Graph definition Degree, in, out degree, oriented graph. Complete, regular, bipartite graph. Graph representation, connectivity, adjacency.

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1 Mária Markošová

2 Graph definition Degree, in, out degree, oriented graph. Complete, regular, bipartite graph. Graph representation, connectivity, adjacency. Isomorphism of graphs. Paths, cycles, trials. Random graph. Graph evolution.

3 Definition: A graph G consists of vertex set V(G) and edge set E(G) and edges are defined by vertices pairs. supgraph loop multiple edge Oriented graph: if edges have orientation. Simple graph: graph without loops and multiple edges, no edge orientation.

4 w Vertex degree a x c b d y e z Degree of a vertex x. k(x) is a number of edges incident with the vertex x. w Oriented graph k in x x b a c x y e d z In degree of vertex x, : Number of edges leading to the vertex x. k out Out degree of vertex x, : Number of edges leading out of vertex x. k x k x k x in out

5 Example. k(x)= 5 x k x k x k x in out k k in out x? x?

6 Oriented graphs: G(V,E), where V is a set of vertices, E is a set of ordered pairs e=(u,v), u, v are endpoints of edge and their order is fixed. Simple graphs: without oriented edges, no loops, no multiple edges: in what follows we shall deal with such graphs, if not given otherwise. Lemma: Unoriented graph has even number of vertices with odd degree.

7 Complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. Complete graph of N vertices is denoted as K N K 4 K K 7 2 The number of edges in the complete graph: E N 2 N N 2 What is the degree of each node in K N?

8 Regular graph is a graph where each vertex has the same number of neighbors, that means each vertex has the same degree. What about the complete graph? Is the complete graph also a regular graph?

9 Complementary graphs What is the complementary graph of? K 2 Are there a graphs, which have the same complementary graphs? Comlementary graphs mapped on each other create complete graph, in which each vertex is connected to each other vertex.

10 Bipartite graph Graphs, in which V(G) (vertex set) is the union of the two disjoint independent sets (no edges between nodes in the set). people jobs What type of the graph is also a bipartite graph? Find an example. men women authors papers Find another examples at home.

11 Tree graph: Undirected graph in which two vertices are connected exactly by one simple path. Is the tree on fig a bipartite graph? Every tree is a bipartite graph.

12 Spanning tree: Spanning tree of the connected undirected graph G is a tree composed of all vertices and some edges of the graph G. A spanning tree of a connected graph G can also be defined as a maximal set of edges of G that contains no cycle, or as a minimal set of edges that connect all vertices.

13 Graph representation Loopless graph: Graph without loops, multiple edges are allowed w x y z a b c d e 2 2 z y x w W x z y A(G) adjacency matrixnumber of edges between defined endpoints z y x w e d c b a M(G) incidence matrix- One, if vertex is the endpoint of an edge, zero if not

14 w x y z a b c d e 2 2 z y x w W x z y A(G)= Every adjacency matrix is symetric (no oriented graphs). The degree of vertex x is a sum of entries of row x in both A(G) and M(G). z y x w e d c b a M(G)= 3 2 x k

15 Isomorphic graphs Two graphs G and H are isomorphic, if we are able to map one to the another (e.g. to find a bijection f: V(G) V(H), and if uv is from E(G) then f(u)f(v) is from E(H)) G H Adjacency matrix of G is the same as of H if we reorder the vertices

16 Self complementary graphs Definition: A graph is self complementary if it is isomorphic to its complement. 5 a b 2 c 4 e 3 d

17 Isomorfism has.reflexive, 2.symetric and 3. transitive property:. Each graph is isomorphic to itself. 2. If f : V G V H is an isomorphism from G to H, then f is an isomorphism from H to G. f : V F V G g V G V H 3. Suppose that, and : are isomorphisms then composition f g is isomorphism from F to G. Isomorphic graphs have isomorphic complements. Sometimes it is easier to test the complements to decide, whether two graphs are isomorfic or not.

18 Are they isomorphic? 6 a b 5 2 e f 4 3 d c 5 6 a b Complementary 2 graphs e f 4 3 d c

19 To prove, that graphs are NOT isomorphic is enough to find, that in some structural properties ( e.g. number of edges, supgraphs, complements etc.) they differ.

20 Isomorphic classes P C n K K n n r, s - path with n vertices - cycle with n vertices - complete graph with n vertices - complete bipartite graph, r,s are indexes of two sets of vertices CK P 5 K 2,3

21 What is the number of simple graphs which can be created on a set of N vertices? m 2 N 2 From a set of four vertices we can create 64 different simple graphs, belonging to isomorphism classes. Here are the representatives of all classes

22 Connected and disconnected graphs Definition: A graph G is connected if it has u,v path whenever u, v are from V(G). Othervise G is disconnected. e e2 x e6 w e5 y e4 e7 e3 z

23 v, e, v,..., e, v k k Definition: A walk is a list of vertices and edges such that for i k the edge has endpoints v, i and v i. Edges and vertices can be repeated. e i A u,v-walk or u,v trail has first vertex u and last vertex v. A trail is a walk with no repeated edge. A u,v- path is a path whose vertices of degree one are u,v (endpoints). Other vertices are internal. Edges and vertices are not repeated. A walk and trial is closed if its endpoints are the same.

