Metric Space - Revisited

1
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.

2
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.

3
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)p1/p.
• makes (Rn,dp) a metric space for 1 p 1.
• For p 2 the usual Euclidean metric is obtained.

4
Metric in C.

• Let z a bi and w c di be two complex
  numbers.
• Define d(z,w) z w. Then (C,d) is a metric
  space.
• Note that (C,d) is isometric to the Euclidean
  plane (R,d2).

5
Similarity I

• Let (M,d) and (M,d) be two metric spaces. A
  mapping hM ! 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(s),h(b))
  d(h(c),h(d)) is called similarity (of type I).

6
Similarity II

• Let (M,d) and (M,d) be two metric spaces and r
  2 R\0. A mapping hM ! 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.

7
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))

8
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. Quotient Max d(u,v)/Min d(u,v) is called
  dilation coefficient.

9
Representation of Graphs

• Let G be a graph and let V be a set. A pair of
  mappings rVV(G) ! V and rEV(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.
• 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. Insuch 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).

10
Representation of Graphs in Metric Space

• There are important and deep results by László
  Lovász et al.
• 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 representation.

11
Point Configuration

• A point configuration S µ V is a collection of
  elements of some space V. Later we will 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 rV(G) ! S is
  a bijection. We are mostly interested in vertex
  faithful representations.

12
Graph Representation An Example

• 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
  0,13.
• 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 eikp0 L k L 7.

13
Extending Representation to Edges

• Usually we try to extend r mapping to the edges.
  In the case V R2 or V R3 finding a
  representation means actually drawing graph G in
  V R2 or V R3 .
• Each edge euv is then represented as the line
  segment connecting r(u) and r(v).
• Hence r(e) conv(r(u),r(v)).
• In general we extend r to the edges r E(G) !
  P(V) and require that for e uv, r(u),r(v) µ
  r(e).
• If nothing is said about edge extension, we
  assume r(e) r(u),r(v).

14
Edge Extensions

• 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(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.

r(u)
r(u)
r(u)
r(u)
r(u)
r
r
r(v)
r(v)
r(v)
r(v)
r(v)
15
Three Classical Results

• Steinitz Theorem, Fary Theorm and Tutte Theorem
  can be interpreted as graph representations.

16
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.

17
The Energy

• Usually we try to find among the representations
  of certain type the one that is optimal in
  cetrain sense.
• To this end we may define an energy function E(r)
  and then seek for representation that minimizes
  the energy.
• There are several such energy functions used in
  various problem areas.

18
Some Energy Models

• Spring embedders
• Molecular mechanics
• Tutte drawing
• Schlegel diagram drawing (B. Plestenjak).
• Connection to Markov Chains
• ...
• Laplace Representation

19
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.
• 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 x2x3 ... xk1
• Let r(vi) rowi(R).

An R3 Laplace representation of a fullerene
(skeleton of a trivalent polyhedron with
pentagonal and hexagonal faces)
20
Nodal Domains

• One dimensional representation defines partition
  of the vertex set in three classes V, V-, V0.
• A nodal domain is a connected component of the
  graph induced by Vor V-. Weak nodal domain V
  V0.

21
Nodal Domains

• The Example on the left represents nodal domains
  obtained from the Laplace representation of
  G(10,4).

22
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 vertices,
  we may scale it in such a way that Min d(u,v)
  1, for u v. Each vertex faithful representation
  is similar to a standard one.

23
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.

24
Homework

• 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.

25
Flat Torus

• Take a unit square and identify two opposite
  pairs of sides. The resulting topological space
  is a torus.In order to make it metric sapce we
  can extend the usual Euclidean distance .
• dT(r,s) Mind(r,s(0,1)), d(r,s(1,0)),
  d(r,s(1,1)), d(r,s(0,-1)). d(r,s(-1,0)),
  d(r,s(-1,1)), d(r,s(1,-1)), d(r,s(-1,-1)).

r (rx,ry)
s (sx,sy)
26
Exercises

• N1. Given a standard drawing of G(n,r) with inner
  radius r and outer radius R, determine the
  dialation coefficient of this planar
  representation.
• N2. Select the optimal quotient R/r in the
  previous exercise.
• N3. In the unit flat torus draw the circle of
  radius ½ centered in the point (¼, ¼).

27
Embeddings vs. Representations

• 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
• Injective mapping ?V(G) ? S
• Family of continuous mappings ?e0,1 ? S, for
  each edge e uv so that ?e( 0) ?(u) and ?e(1)
  ?(v).
• In the interior of the interval ?e is
  injective.
• Each embedding would qualify.
• Note that ? defines a representation of G in S.

28
Embeddings are Representations

• Think of K3 K3 embedded in torus, that, in
  turn, is embedded in R3. We obtain a
  representation of this graph in torus and another
  one in R3.

29
Stereographic Projection

• 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.

N
T0
T1
30
Stereographic Projection

• 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)

N
T0
T1
31
Stereographic projection and representations

• We may use stereographic projection to get a R2
  drawing from a R3 drawing.
• Note that the representation of edges is computed
  anew!

32
Example

• Take the Dodecahedron and a random point N on a
  sphere.
• Stereographic projection is depicted below.
• A better strategy is to take N to be a face
  center.

33
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.

34
Schlegel Diagram

• Usually Schlegel diagram is defined for a
  (convex) polyhedron. Normally, it is definded as
  a projection of the polyhedron on one of its
  faces.
• We understand this notion in a broader sense.
  This is a drawing of a graph G within the convex
  region defined by some of its vertices S ½ V(G).

35
Exercises

• N1. Use Laplace representation followed by
  stereographic projection to get schlegel diagrams
  of platonic graphs.
• N2. Use a generalization of Tuttes method to
  slove the same problem.
• N3. Repeat the the two exercises for some of the
  archimedean solids and their duals.
• N4. Is there a unit distance representation for
  the subdivision graph S(K4)?