unit 5 graphs & trees 1 it discipline itd1111 discrete mathematics & statistics stdtlp unit...

IT Discipline ITD1111 Discrete Mathematics & Statistics STDTLP

Unit 5 Graphs & Trees

Unit 5 Discrete Mathematics and Statistics

Graphs and Trees

Because many situations and structures give rise to graphs, graph theory has become an important mathematical tool in a wide variety of subjects, ranging from operations research, computing science and linguistics [ 語言學 ] to chemistry and genetics

[ 基因 ].

Recently, there has been considerable interest in tree structures arising in computer science and artificial intelligence. We often organize data in a computer memory store or the flow of information through a system in tree structure form. Indeed, many computer operating systems are designed to be tree structures.

Introduction

road map

Diagrammatic Representation

5.1 What is a Graph

electrical network

An undirected graph G consists of a non-empty set of

elements, called vertices [ 頂點 ],

and a set of edges [ 邊 ], where each edge is associated with

a list of unordered pairs of either one or two vertices called

its endpoints. The correspondence from edges to end-points

is called an edge-endpoint function.

5.2 Undirected Graph

Edge e1 e2 e3 e4

Endpoints {u, v} {u, w} {v, w} {w, z}

vertex-set V(G) = {u, v, w, z}

edge-set E(G) = {e1, e2, e3, e4}

edge-endpoint function

Example 5.2-1 (Undirected Graph)

A directed graph G consists of vertices, and a

set of edges, where each edge is associated with a

list of ordered pair endpoints

5.3 Directed Graph

Edge e1 e2 e3 e4

Endpoints (u, v) (w, u) (w, v) (z, w)

vertex-set V(G) = {u, v, w, z}

edge-set E(G) = {e1, e2, e3, e4}

Example 5.3-1 (Directed Graph)

Two or more edges joining the same pair of vertices are called multiple edges and an edge joining a vertex to itself is called a loop.

The degree of a vertex is the number of edges meeting at a given vertex.

5.4 Multiple edges , Loop and Degree

V(G) = {u, v, w, z}

E(G) = {e1, e2, e3, e4, e5, e6, e7}

Multiple edges

e2 e3 e4

e5e6e7

deg u = 6, deg v = 5, deg w = 2, deg z = 1

total degree = 6 + 5 + 2 + 1 = 14

Edge e1 e2 e3 e4 e5 e6 e7

Edge points {u, u} {u, w} {v, w} {v, z} {u, v} {u, v} {u, v}

Example 5.4-1 (Multiple edges , Loop and Degree)

A graph G is connected if

there is a path in G between

any given pair of vertices, and

disconnected otherwise.

Connected Graph

Disconnected Graph

components

5.5 Connected and Disconnected Graphs

K2K3 K4

A complete graph on n vertices , denoted Kn, is a simple graph with n vertices v1, v2, ……vn whose set of edges contains exactly one edge for each pair of distinct vertices.

5.6 Complete graph

In any graph, the sum of all the vertex-degrees is equal to twice the number of edges.(see

e.g.3)

No. of edges =7

total degree = 6 + 5 + 2 + 1 = 14

As the consequences of the Handshaking Lemma, the

sum of all the vertex-degrees is an even number and the

number of vertices of odd degree is even.(e.g. 3 odd deg

vertices are:v and z)

5.7 Handshaking Lemma [輔助定理 ]

2010v 1

v 1 v2 v3 v4

A(G) =

Adjacency [ 毗鄰 ] Matrix A(G)

5.8 Matrix Representations of Undirected Graph

Incidence [ 關聯 ] Matrix I(G)

1111100

0000110

1100011

0011001e1 e2 e3 e4 e5 e6 e7

I(G) =

Adjacency Matrix A(G)

e1e2 e3

010v 1

v 1 v2 v3

A(G) =

5.9 Matrix Representations of Directed Graph

Incidence Matrix I(G)

e1e2 e3

e1 e2 e3 e4 e5

I(G) =

A walk of length k between v1 and v8 in a graph G is a succession of k edges of G of the form v1e1v2, v2e2v3, v3e3v4, … ,v7e7v8.

We denote this walk by v1e1v2e2v3e3v4e4v5e5v6e6v7e7v8.

5.10 Walk

If all the edges (but not necessarily all the vertices) of a walk are different, then the walk is called a path.

If , in addition, all the vertices are different, then the trail is called simple path.

A closed path is called a circuit.

A simple circuit is a circuit that does not have any other repeated vertex except the first and last.

5.11 Paths and circuits

v4v2simple path

v1 v2 v4 v1 v3 v5 is a path v1 v2 v4 v3 v5 is a simple path

Example 5.11-1 (Paths)

circuit

simple circuit

v1 v2 v4 v3 v5 v4 v1 is a circuit v1 v2 v4 v5 v3 v1 is a simple circuit

Example 5.11-2 (Circuits)

Repeated Edge Repeated Vertex

Starts and Ends at Same Point

WalkPathSimple pathClosed walkCircuitSimple circuit

AllowedNoNo

AllowedAllowed

NoAllowedAllowed

First and last only

AllowedAllowed

NoYesYesYes

Summarized Table

If G is a graph with vertices v1, v2, …… vm and A(G) is the adjacency matrix of G, then for each positive integer n,

the ijth entry of An = the number of walks of length n from vi to vj.

