2
• N6: Write an LCF code for the Dürer graph.
• N7: Write an LCF code for K4.
• N8: Write an LCF code for M3 = K3,3.
Generalize to the Möbius ladder Mn.
Exercises 6-3
• N9: Prove that Z3 £ Z3 À Z9.
• N10: Prove that Z2 £ Z3 @ Z6.
• N11(*): Prove that any finite abelian group A is
isomorphic to the direct product A(n1,n2,...,nk) =
Zn1 £ Zn ... £ Znk, where n1|n2|...|nk.
• N12(*): Prove that the groups A(n1,n2,...,nk) and
A(m1,m2,...,mj) with n1|n2|...|nk and m1|m2|...|mj
are equal if and only if j= k and nt=mt, for each t.
Homework 6-1
• H1. Find a better
drawing of Gewirtz
graph.
Homework 6-2
• H2. Find the definition and a drawing of
any missing quartic graph in the previous
theorem.
• H3. Determine all groups that have a cycle
Cn their Cayley graph.
7. Representations of Graphs and
their Symmetries
• Graph drawings in a vector space or on a surface may be
considered graph representations.
• Graphs obtained from applications may have
interpretations for their vertices and edges, yielding
various graph representations.
Representation of Graphs
• Let G be a graph and let V be a set. A pair of
mappings
• rV:V(G) ! V and
• rE:V(G) ! P(V) is called a
• V-representation of graph G
• if for any edge e = uv 2 E(G)
• we have {rV(u),rV(v)} µ rE(uv).
• If there is no danger of confusion we will drop
the subscripts and denote both mappings simply
by r.
Representation of Graphs
• Usually we require V to be a vector space (this is what
C. Godsil and G. Royle do in their book Algebraic
Graph Theory, Springer, 2001). But that is not always
the case.
• In their definition Godsil and Royle use a single
mapping defined on the vertices.
• In such a case we may extend the mapping on the edge
set in an arbitrary way, for instance by taking rE(uv) :=
{rV(u),rV(v)}. [This works for all representations.]
Graph Representation Examples
• For the cube graph Q3 there are several
useful representations:
• [3 dimensional real representation] In R3 the eight
vertices are mapped to the eight points of {0,1}3.
• The two drawings of Q3 in the Euclidean
plane can be interpreted as representations in
• [2 dimensional real representation] R2 or in
• [1 dimensional complex representation] C.
• In the latter case, the points in the complex
plane are given by
{eikp|0 · k · 7}.
Edge Representation
r(u)
r(v)
• Let e = uv 2 E(G). For a representation in
Rn or Cn it makes sense to define
• rE(e) = conv(rV(u),rV(v)).
• Hence an edge is represented as the
segment joining the two endpoints.
• For other representation spaces a different
edge representation may be of interest.
Edge Extensions
•
•
r(u)
r(u)
r(u)
r(u)
•
•
r(u)
Let e = uv 2 E(G).
There are several possible edge
extensions:
r(e) = {r(u),r(v)}.
r(e) = {r(u),r,r(v)}.
• r = (r(u)+r(v))/2.
r
r
r(v)
r(v)
r(v)
r(v)
r(v)
•
•
•
•
r(e) = conv(r(u),r(v)).
r(e) = aff(r(u),r(v))
We may speak of barycentric,
convex and affine edge extensions,
respectively.
But there are several other
interpretations of r and a variety of
possible edge extensions.
Graph Representation vs. Graph
Drawing
There is some overlap but there are many differences.
•
• In graph drawing (in the broad sense of the word) the object is to
find algorithms to draw a graph (usually in the plane) with
certain restrictions or with some optimization criterion.
[Computer Science Approach.] See for example: Annotated
bibliography on graph drawing algorithms, by Di Battista,
Eades, Tamassia and Tollis.
• In graph representation we label vertices (= add coordinates). We
may look at this as a functor from the category of graphs to the
category of coordinatized graphs. [Mathematical Approach].
• We will use the word graph drawing in a narrow sense of the
word (as a special representation).
One Dimensional Real
Representation
• Let r: V(G) ! R be a graph representation. Then
such a representation is sometimes called a fitness
landscape over G and r is a cost function or
potential function.
• For many problems of combinatorial optimization
we may model a solution space by a graph. To
each vertex we assign a cost and try to find a
feasible solution yielding a minimum cost.
