Foundations of Discrete Mathematics
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1 Foundations of Discrete Mathematics Chapters 9 By Dr. Dalia M. Gil, Ph.D.
2 Graphs Graphs are discrete structures consisting of vertices and edges that connect these vertices.
3 Graphs A graph is a pair (V, E) of sets, V nonempty and each element of E a set of two distinct elements of V. The elements of V are called vertices; the elements of E are called edges.
4 Graphs If e is an edge, then e = {v, w}, where v and w are different elements of V called the end vertices of ends of e. The vertices v and w are said to be incident with the edge vw. The edge vw is incident with each vertex.
5 Graphs Two vertices are adjacent if they are the end vertices of an edge. Two vertices are adjacent if they have a vertex in common. The number of edges incident with a vertex v is called the degree of that vertex and is denoted deg v.
6 Graphs If deg v is an even number, then v is said to be an even vertex. If deg v is an odd number, then v is odd vertex. A vertex of degree 0 is said to be isolated.
7 Subgraph A graph G 1 is a subgraph of another graph G if and only if the vertex and edge sets of G 1 are, respectively, subsets of the vertex and edge sets of G.
8 Example: Subgraph A graph G and three subgraphs G 1, G 2, and G 3 Discrete Mathematics with Graph Theory. Third Edition, by E. G. Goodaire and M. Parmenter Prentice Hall, page 290
9 A Bipartite Graph A bipartite graph is one whose vertices can be partitioned into two (disjoint) sets V 1 and V 2, called bipartition sets in such a way that every edge joins a vertex in V 1 and a vertex in V 2.
10 A Bipartite Graph The complete bipartite graph on bipartition sets of m vertices and n vertices, respectively, is denoted K m,n.
11 Bipartite Graphs Three bipartite graphs, two of which are complete. Discrete Mathematics with Graph Theory. Third Edition, by E. G. Goodaire and M. Parmenter Prentice Hall, page 291
12 Bipartite Graphs A graph and a way to show it is bipartite. Discrete Mathematics with Graph Theory. Third Edition, by E. G. Goodaire and M. Parmenter Prentice Hall, page 291
13 Euler The sum of the degrees of the vertices of a pseudograph is an even number equal to twice the number of edges. In symbols, if G(V, E) is a pseudograph, then deg v = 2 E v V
14 Example: Euler The graph has 8 vertices of degree 3 deg v = 8(3) = 24 =2 E v V it must have 12 edges. Discrete Mathematics with Graph Theory. Third Edition, by E. G. Goodaire and M. Parmenter Prentice Hall, page 291
15 Graphs There are several different types of graphs that differ with respect to the kind and number of edges that connect a pair of vertices. Problems in almost every conceivable discipline can be solved using graph models.
16 Types of Graphs Simple graph. Multigraph. Pseudograph. Directed graph. Direct multigraph.
17 A Simple Graph A simple graph G = (V, E) consists of V, a nonempty set of vertices, and E, a set of unordered pairs of distinct elements of V called edges.
18 Example of a Simple Graph A computer network that represents computers (vertices) and telephone lines (undirected edges) that connect two distinct vertices. Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, page 538
19 A Multigraph A multigraph G = (V, E) consists of a set V of vertices, a set E of edges, and a function f from E to {{u, v} u, v V, u v}. The edges e 1 and e 2 are called multiple or parallel edges if f(e 1 ) = f(e 2 ).
20 Example of a Multigraph This graph consist of vertices and undirected edges between these vertices with multiple edges between pairs of vertices allowed (two or more edges may connect the same pair of vertices). Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, page 538
21 A Pseudograph A pseudograph G = (V, E) consists of a set V of vertices, a set E of edges, and a function f from E to {{u, v} u, v V}. An edge is a loop if f (e) = {u, v} = {u}
22 Example of a Pseudograph A computer network may contain vertices with loops, which are edges from a vertex to itself. Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, page 539
23 A Direct Graph A direct graph (V, E) consists of a set of vertices V and a set of edges E that are ordered pairs of elements of V.
24 Example of a Direct Graph A network may not operate in both directions. In this case an arrow pointing from u to v to indicate the direction of the edge (u, v) is used. Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, page 539
25 A Directed Multigraph A directed multigraph G = (V, E) consists of a set V of vertices, a set E of edges, and a function f from E to {(u, v) u, v V}. The edges e 1 and e 2 are multiple edges if f(e 1 ) = f(e 2 ).
