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- 1. Graph Theory Graphs are discrete structures consisting of vertices and edges that connects these vertices. There are several types of graphs that differ with respect to the kind and number of edges that can connect a pair of vertices. A simple graph G = (V, E) consists of V, a nonempty set of vertices, and E, a set of unordered pairs of distinct elements V called edges. A multigraph G = (V, E) consists of a set V of vertices, a set of E of edges, and a function f from E to {{u, v} | u, v є V, u ≠ v}. The edges e1 and e2 are called multiple or parallel edges if f(e1) = f(e2).
- 2. 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 є V}. An edge is loop of f(e) = {u, u} = {u} for some u є V. A directed graph (V, E) consists of a set of vertices V and a set of edges E that are ordered pairs of elements V. A directed multigraph G = (V, E) consists of a set of vertices V and a set of edges E, and d function f from E to {(u, v) | u, v ε V}. The edges e1 and e2 are multiple edges if f(e1) = f(e2).
- 3. Summary Graph Terminology Type Edges Multiple Loops edges Simple Graph Undirected No No Multigraph Undirected Yes No Pseudograph Undirected Yes Yes Directed Graph Directed No Yes Directed multigraph Directed Yes Yes
- 4. Basic Terminology Two vertices u and v in an undirected graph G are called adjacent (or neighbors) in G if {u, v} is an edge of G. If e = {u, v}, the edge e is called incident with the vertices u and v. The edge e is also said to connect u and v. The vertices u and v are called endpoints of the edge {u, v} The degree of a vertex in an undirected graph is the number of edges incident with it, except that a loop at a vertex contributes twice to the degree of that vertex. The degree of the vertex v is denoted by deg(v). A vertex of degree 0 is called isolated. A vertex is pendant if and only if it has degree 1.
- 5. Example: What are the degrees of the vertices in graphs G and H. b G a c b H e g e f a d d c
- 6. Theorem 1 The Handshaking Theorem Let G = (V, E) be an undirected graph with e edges. Then 2e = ∑ deg( v ) vεV Example: How many edges are there in a graph with 10 vertices each of degree 6. 2e = (10)(6) 2e = 60 e = 30
- 7. Theorem 2 An undirected graph has an even number of vertices of odd degree. When (u, v) is an edge of the graph G with directed edges, u is said to adjacent to u and v is said to be adjacent from u. The vertex u is called the initial vertex of (u, v), and v is called the terminal or end vertex of (u, v). The initial vertex and terminal vertex of a loop are the same.
- 8. In a graph with directed edges the in-degree of a vertex v, denoted by deg – (v), is the number of edges with v as their terminal vertex. The out-degree of v, denoted by deg + (v), is the number of edges with v as their initial vertex. (Note that a loop at a vertex contributes 1 to both the in-degree and the outdegree of this vertex.) Let G = (V, E) be a graph with directed edges. Then − + ∑ deg ( v ) = ∑ deg ( v ) = E vεV vεV
- 9. Example: Find the in-degree and out-degree of each vertex in the graph shown below with directed edges. a b c f e d
- 10. Cycles – the cycle Cn, n ≥ 3, consists of n vertices v1, v2,…., vn and edges {v1, v2}, {v2, v3},…, {vn-1, vn}, and {vn, v1}. C3 C4 C5 C6
- 11. Wheels – we obtain the wheel Wn when we add an additional vertex to the cycle Cn, for n ≥ 3, and connect this new vertex to each of the n vertices C n, by new edges. W3 W4 W5 W6
- 12. Representing Graphs Adjacency lists – is used to represent a graph with no multiple edges which specify the vertices that are adjacent to each vertex of the graph. B A E B C A D C E D Vertex Adjacent Vertices Vertex Adjacent Vertices A B C D E A B C D E B, C, E A A, D, E C, E A, C, D B, C, D, E B, D A, C, E B, C, D
- 13. Adjacency matrix (Ag), with respect to this listing of the vertices, is the n x n zero-one matrix with 1 as its (i, j)th entry when vi and vj are adjacent, and 0 as its (i, j) entry when they are not adjacent. In other words, if its adjacency matrix is A = [aij], then 1 aij = 0 a c If {vi, vj} is an edge of G, otherwise. 0 b 1 1 d 1 1 0 1 0 1 1 0 0 1 0 0 0 a d 0 b 1 1 c 0 1 0 0 1 1 0 0 1 0 1 1 0