24 Euler s question Konigsberg bridges Walk of lenght 4: x.e2,w,e5,y,e6,x,e2,w Closed walk of lenght 5: x x.e2,w,e5,y,e6,x,e,w,e2,x e e2 e6 Trail of length 4 w e5 walk cycle trail y x.e2,w,e5,y,e6,x,e,w e3 e4 z e7 Cycle of lenght 3: Subgraph consisting of edges e,e6,e5 and vertices w,x,y. Deleting one of its edges we get a path of length 2.

25 Eulerian trial: Trial which visits every edge at most once. Euler s lemma: Connected graph has closed eulerian trial if and only if all of its vertices have even degree. x y z w e e 2 e 3 e 4 e 5 e 6 e 7 x y z w e e 3 e 6 e 7 Why?

26 Connectivity and adjacency G has w,z- path xw E G w and z are connected w is connected to z x and w are adjacent x is joined to w x is adjacent to w w a x c b d y e z

27 Component of graph Definitions: Maximal connected subgraph of graph G is a subgraph, which is connected and is not contained in any other connected subgraph of G. The components of graph G are its maximal connected subgraphs. 4 components of graph G

28 Proposition: Every graph with N nodes and m edges has at least N-m components. Proof: N isolated vertices create a graph with N components. Each added edge links vertices in the same component, or in different components merging them in one. So adding an edge decreases the number of components by or one. If we add m edges, the number of components is at least N-m. Definition: Cut edge or cut vertex is an edge or vertex whose deletion increases the number of components.

29 Cut vertex increases the number of subgraphs by many Cut edge increases the number of subgraphs always by one

30 Vertex degrees Degree of vertices are fundamental parameters of graph. k G v G G Definitions: Degree of a vertex v of graph G Maximum degree in graph G Minimum degree in graph G Graph G is regular if all degrees are equal. Graph G is k-regular if the common degree is k. N(v) neigborhood of vertex v, set of vertices adjacent to v N(G)- order of G, number of vertices in G e(g)- size of G, number of edges in G

31 First teorem of graph theory If G is a graph, then : Proof: each edge adds to the degree of two vertices. What can be concluded from the theorem?. Average vertex degree is: 2. Every graph has an even number of vertices of odd degree. No graph of odd order k is regular with the odd degree. 3. A k regular graph of with N vertices has Nk/2 edges. G e v k G V v 2 G G N G e G, G N G e 2 2

32 Proposition: If k is greather then zero, then a k regular bipartite graph has the same number of vertices in each partite set. Proof : Let G be an X,Y bigraph. Counting the edges according to their endpoints in X yealds e G k X.Counting them by their endpoints in Y yealds e G k Y. Thus k X k Y X Y, k And therefore. X Y

33 Extremal problems Extremal problem: asks for maximal or minimal value of a function over a class of objects.. Maximal number of edges in a simple graph (having no loops and multiple edges )is N N! N. N 2 N 2! 2! 2 2. The minimum number of edges in a connected graph with N vertices is N-. 3. If G is a simple N vertex graph with, then G is connected. G N 2

34 Model A: Let us have a set of N vertices and the probability p(n)=p, that each pair of vertices is connected by an edge. Each graph with m edges has a probability of occurrence p m N m p 2 The random variable G p denotes a realization of such graph. m=4

35 Model B: Let us have N vertices and m=m(n) edges and each such graph occurs with probability, where n is a possible number of edges if each vertex is connected to each and. Random variable denotes a graph generated in this way. Both models are the same for large number of vertices and probability m n 2 N n m G 2 / / N m n m p

36 Random graph theory studies the properties of the probability space associated with graphs with N nodes,. N Eordos, Renyi: Almost every graph has a property Q if the probability of having Q approaches one, as. N What is Q? Examples: -is a typical graph connected -does typical graph contain triangles, trees, etc. -etc.

37 Evolution of graph: One starts with isolated points and edges are succesively added. This corresponds to groving probability p in the A model. Goal: At what probability p particullar property Q in a graph arises? Many important properties appear suddenly, at certain critical probability p c N. lim N P N, p Q Probability to have graph with quality Q, number of nodes N and edge probability between nodes p, if, if p p c p p c N N N N

38 It can be shown, that the number of subgraphs F contained in graph G having number of nodes N and probability p of two vertices being connected is: E n F N a p m a m That is: The number of subgraph is finite if -number of nodes in subgraph -number of isomorfic subgraphs -number of edges p cn m

39 Evolutionary stage of graphs: It is a range of values for m(n) or p(n) in which a structural description of a typical graph does not change too much.. If there is a huge amount of vertices compared to the number of edges, then adding an edge to a randomly chosen pair makes an isolated edge. 2. Then, for certain threshold probabilities (or threshold number of edges) subtrees with vertices appear. Graph has no cycles. p c / N cn Number of nodes in tree Number of edges in tree

40 N c N p c / c 3. For probability with cycles start to appear. 4. The critical probability of having a complete subgraph of order is p c 2 / N cn cycle

41 Evolution of graphs intuitively

42 Conclusion: What to remember: Types of graphs (unoriented, oriented, simple, complete, bipartite... Important properties (degree, connectivity... ) What you need to understand: Lemmas at least on the intuitive level

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