26 Example of a Directed Multigraph The algorithm uses a finite number of steps, since it terminates after all the integers in the sequence have been examined. Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, page 540
27 Graph Terminology Type Edges Multiple Edges Allowed? Simple Graph Undirected No No Multigraph Undirected Yes No Pseugograph Undirected Yes Yes Directed Graph Directed No Yes Directed Multigraph Directed Yes Yes Loops Allowed? Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, page 540
28 Niche Overlap Graphs in Ecology The competition between species in an ecosystem can be modeled using a niche overlap graph. Each species is represented by a vertex.
29 Niche Overlap Graphs in Ecology An undirected edge connects two vertices if the two species represented by these vertices compete. Two species are connected if the food resources they use are the same. Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, page 541
30 Acquaintanceship Graph To represent whether two people know each other (whether they are acquainted) Each person is represented by a vertex. An undirected edge connects two people when they know each other. Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, page 541
31 Influence Graph In studies of group behavior it is observed that certain people can influence the thinking of others. A directed graph can model this behavior. Each person is represented by a vertex. There is a directed vertex from vertex a to vertex b when person a influences person b.
32 Influence Graph (A Directed Graph) Deborah can influence Brian, Fred, and Linda, but no one can influence her. Yvonne and Brian can influence each other. Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, page 542
33 Round-Robin Tournaments A tournament where each team plays each other team exactly once is called a round-robin tournament. In this case a directed graph is used. Each team is represented by a vertex.
34 Round-Robin Tournaments (a, b) is an edge if team a beats team b. Team 1 is undefeated in this tournament Team 3 is winless. Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, page 542
35 Call Graphs Graphs can be used to model telephone calls made in a network, such as a longdistance telephone network. A directed multigraph can be used. Each telephone is a vertex Each telephone call is represented by a directed edge.
36 Call Graph using Directed Graph Three calls have been made from to and two in the other direction. Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, page 542
37 Call Graph using Directed Graph One call has been made from to Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, page 542
38 Call Graph using Undirected Graph When we care only whether there has been a call connecting two telephone numbers, an undirected graph is used. Each edge tells us whether there has been a call between two numbers. Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, page 542
39 Precedence Graph and Concurrent Processing Computer programs can be executed more rapidly by executing certain statements concurrently. It is important not to execute a statement that requires results of statement not yet executed.
40 Precedence Graph and Concurrent Processing The dependence of statements on previous statements can be represented by a directed graph. There is an edge from one vertex to a second vertex if the statement represented by the second vertex cannot be executed before the statement represented by the first vertex has been executed.
41 Precedence Graph and Concurrent Processing In this section of a computer program the statement S 5 cannot be executed before S 1, S 2, and S 4 are executed. Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, page 543
42 Representing Graphs One way to represent a graph without multiple edges is to list all the edges of this graph. Another way to represent a graph with no multiple edges is to use adjacent list, which specify the vertices that are adjacent to each vertex of the graph.
43 Example: Representing Graphs Use adjacent lists to describe the simple graph given in the figure. Vertex a b c d e Adjacent Vertices b, c, e a a, d, e c, e a, c, d Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, page 557
44 Example: Representing Graphs Represent the directed graph shown in the figure by listing all the vertices that are the terminal vertices of edges starting at each vertex of the graph in the figure. Initial Vertex a b c d e Terminal Vertices b, c, d, e b, d a, c, e b, c, d Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, page 558
45 The Adjacency Matrix A (A G ) If A = {a ij } is the adjacency matrix A of G, then a ij = 1 if {v i, v j } is an edge of G and a ij = 0 otherwise. The adjacency matrix of a simple graph is symmetric if a ij = a ji. A simple graph has no loops, so each entry a ii = 1, 2,, n is 0.
46 Example: Adjacent Matrix Use an adjacent matrix to represent the graph. Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, page 558
47 Example: Adjacent Matrix draw a graph with the adjacent matrix with respect to ordering of vertices a, b, c, d. Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, page 559
48 Example: Adjacent Matrix Use an adjacent matrix to represent the pseudograph shown in the figure. Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, page 559
49 Incidence Matrices Let G = (V, E) be an undirected graph. Suppose that v 1,v 2,, v n are the vertices and e 1, e 2,, e m are the edges of G. Then the incidence matrix with respect to this ordering of V and E is n x m matrix M = {m ij }, where m ij = 1 when edge ei is incident with vi, m ij = 0 otherwise
50 Example: Incidence Matrix Represent the graph with an incidence matrix. Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, page 560
51 Example: Incidence Matrix Represent the pseudograph with an incidence matrix. Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, page 560
52 Isomorphism of Graphs We often need to know whether it is possible to draw two graph in the same way. In chemistry different compounds can have the same molecular formula but can differ in structure. Such compounds will be represented by graph that cannot be drawn in the same way.
