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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.

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.

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).

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).

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.

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

dilation coefficient.

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).

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.

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.

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.

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).

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)

Three Classical Results

- Steinitz Theorem, Fary Theorm and Tutte Theorem

can be interpreted as graph representations.

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.

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.

Some Energy Models

- Spring embedders
- Molecular mechanics
- Tutte drawing
- Schlegel diagram drawing (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. - 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)

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.

Nodal Domains

- 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 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.

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.

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.

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)

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).

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.

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.

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

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

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!

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.

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.

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).

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)?