Nodal Domains
• A one dimensional
representation defines a
partition of the vertex set
into three classes: V+, V-,
and V0.
• V+ = {v 2 V(G)| r(v) > 0}.
• V- = {v 2 V(G)| r(v) < 0}.
• V0 = {v 2 V(G)| r(v) = 0}.
• A nodal domain is a
connected component of
the graph induced by V+or
V-. [Weak nodal domain
V+ [ V0].
Local Search
• Idea: Feasible solutions
form a metrc space. For
any solution x choose a
neighborhood N(x).
• In N(x) find a solution x’
with smaller cost value:
r(x’) < r(x).
• Repeat the procedure as
long as possible.
r-Neighborhoods
x
• Y – feasible solutions
form a metric space.
• Select radius r > 0.
• Declare:
• N(x) := B(x,r) = {y 2 Y | d(x,y) ·
r}
N(x)
• (Y,d,r) defines a graph G:
• V(G) := Y
• x ~ y if and only if d(x,y) · r.
Neighborhoods in Graphs
x
• For graphs we usually
take N(x) the set of
neighboors of x.
• For a connected graph
this is equivalent to
N(x)
• N(x) = B(x,1) = {y 2 Y |
d(x,y) · 1}
Local Search Algorithm
Input: Connected graph G, representation
V(G) ! R, initial value x0 2 V(G).
• Output: Local minimum at x.
• x à x0;
• Repeat:
• finish à true;
• For each y 2 N(x) if r(y) < r(x) then
» x à y; finsh à false
• Until finish
r:
Simulated Annealing - Idea
• Use local search to move downhill (Df < 0).
• Possible temperature dependent jumps uphill
> 0).
f(y)
• Lower the temperature.
Df
f(x)
(Df
Characteristic Vector
• Let G be a graph and S µ
V(G) a set of vertices. Let
GS denote the induced
subgraph of G, defined by
set S.
• The characteristic vector
• S: V(G) ! {0,1} with
• S(x) = 1 if x 2 S and
• S(x) = 0 otherwise,
• is a graph representation.
Vertex Coloring as Graph
Representation
• Let r: V(G) ! {1,2,...,k} be a representation
of graph G and for each edge let r(uv) =
{r(u),r(v)}.
• Then r is a proper vertex coloring if and
only if for each edge |r(uv)| = 2.
Identity Representation
• Each graph G has a trivial, or identity
representation:
• rV = id: V(G) ! V(G).
• rE: uv a {u,v}.
• For simple graphs, rE is an injective map.
Point Configuration
• A point configuration S µ V
is a collection of elements of
some space V. Usually we
consider point configurations
in R2.
• If r is a V-representation of G
then the image S = r(V(G)) is
a point configuration.
• We say that r is vertex
faithful is r:V(G) ! S is a
bijection. We are mostly
interested in vertex faithful
representations.
Graph Construction from Point
Configurations
• Given point configuration S
µ V and a symmetric
predicate
• P : V ! {true,flase}
• P(p,q) = P(q,p).
• We define a graph:
• G = G(S,P) as follows:
• V(G) := S.
• x ~ y if and only if P(x,y) =
true.
• Example:
• V = Rn,
• P(x,y) = (d(x,y) · r)
•
Let S be a “nice” topological space such as metric space and G
be a general graph. A mapping :G  S is defined as follows:
1. Injective mapping :V(G)  S
2. Family of continuous mappings e:[0,1]  S, for each edge
e = uv so that e( 0) = (u) and e(1) = (v).
3. In the interior of the interval e is injective.
•
Each embedding would qualify. For embeddings we
need more!
Note that  defines a representation of G in S.
•
Embeddings are Representations
• Think of K3 ¤ K3
embedded in a torus,
the torus, in turn, is
embedded in R3. We
obtain a representation
of this graph in the
torus and another one
in R3.
New Representations from Old
• We are investigating
techniques that use
existing
representations for
obtaining new, more
sophisticated
representations.
Stereographic Projection
N
T0
T1
• There is a homeomorphic
mapping of a sphere
without the north pole N
to the Euclidean plane R2.
It is called a stereographic
projection.
• Take the unit sphere
x2 + y2 + z2 = 1 and the
plane z = 0.
