Graphs

1. Graphs • Simple graph G=(V,E) • V = V(G) ={1,2,3,4} – vertices • E = E(G) = {a,b,c,d,e} – edges • Edge a has end-vertices 1 and 2. Vertices 1 and 2 are adjacent: 1 ~ 2. a 1 2 c b d e 3 4

2. Simple Graph • Definition: Graph X is composed of the set of vertices V(X) endowed with irreflexive symmetric relation ~ (adjacency). An unoredered pair of adjacent vertices uv = vu forms an edge. The set of edges is denoted by E(X). Sometimes we write X = (V,E) or X(V,E).

3. Families of Graphs

4. Cycle Cn on n vertices. V – vertices of a regular n-gon E – edges • |V|=n • |E|=n 1 2 3 4 C4

5. Small Cycles • Some cycles as drawn by VEGA. • It makes sense to define cylces C1 (a loop) and C2 (parallel edges), that are NOT simple. C3 C4 C1 C5 C6 C2

6. 1 2 3 4 Path Pn on n vertices. V – vertices of polygonal line. E – segments. The endpoints of the polygonal line are called the endpoint of the path. For instance, 1 and 4 are the endpoints of the path on the left. • |V|=n • |E|=n-1 P4 1 2 3 4

7. Complete graph on n vertices Kn. V – vertices of a regular n-gon E – edges and diagonals. • |V|=n • |E|=n(n-1)/2 1 2 3 4 K4

8. Complete Bipartite Graph on n+m vertices Kn,m. V = U1 U2 , U1Å U2 = ; |U1| = m, |U2 | = n. E = U1 U2 • |V|=n + m • |E|=n m 1 2 3 4 K2,2

9. Metric Space • Space V, with mapping d (distance): • d:V  V R with the following properties: • d(u,v) ¸ 0, d(u,v) = 0, iff u = v. • d(u,v) = d(v,u) • d(u,v) · d(u,w) + d(w,v) • is called a metric space with distance d.

10. Example: Hamming Distance {0,1}n is a metric space if distance between u and v is the number of components in which the two vectors differ. • E.g. d([0,0,0,1,0,1],[1,1,0,1,1,1]) = 3. • d is called the Hamming distance.

11. Hypercube Qn. • Hypercube of dimension d is the graph Qn, with: • V(Qn) = {0,1}n. • u ~ v, if d(u,v) = 1. • |V(Qn)| = 2n • |E(Qn )|= n 2n-1 Q2 Q3 Q1 Q5 Q4

12. 1 2 3 Vertex Valence • G = (V,E) • V(G) ={1,2,3,4} • E(G) = {a,b,c,d,e} • Number of edges incident with vertex v is called the valence or degree of v: deg(v). • deg(1) = deg(4) = 3, deg(2) = deg(3) = 2. • Vertex of valence 1 is called a leaf, vertrex of valence 0 is isolated. • d(G) – minimal valence. • D(G) – maximal valence. a c b d e 4

13. Regular Graphs Graph G is regular (of valence k), if d(G) = D(G) = k. Zgled: • Regular graphs: Kn, Cn, Kn,n • Nonregular graphs: Pn, n > 2, Kn,m, n ¹ m. 1-valent and 2-valent graphs have simple structure. Trivalent graphs have special name: cubic graphs. (See example on the left)

14. Girth • Girth g(G) of graph G is the number of vertices of the shortest cycle in G. If G has no cycles, its girth is infinite.

15. Cages • Graph G is a g-cage, if the following holds: • Trivalent • Has girth g • Has the least number of vertices among the graphs satisfying 1 and 2.

16. Exercises 01 • N1. Deterimine the 3-cage. • N2. Determine the 4-cage. • N3. Determine the 5-cage. • N4. Determine the 6-cage.

17. The Petersen Graph and its Generalizations G(n,k) • Petersen graph G(5,2) is an example of a generalized Petersen graph G(n,k). • V(G(n,k)) consists of • ui, vi, i = 1,2, ..., n. Edges: • ui ~ ui+1 • ui ~ vi • vi ~ vi+k (Warning! Addition mod n)

18. Examples of Generalized Petersen graphs • G(10,2) Dodecahedron • G(10,3) Desargues graph. • G(8,3) Möbius-Kantor graph. • G(6,2) Dürer graph.

