graph.pptx
- 1. 1 Kenneth H. Rosen Chapter 7 Graphs Discrete Mathematics and It s Applications
- 2. 2 Definition 1. A simple graph G = (V, E) consi sts of V, a nonempty set of vertices, and E, a set of unordered pairs of distinct ele ments of V called edges. Simple Graph
- 3. 3 A simple graph San Francisco Denver Los Angeles New York Chicago Washington Detroit How many vertices? How many edges?
- 4. 4 A simple graph V = { Chicago, Denver, Detroit, Los Angeles, New York, San Francisco, Washington } SET OF VERTICES E = { {San Francisco, Los Angeles}, {San Francisco, Denver}, {Los Angeles, Denver}, {Denver, Chicago}, {Chicago, Detroit}, {Detroit, New York}, {New York, Washington}, {Chicago, Washington}, {Chicago, New York} } SET OF EDGES
- 5. 5 A simple graph San Francisco Denver Los Angeles New York Chicago Washington Detroit The network is made up of computers and telephone lines between computers. There is at most 1 telephone line between 2 computers in the network. Each line operates in both directions. No computer has a telephone line to itself. These are undirected edges, each of which connects two distinct vertices, and no two edges connect the same pair of vertices.
- 6. 6 Definition 2. In a multigraph G = (V, E) tw o or more edges may connect the same pair of vertices. A Non-Simple Graph
- 7. 7 A Multigraph San Francisco Denver Los Angeles New York Chicago Washington Detroit THERE CAN BE MULTIPLE TELEPHONE LINES BETWEEN TWO COMPUTERS IN THE NETWORK.
- 8. 8 Multiple Edges San Francisco Denver Los Angeles New York Chicago Washington Detroit Two edges are called multiple or parallel edges if they connect the same two distinct vertices.
- 9. 9 Definition 3. In a pseudograph G = (V, E) t wo or more edges may connect the sa me pair of vertices, and in addition, an edge may connect a vertex to itself. Another Non-Simple Graph
- 10. 10 A Pseudograph San Francisco Denver Los Angeles New York Chicago Washington Detroit THERE CAN BE TELEPHONE LINES IN THE NETWORK FROM A COMPUTER TO ITSELF (for diagnostic use).
- 11. 11 Loops San Francisco Denver Los Angeles New York Chicago Washington Detroit An edge is called a loop if it connects a vertex to itself.
- 13. 13 Definition 4. In a directed graph G = (V, E) the edges are ordered pairs of (not nec essarily distinct) vertices. A Directed Graph
- 14. 14 A Directed Graph San Francisco Denver Los Angeles New York Chicago Washington Detroit SOME TELEPHONE LINES IN THE NETWORK MAY OPERATE IN ONLY ONE DIRECTION . Those that operate in two directions are represented by pairs of edges in opposite directions.
- 15. 15 Definition 5. In a directed multigraph G = (V, E) the edges are ordered pairs of (not necessari ly distinct) vertices, and in addition there ma y be multiple edges. A Directed Multigraph
- 16. 16 A Directed Multigraph San Francisco Denver Los Angeles New York Chicago Washington Detroit THERE MAY BE SEVERAL ONE-WAY LINES IN THE SAME DIRECTION FROM ONE COMPUTER TO ANOTHER IN THE NETWORK.
- 17. 17 Types of Graphs TYPE EDGES MULTIPLE EDGES LOOPS ALLOWED? ALLOWED? Simple graph Undirected NO NO Multigraph Undirected YES NO Pseudograph Undirected YES YES Directed graph Directed NO YES Directed multigraph Directed YES YES
- 18. 18 Definition 1. Two vertices, u and v in an undirected gr aph G are called adjacent (or neighbors) in G, if {u, v} is an edge of G. An edge e connecting u and v is called incident with v ertices u and v, or is said to connect u and v. The v ertices u and v are called endpoints of edge {u, v}. Adjacent Vertices (Neighbors)
- 19. 19 a Degree of a vertex Definition 1. The degree of a vertex in an undirecte d graph is the number of edges incident with it, e xcept that a loop at a vertex contributes twice to the degree of that vertex. b g f e c d deg( d ) = 1
- 20. 20 a Degree of a vertex Definition 1. The degree of a vertex in an undirecte d graph is the number of edges incident with it, e xcept that a loop at a vertex contributes twice to the degree of that vertex. b g f e c d deg( e ) = 0
- 21. 21 a deg( b ) = 6 Degree of a vertex Definition 1. The degree of a vertex in an undirecte d graph is the number of edges incident with it, e xcept that a loop at a vertex contributes twice to the degree of that vertex. b g f e c d