• The mapping
p: T0(x0,y0,z0) a T1(x1,y1)
is shown on the left.
Stereographic Projection
N
T0
T1
• The mapping
p: T0(x0,y0,z0) a
T1(x1,y1)
is shown
on the left.
• r1 = r0/(1-z0)
• x1 = x0/(1-z0)
• y1 = y0/(1-z0)
Stereographic projection and
representations
• We may use stereographic projection to get
an R2 drawing from an R3 drawing.
• Note that the representation of edges is
computed anew!
Example
• Take the dodecahedron
and a random point N on a
sphere.
• The resulting
stereographic projection is
depicted below.
• A better strategy is to take
N to be a face center.
Example
• A better strategy is to
take N to be a face
center as shown on the
left.
• Only vertices are
projected. The edges
are re-computed.
Representation of Graphs in
Metric Space
• Sometimes we may take V to be a metric space,
projective space or some other structure.
• If (V,d) is a metric space we may define the
energy of the representation r.
• Ep(r) = [Suv 2 Ed(r(u),r(v))p]1/p, for 1 · p · 1.
Euclidean metric in Rn.
• The set of real n-tuples
• Rn := {x = (x1,x2,...,xn)|xi 2 R, 1 · i · n}
• carries a number of important mathematical
structures. The mapping
• dp(x, y) = [(x1 – y1)p + (x2 – y2)p + ... + (xn – yn)p]1/p.
• makes (Rn,dp) a metric space for 1 · p · 1.
• For p = 2 the usual Euclidean metric is obtained.
• There are important and deep results by László
Lovász et al.
Metric Space - Revisited
• If (M,d) is a metric space, then for any
A
µ M with the induced metric (A,d) is also a
metric space, a subspace.
• A natural question is: when are two metric
spaces (M,d) and (M’,d’) considered
isomorphic?
• There are two types of mappings that are
candiates for “isomorphism”.
Isometries
• Let (M,d) and (M’,d’) be two metric spaces.
A bijective mapping s: M ! M’ is called is
isometry, if for every pair of points u,v 2 M
we have:
• d(u,v) = d’(s(u),s(v)).
• Clearly, isometric spaces are
indistingushable as far as metric properties
are concerned.
Similarity I
• Let (M,d) and (M’,d’) be two metric spaces.
A mapping h:M ! M’ with the property that
for any four points a,b,c,d 2 M we have:
• If d(a,b) = d(c,d) then d(h(a),h(b)) =
d(h(c),h(d)) is called similarity (of type I).
Similarity II
• Let (M,d) and (M’,d’) be two metric spaces
and r 2 R\{0}. A mapping h:M ! M’ with
the property that for any pair of points a,b, 2
M we have:
• If d(a,b) = r d(h(a),h(b)) then h is called
similarity (of type II) and r is called the
dilation factor.
Type I vs. Type II
• Clearly each similarity of type II is also a
similarity of type I. In general, the converse
is false.
• Theorem. A similarity on (Rn,d2) of type I
is also of type II. (Proof can be found in
Paul B. Yale: Geomerty and Symmetry,
Dover, 1988 (reprint from 1968))
Finite Metric Space
• In a finite metric space (M,d) we may
assume that min d(u,v) = 1. Max d(u,v) is
called the diameter of M. The quotient Max
d(u,v)/Min d(u,v) is called dilation
coefficient.
Three Classical Results
• The Steinitz Theorem,
Fary’s Theorem and
Tutte’s Theorem can
be interpreted as graph
representations.
The Energy
• Usually we try to find among the
representations of a certain type the one that
is “optimal” in certain sense.
• To this end we may define an energy
function E(r) and then seek a representation
that minimizes the energy.
• There are several such energy functions
used in various problem areas.
Some Energy Models
•
•
•
•
Spring embedders
Molecular mechanics
Tutte drawing
Schlegel diagram (B.
Plestenjak).
• [Connection to Markov
Chains]
• ...
• Laplace Representation
The Laplace Representation
•
Let r be a representation in Rk. Define
E(r) = Suv 2 E(G) ||r(u)-r(v)||2
•
It turns out that the minimum (under some
reasonable conditions) is achieved as
follows.
1.
2.
3.
4.
Take the Laplace matrix of G.