19. Incidence Matrix M(G). • To G=(V,E) we associate a rectangle matrix M=M(G) with |V| rows and |E| columns: { 1 ... v is the endpoint of e Mv,e = 0 ... otherwise

20. Incidence Matrix - Example • G=(V,E) • VG ={1,2,3,4} • EG = {a,b,c,d,e} MG = a 1 2 c b d e 3 4

21. Handshaking Lemma • In each graph G=(V,E) : • 2 |E(G)| = Sv 2 V(G) deg(v), • The proof uses the so-called bookkeepers rule in the incidence matrix of graph G.

22. Graph Invariant • It is well-known that we associate numbers to mathematical objects in various ways. For instance: Determinant is assicated to a matrix, degree is associated to a polynomial, dimension is associated to a space, length is associated to a vector, etc. • There are several numbers that can be associated with a graph. Such a number is usually called graph invariant. One may argue that the main topic of graph theory is the study of graph invariants. • In addition to numbers other objects may be graph invariants.

23. Isomorphisms and Graph Invariants Isomorphisms(G) = H is a bijective mapping: • s: V(G) ! V(H). that preserves adjacency: • u ~ v if and only if s(u)~s(v). Graph invariant is a property, (usually a number), that is preserved under an isomorphism.

24. Isomorphism - Exercises B A • N1. Determine an isomorphism between graphs A and B. • N2. Determine an isomorphism between graphs C and D. D C

25. Adjacency Matrix A(G). • To each graph G=(V,E) with V={1,2,3,...,n} we can associate the adjacency matrix A=A(G) as follows: { 1 ... i ~ j Ai,j = 0 ... sicer

26. Adjacency Matrix - Example • G=(V,E) • VG ={1,2,3,4} • EG = {a,b,c,d,e} AG = a 1 2 c b d e 3 4

27. Adjacency Matrix is Not an Invariant • Adajcency matrix is not an invariant. It depends on the numbering of vertices. • Incidence matrix is not an invariant

28. Some Graph Invariants • |V(G)| = number of vertices • |E(G)| = number of edges • d(G) = minimal valence. • D(G) = maximal valence

29. Invariants - Example • |V(G)| = 4 • |E(G)| = 5 • d(G) = 2 • D(G) = 3 a 1 2 c b d e 3 4

30. Trees • A tree is a connected graph with no cycles • There are several characterizations of tree, such as: • A tree is a connected graph with n vertices and n-1 edges. • A tree is a connected graph that is no longer connected after removal of any edge.

31. Disjoint Union of Sets • Let A and B be sets. By A t B we denote the disjoiont union of A and B. If A Å B = ;, then A t B is simply the union of the two sets. Otherwise we defne formally A t B = A £ {0} [ B £ {1}.

32. Disjoint Union of Graphs • Let G’ and G” be graphs. By G’ t G” we denote the disjoiont union of graphs G’ and G”. This means • V(G’ t G”) := V(G’) t V(G”) and • E(G’ t G”) := E(G’) t E(G”).

33. The Empty Graph • Empty graphf = (f,f) has no vertices and no edges.

34. Connectivity in Graphs - Theory • Graph G is connected, if and only if it cannot be written as a disjoint union of two non-empty graphs.

35. Connectivity of Graphs - Practice • Graph is connected, if we grab and shake the “model” made of balls and strings, and nothing falls down the earth. (No knotting of strings is permitted!)

36. Equivalence Relation @. • Let G be a graph. On V(G) define @ as follows: For any u,v 2 V(G) let u @ v, if and only if there exists a subgraph, isomorphic to a path that has the endponts u and v. • Proposition. @ is an equivlanece relation on V(G). • Proof. Obviously reflexive and symmetric. Proof of transitivity – Homework.

37. Path Connectivity of Graphs • G is connected by paths, if the equivalence relation @ has a single equivalence class.

38. Homework • H1: Prove that the relation @ is transitive. • H2: Prove that for finite graphs the notions of connectedness and path connectedness coincide.