53 Isomorphism of Graphs The simple graphs G 1 = (V 1, E 1 ) and G 2 = (V 2, E 2 ) are isomorphic if there is a one-to-one and onto function f from V 1 to V 2, with the property that a and b are adjacent in G 1 if and only if f(a) and f(b) are adjacent in G 2, for all a and b in V 1. Such a function f is called an isomorphism.
54 Isomorphism of Graphs Two simple graphs are isomorphic, if there is a one-to-one correspondence between vertices of the two graphs that preserves the adjacency relationship. Isomorphism of simple graph is an equivalence relation. Isomorphism comes from the Greek root isos for equal and morphe for form.
55 Example: Isomorphism Show that the graphs G = (V, E) and H = (W, F) are isomorphic The function f with f(u 1 )= v 1, f(u 2 )= v 4, f(u 3 )= v 3, f(u 4 )= v 2 is a one-to-one correspondence between V and W. Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, page 561
56 Example: Isomorphism (cont.) Show that the graphs G = (V, E) and H = (W, F) are isomorphic The adjacent vertices in G are u 1 and u 2, u 1 and u 3, u 2 and u 4, u 3 and u 4 Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, page 561
57 Example: Isomorphism (cont.) Show that the graphs G = (V, E) and H = (W, F) are isomorphic Each of the pairs f(u 1 )= v 1 and f(u 2 )= v 4 f(u 1 )= v 1 and f(u 3 )= v 3 f(u 2 )= v 4 and f(u 4 )= v 2 f(u 3 )= v 3 and f(u 4 )= v 2 are adjacent in H. Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, page 561
58 Isomorphism of Graphs Two simple graphs are not isomorphic, if they do not share a property that isomorphic simple graphs must both have. This property is call an invariant with respect to isomorphism of simple graph.
59 Isomorphism: Some Invariants Isomorphic simple graphs must have the same number of vertices, since there is a one-to-one correspondence between the sets of vertices of the graphs.
60 Isomorphism of Graphs Isomorphic simple graphs must have the same number of edges, because the one-to-one correspondence between vertices establishes a one-to-one correspondence between edges.
61 Isomorphism of Graphs The degree of the vertices in isomorphic simple graph must be the same. A vertex v of degree d in G must correspond to a vertex f(v) of degree d in H, since a vertex w in G is adjacent to v if and only if f(v) and f(w) are adjacent in H.
62 Examples of Graphs not Isomorphic Show that the graphs are not isomorphic. G and H have 5 vertices and 6 edges H has a vertex of degree one (e). G has no vertices of degree one. G and H are not isomorphic. Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill,
63 Examples of Graphs not Isomorphic Determine whether the graphs are isomorphic. G and H have 8 vertices and 10 edges. G and H have 4 vertices of degree 2 and 4 of degree 3. Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, page 561
64 Examples of Graphs not Isomorphic Determine whether the graphs are isomorphic. In G deg(a) = 2, and a must correspond to either t, u, x, or y in H. t, u, x, and y are adjacent to another vertex of degree 2, which is not true for a in G. G and H are not isomorphic. Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, page 561
65 Examples of Graphs not Isomorphic Determine whether the graphs are isomorphic using subgraphs. In G deg(a) = 2, and a must correspond to either t, u, x, or y in H. t, u, x, and y are adjacent to another vertex of degree 2, which is not true for a in G. Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, page 561
66 Examples of Graphs not Isomorphic Determine whether the graphs are isomorphic using subgraphs. Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, page 562
67 Examples of Graphs not Isomorphic Determine whether the graphs are isomorphic using subgraphs. The subgraphs of G and H made up of vertices of degree 3 and the edges connecting then must be isomorphic if these two graphs are isomorphic. G and H are not isomorphic. Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, page 562
68 Examples of Graphs Isomorphic Determine whether the graphs are isomorphic using adjacency matrix. Both G and H have 6 vertices and 7 edges. Both have four vertices of degree 2 and 2 vertices of degree three. Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, page 562
69 Examples of Graphs Isomorphic Determine whether the graphs are isomorphic using adjacency matrix. Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, pp
70 Examples of Graphs Isomorphic Determine whether the graphs are isomorphic using adjacency matrix. Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, pp
71 Examples of Graphs Isomorphic Determine whether the graphs are isomorphic using adjacency matrix. A G = A H, G and H are isomorphic. Discrete Mathematics and its Applications. Fifth Edition, by Kenneth H. Rosen. McGraw Hill, p 563
72 Topics covered Graph. Definitions and basic properties. Adjacency matrices. Isomorphism.
73 Reference Discrete Mathematics with Graph Theory, Third Edition, E. Goodaire and Michael Parmenter, Pearson Prentice Hall, pp
74 Reference Discrete Mathematics and Its Applications, Fifth Edition, Kenneth H. Rosen, McGraw- Hill, pp
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