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  : M ! M’ is called is isometry, if for every pair of points u,v 2 M we have: d(u,v) = d’(  (u),  (v)). Clearly, isometric spaces are indistingushable as far as metric properties are concerned.

3. Euclidean metric in R n. The set of real n-tuples R n := {x = (x 1,x 2,...,x n )|x i 2 R, 1 · i · n} carries a number of important mathematical structures. The mapping d p (x, y) = [(x 1 – y 1 ) p + (x 2 – y 2 ) p +... + (x n – y n ) p ] 1/p. makes ( R n,d p ) 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,d 2 ).

5. 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(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 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.

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 ( R n,d 2 ) 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  V :V(G) ! V and  E :V(G) ! P (V) is called a V- representation of graph G if for any edge e = uv 2 E(G) we have {  V (u),  V (v)} µ  E (uv). If there is no danger of confusion we will drop the subscripts and denote both mappings simply by . 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  E (uv) := {  V (u),  V (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 R 2. If  is a V-representation of G then the image S =  (V(G)) is a point configuration. We say that  is vertex faithful is  :V(G) ! S is a bijection. We are mostly interested in vertex faithful representations.

12. Graph Representation – An Example For the cube graph Q 3 there are several useful representations: [3 dimensional real representation] In R 3 the eight vertices are mapped to the eight points of {0,1} 3. The two drawings of Q 3 in the Euclidean plane can be interpreted as representations in [2 dimensional real representation] R 2 or in [1 dimensional complex representation] C. In the latter case, the points in the complex plane are given by {e ik  |0  k  7}.

13. Extending Representation to Edges Usually we try to extend  mapping to the edges. In the case V = R 2 or V= R 3 finding a representation means actually drawing graph G in V = R 2 or V= R 3. Each edge e=uv is then represented as the line segment connecting  (u) and  (v). Hence  (e) = conv(  (u),  (v)). In general we extend  to the edges  : E(G) ! P (V) and require that for e = uv, {  (u),  (v)} µ  (e). If nothing is said about edge extension, we assume  (e) = {  (u),  (v)}.

14. Edge Extensions Let e = uv 2 E(G). There are several possible edge extensions:  (e) = {  (u),  (v)}.  (e) = {  (u),r,  (v)}. r = (  (u)+  (v))/2.  (e) = conv(  (u),  (v)).  (e) = aff(  (u),  (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.  (u)  (v)  (u)  (v) r  (u)  (v)  (u)  (v) r

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 (  ) 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  be a representation in R k. Define E (  ) =  uv 2 E(G) ||  (u)-  (v)|| 2 It turns out that the minimum (under some reasonable conditions) is achieved as follows. 1.Take the Laplace matrix of G. 2.Q(G) = D(G)-A(G) 3.Find the eigenvalues 0 = 1 · 2 ·... · n. 4.Find the corresponding orthonormal eigenvectors x 1, x 2,..., x n. 5.Form a matrix R =[x 2 |x 3 |... |x k+1 ] 6.Let  (v i ) = row i (R). An R 3 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 -, V 0. A nodal domain is a connected component of the graph induced by V + or V -. [Weak nodal domain V + [ V 0 ].

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 R n, can be scaled without “being changed too much”, If  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  be a representation in R k. Define E p (  ) = (  uv 2 E(G) ||  (u)-  (v)|| p ) (1/p) We assume that Min uv 2 E(G) ||  (u)-  (v)|| = 1 In the limit when p ! 1 we get E1 (  ) = Max uv 2 E(G) ||  (u)-  (v)|| The number E1 (  ) is called dilation coefficient. Hence E1 (  ) ¸ 1. In the special case: E1 (  ) = 1 we call this representation a unit distance graph.

24. Homework H1. It is easy to verify that K 4 is not a unit distance graph in the plane. Consider a drawing of K 4 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. d T (r,s) := Min{d(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 = (r x,r y ) s = (s x,s y )

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  be a “nice” topological space such as metric space and G be a general graph. A mapping  :G   is defined as follows: 1.Injective mapping  :V(G)   2.Family of continuous mappings  e :[0,1]   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. Note that  defines a representation of G in .

28. Embeddings are Representations Think of K 3 ¤ K 3 embedded in torus, that, in turn, is embedded in R 3. We obtain a representation of this graph in torus and another one in R 3.

29. Stereographic Projection There is a homeomorphic mapping of a sphere without the north pole N to the Euclidean plane R 2. It is called a stereographic projection. Take the unit sphere x 2 + y 2 + z 2 = 1 and the plane z = 0. The mapping p: T 0 (x 0,y 0,z 0 )  T 1 (x 1,y 1 ) is shown on the left. N T0T0 T1T1

30. Stereographic Projection The mapping p: T 0 (x 0,y 0,z 0 )  T 1 (x 1,y 1 ) is shown on the left. r 1 = r 0 /(1-z 0 ) x 1 = x 0 /(1-z 0 ) y 1 = y 0 /(1-z 0 ) N T0T0 T1T1

31. Stereographic projection and representations We may use stereographic projection to get a R 2 drawing from a R 3 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 Tutte’s 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(K 4 )?