Q(G) = D(G)-A(G)
Find the eigenvalues
0 = l1 · l2 · ... · ln.
Find the corresponding orthonormal eigenvectors x1,
x2, ..., xn.
Form a matrix R =[x2|x3| ... |xk+1]
Let r(vi) = rowi(R).
5.
6.
An R3 Laplace representation of a
fullerene (skeleton of a trivalent
polyhedron with pentagonal and
hexagonal faces)
Nodal Domains - Revisited
• The Example on the
left represents nodal
domains obtained
from the Laplace
representation of
G(10,4).
Congruence and Similarity
• A representation in any metric sapce, in
particular in Rn, can be scaled without
“being changed too much”. If r is injective
on the vertices, we may scale it in such a
way that Min d(u,v) = 1, for all u ~ v. Each
vertex faithful representation is similar to a
standard one.
Similar Representations
• Let r,s:G ! M be graph representations into
a metric space M. We say they are similar,
if there exists a similarity h 2 Sim(M) such
that for each v 2 V(G) we have s(v) :=
h(r(v)).
• We would like to assign the same energy to
similar representions.
Unit Distance Graphs
• Let r be a representation in Rk. Define
Ep (r) = (Suv 2 E(G) ||r(u)-r(v)||p) (1/p)
• We assume that Min uv 2 E(G) ||r(u)-r(v)|| = 1
• In the limit when p ! 1 we get
E1 (r) = Maxuv 2 E(G) ||r(u)-r(v)||
• The number E1 (r) is called dilation coefficient.
• Hence E1 (r) ¸ 1. In the special case: E1 (r) = 1 we
call this representation a unit distance graph.
Generalized Petersen Graphs
• Some generalized Petersen
graphs admit unit distance
representations in R2 with
considerable symmetry.
• Several questions:
• Which G(n,k) have unit
distance representations?
• Connection to symmetry.
• Change dimension.
Symmetry of Representation
• Let r:G ! M be a graph representation into a
metric space M. Let Aut r be the group of
symmetries of this representation. Namely
g 2 Aut G is a symmetry of r (and
therefore g 2 Aut r) if there exists an
isometry h 2 Iso(M) such that for each v 2
V(G) we have r(g(v)) = h(r(v)) and for
each e=uv 2 E(G) we have d(r(u),r(v)) =
d(r(g(u)),r(g(v)).
Representations with Symmetry
(Motivation: Recent work on regular polygons and
regular polyhedra by Branko Grünbaum)
• Let G be a graph and let Aut(G) be its automorphism
group.
• Let Iso(Rk) be the group of Euclidean isometries.
• We say that an automorphism a 2 Aut(G) is preserved by
the representation r if there exists an isometry a 2 Iso(Rk)
such that
• for each vertex v 2 V(G) it follows that a(r(v)) = r(a(v)).
• The set of all automorhpisms Gr 2 Aut(G) that are
preseved by r forms a group that we call the symmetry
group of the representation r.
• A representation with trivial symmetry group is called
rigid.
An Example
(13)
(23)
(12)
(13)
(23)
(12)
3
• Consider the one-dimensional
representation of the triangle C3 with
V(C3) = {1,2,3}.
• Aut(C3) = S3 =
{id,(12),(13),(23),(123)
,(132)}.
• Let ri = r(i). W.l.o.g. assume r3 = 0.
Hence each representation can be
viewed as a point in the (r1,r2) –
plane.
• The points not lying on any of the
axes or lines determine a rigid
representation. Each line is labeled
by its symmetry group. The origin
retains the whole symmetry.
• Note that the underlined
representations are non-singular
(meaning that r is one-to-one)..
r1
1
2
0 =r3
r2
A General Problem
• For an arbitray graph
G find a non-singular
representation in R2
minimizing the
number of vertex
orbits or edge orbits.
• There are several
obvious variations to
this problem.
Homework 7
• H1. It is easy to verify that K4
is not a unit distance graph in
the plane. Consider a drawing
of K4 in the plane with only two
distinct edge lengths. How
many such non-isomorphic
drawings are there? (Hint: there
are six). Compute the dilation
coefficient for all such
drawings.
Chapter 2. Statistics Page
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•
•
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Number of slides:282
Number of sections:7
Number of exercises:32
Number of homeworks:21