- 14. Adjacency matrices can also be used to represent undirected graphs with loops and with multiple edges. A loop at the vertex ai is represented by 1 at the (i, j)th position of the adjacency matrix. When multiple edges are present, the adjacency matrix is no longer zero-one matrix, since the (i, j) entry of this matrix equals the number of edges that are associated to {ai, aj}. All undirected graphs, including multigraphs and pseudographs, have symmetric adjacency matrices 0 3 0 2 a b d c 3 0 0 1 0 0 1 2 2 1 2 0
- 15. Incidence matrices – Let G = (V, E) be an undirected graph. Suppose that v1, v2, …, vn are the vertices and e1, e2, …., em are the edges of G. Then the incidence matrix with respect to this ordering of V and E is the n x m matrix M = [mij], where 1 mij = 0 v1 e1 v4 e3 e2 when edge ej is incident with vi, otherwise. v 2 e6 e4 v5 v3 e5 1 0 0 1 0 1 0 0 0 0 0 1 1 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 1 0
- 16. Exercises: I. Represent the following adjacency and incidence matrix. e1 e2 b a a b 1. 2. e3 e3 e1 e4 e4 e5 e c e2 d d e9 a 3. e1 d e11 e7 e5 e2 e3 e6 e8 e10 b e4 c e12 e6 e7 e8 5 c using
- 17. II. Draw the graph represented by the given adjacency matrix. 1 0 1 1 2 0 0 0 1 4. 5.2 0 2 1 1 1 0 2 1 1 2 6. 0 1 2 0 1 0 3 0 3 1 1 0 1 0 III. Draw the graph and represent it using adjacency matrix. 7. K4 8. K6 9. C10 10. W6
- 18. Shortest Path Problems Applications: mileage, flight times, fares, distance, response time, least rates, and so on. Weighted graphs are graphs that have a number assigned to each edge. The length of a path in a weighted graph is the sum of the weights of the edges of this path.
- 19. Dijktra’s Algorithm (a shortest path algorithm) Theorem 1 Dijktra’s algorithm finds the length of a shortest path between two vertices in a connected simple undirected weighted graph. Theorem 2 Dijktra’s algorithm uses O(n2) operations (additions and comparisons) to find the length of the shortest path between two vertices in a connected simple undirected weighted graph.
- 20. Find the length of the shortest path between a and z in the given weighted graph. b d 5 f 5 7 4 3 2 a 1 2 z 3 4 c 6 e 5 g
- 21. The traveling salesman problem It asks for the circuit of minimum total weigh in a weighted, complete, undirected graph that visits each vertex exactly once and returns to its starting point. Saginaw 113 Ground Rapids 56 137 142 98 147 167 Detroit 135 Kalamazoo 58 133 Toledo
- 22. Route Total distance (miles) D–T–G–S–K–D D–T–G–K–S–D D–T–K–S–G–D D–T–K–G–S–D D–T–S–K–G–D D–T–S–G–K–D D–S–T–G–K–D D–S–T–K–G–D D–S–K–T–G–D D–S–G–T–K–D D–G–S–T–K–D D–G–T–S–K–D 610 516 588 458 540 504 598 576 682 646 670 728
- 23. Euler and Hamilton Paths An Euler circuit in a graph G is s simple circuit containing every edge of G. An Euler path in G is a simple path containing every edge of G. Theorem 1 A connected multigraph has an Euler circuit if and only if each of its vertices has even degree. Theorem 2 A connected multigraph has an Euler path but not Euler circuit if and only if it has exactly two vertices of odd degree.
- 24. A path x0, x1, …., xn-1, xn in the graph G = (V, E) is called a Hamilton path if V = {x0, x1, …, xn-1, xn} and xi ≠ xj for 0 ≤ i , j ≤ n. A circuit x0, x1, …, xn-1, x0 (with n > 1) in a graph G = (V, E) is called Hamilton circuit if x0, x1, …, xn-1, xn is a Hamilton path. Theorem 3 If G is connected simple graph with n vertices where n ≥ 3, then G has a Hamilton circuit if the degree of each vertex is at least n/2.
- 25. Exercises: 1. Find the shortest path from a to z and a to x. d 30 a 20 42 b 35 75 40 x c 55 75 85 e 45 z
- 26. 2. Solve the traveling salesman problem for the following graph by finding the total weight of all the circuits and determining a circuit with minimum total weight. 3 a b 7 10 2 8 5 e 1 4 c 9 d 6

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