- 22. 22 a deg( b ) = 6 Degree of a vertex Find the degree of all the other vertices. deg( a ) deg( c ) deg( f ) deg( g ) b g f e c d deg( d ) = 1 deg( e ) = 0
- 23. 23 a deg( b ) = 6 Degree of a vertex Find the degree of all the other vertices. deg( a ) = 2 deg( c ) = 4 deg( f ) = 3 deg( g ) = 4 b g f e c d deg( d ) = 1 deg( e ) = 0
- 24. 24 a deg( b ) = 6 Degree of a vertex Find the degree of all the other vertices. deg( a ) = 2 deg( c ) = 4 deg( f ) = 3 deg( g ) = 4 TOTAL of degrees = 2 + 4 + 3 + 4 + 6 + 1 + 0 = 20 b g f e c d deg( d ) = 1 deg( e ) = 0
- 25. 25 a deg( b ) = 6 Degree of a vertex Find the degree of all the other vertices. deg( a ) = 2 deg( c ) = 4 deg( f ) = 3 deg( g ) = 4 TOTAL of degrees = 2 + 4 + 3 + 4 + 6 + 1 + 0 = 20 TOTAL NUMBER OF EDGES = 10 b g f e c d deg( d ) = 1 deg( e ) = 0
- 26. 26 Theorem 1. Let G = (V, E) be an undirected graph G with e edges. Then deg( v ) = 2 e v V “The sum of the degrees over all the vertices equa ls twice the number of edges.” NOTE: This applies even if multiple edges and loops are pre sent. Handshaking Theorem
- 27. 27 Definition 6. A subgraph of a graph G = (V, E) is a graph H = (W, F) wher e W V and F E. Subgraph
- 28. 28 C5 is a subgraph of K5 C5 K5
- 29. 29 Definition 7. The union of 2 simple graphs G 1 = ( V1 , E1 ) and G2 = ( V2 , E2 ) is the simpl e graph with vertex set V = V1 V2 and edg e set E = E1 E2 . The union is denoted by G1 G2 . Union
- 30. 30 W5 is the union of S5 and C5 C5 W5 S5 a b c e d a c e a b c e d a b c e d f f
- 31. 31 p. 443 # 1 a, 2 a. p. 454 # 1-5, 12 adef, 19 abce, 44. Homework
- 32. 32 A simple graph G = (V, E) with n vertices can be represented by its adjacency matrix, A, where entry aij in row i and column j is 1 if { vi, vj } is an edge in G, aij = 0 otherwise. Adjacency Matrix
- 33. 33 Finding the adjacency matrix This graph has 6 vertices a, b, c, d, e, f. We can arrange them in that order. d W5 a b c e a c e f
- 34. 34 Finding the adjacency matrix a b c d e f d a 0 1 0 0 1 1 b c d e f FROM TO There are edges from a to b, from a to e, and from a to f W5 a b c e a c e f
- 35. 35 Finding the adjacency matrix a b c d e f d a 0 1 0 0 1 1 b 1 0 1 0 0 1 c d e f FROM TO There are edges from b to a, from b to c, and from b to f W5 a b c e a c e f
- 36. 36 Finding the adjacency matrix a b c d e f d a 0 1 0 0 1 1 b 1 0 1 0 0 1 c 0 1 0 1 0 1 d e f FROM TO There are edges from c to b, from c to d, and from c to f W5 a b c e a c e f
- 37. 37 Finding the adjacency matrix a b c d e f a 0 1 0 0 1 1 b 1 0 1 0 0 1 c 0 1 0 1 0 1 d e f FROM TO COMPLETE THE ADJACENCY MATRIX . . . d W5 a b c e a c e f
- 38. 38 Finding the adjacency matrix a b c d e f d a 0 1 0 0 1 1 b 1 0 1 0 0 1 c 0 1 0 1 0 1 d 0 0 1 0 1 1 e 1 0 0 1 0 1 f 1 1 1 1 1 0 FROM TO Notice that this matrix is symmetric. That is aij = aji Why? W5 a b c e a c e f
- 39. 39 Definition 1. A path of length n from u to v in an undirected graph is a sequence of edges e1, e2, . . ., en of the graph such that edge e1 has endpoints xo and x1 , edge e2 has endpoints x1 and x2 , . . . and edge en has endpoints xn-1 and xn , where x0 = u and xn = v. Path of Length n
- 40. 40 One path from a to e This path passes through ve rtices f and d in that order. d W5 a b c e a c f e
- 41. 41 One path from a to a This path passes through verti ces f, d, e, in that order. It ha s length 4. It is a circuit because it begins and ends at the same vertex. It is called simple because it d oes not contain the same edge more than once. d W5 a b c e a c f e
- 42. 42 Definition 3. An undirected graph is called connect ed if there is a path between every pair of distinc t vertices of the graph. IS THIS GRAPH CONNECTED? Path of Length n W5 a b c e a c e f
- 43. 43 Theorem 1. There is a simple path between ev ery pair of distinct vertices of a connected u ndirected graph. Theorem 1
- 44. 44 Theorem 2. Let G be a graph with adjacency m atrix A with respect to the ordering v1 , v2 , . . . , vn . The number of different paths of length r from vi to vj , where r is a posti ve integer, equals the entry in row i and col umn j of Ar. NOTE: This applies with directed or undirected edges, with multiple edges and loops allowed. Paths of Length r between Vertices