5.12 Counting Walks of Length N

Adjacency matrix A(G)

Example 5.12-1 (Counting Walks)

one walk of length 2 connecting v1 to v1

(v1e1v2e1v1)

two walks of length 2 connecting v1 to v3

(v1e1v2e3v3, v1e1v2e4v3)

Example 5.12-1 (cont.)

The objective of this algorithm is to find the shortest

path from vertex S to vertex T in a given network.

We demonstrate the steps of the algorithm by

finding the shortest distance from S to T in the

following network:

5.13 The Shortest Path Algorithm

Example 5.13-1 (Shortest Path Alg.)

Example 5.13-1(Cont.)

Therefore, the shortest path from S to T is SBCT with path length 10

Initialization

Assign to vertex S potential 0.

Label each vertex V reached directly from S with distance from S to V.

Choose the smallest of these labels, and make it the potential of the corresponding vertex or vertices.

5.13 The Shortest Path Algorithm Extra Notes

Step Visited A B C D T

Init B 7S 4S 9S 7S X

1 A 5SB 4S 7SB 7S X

2 D 5SB 4S 7SB 7S 11SBA

1 A 5SB 4S 7SB 7S X

3 C 5SB 4S 7SB 7S 11SBA

1 A 5SB 4S 7SB 7S X

3 C 5SB 4S 7SB 7S 11SBA

4 T 5SB 4S 7SB 7S 10SBC

A connected graph G is Eulerian if there is a closed path which includes every edge of G; such a path is called an Eulerian path, and if the path which starts and ends on the same vertex, then it is called Eulerian circuit.

5.14 Eulerian Graphs

Theorem (Eulerian Graph)

Let G be a connected graph. Then G has an Eulerian circuit if and only if every vertex of G has even degree,and has an Eulerian path if no more than two vertices have an odd degree.

Which of the following graph(s) is/are Eulerian?

(a) (b) (c) (d)

The four places (A, B, C and D) in the city of Königsberg were interconnected by seven bridges (p, q, r, s, t, u and v) as shown in the following diagram. Is it possible to find a route crossing each bridge exactly once ?

Königsberg

5.15 Königsberg bridges problem

By the theorem, it is not a Eulerian graph. It follows that there is no route of the desired kind crossing the seven bridges of Königsberg.

We can express the Königsberg bridges problem in terms of a graph by taking the four land areas as vertices and the seven bridges as edges joining the corresponding pairs of vertices.

Answer (Königsberg bridges problem)

STEP 1 Choose a starting vertex u with a vertex of odd degree.

STEP 2 At each stage, traverse [橫越 ] any available edge, choosing a bridge only if there is no alternative.

STEP 3 After traversing each edge, erase [擦掉 ] it (erasing any

vertices of degree 0 which result), and then choose

another available edge.

STEP 4 STOP when there are no more edges.

5.16 Fleury’s Algorithm

Find the Eulerian path of the following graph.

Example 5.16-1 (Fleury’s Algorithm)Constructing Eulerian paths and circuits

Starting at u, we may choose the edge ua, followed by ab. Erasing these edges and the vertex a give us graph (b)

Example 5.16-1 (Cont.)

We cannot use the edge bu since it is a bridge, so we choose the edge bc, followed by cd and db. Erasing these edges and the vertices c and d give us graph (c)b f

We have to traverse the bridge bu. Traversing the cycle uefu completes the Eulerian path. The path is therefore uabcdbuefu.

A connected graph G is Hamiltonian if there is a cycle which includes every vertex of G ; such a cycle is called a Hamiltonian cycle.

Hamiltonian Graphrs

Hamiltonian Cycle

5.17 Hamiltonian Graphs

Let G be a simple graph with n vertices, where n 3. If deg v n/2 for each vertex v, then G is Hamiltonian.

For the above graph, n = 6 and deg v = 3 for each vertex v, so this graph is Hamitonian by Dirac’s theorem.

5.18 DIRAC’S THEOREM

Let G be a simple graph with n vertices, where n 3.

If deg v + deg w n,

for each pair of non-adjacent vertices v and w, then G is Hamiltonian.

5.19 ORE’S THEOREM

n = 5 but deg u = 2, so Dirac’s theorem does not apply.

However, deg v + deg w 5 for all pairs of non-adjacent vertices v and w, so this graph is Hamiltonian by Ore’s theorem.

Example 5.19-1 (Ore’s Thm)

A tree is a connected graph which contains no cycles.

Properties of Tree• Every tree with n vertices has exactly n 1 edges.

• Any two vertices in a tree are connected by exactly one path.

• Each edge of a tree is a bridge.

5.20 Trees

• T is connected and has n 1 edges.• T has n 1 edges and contains no cycles.• T is connected and each edge is a bridge.• Any two vertices of T are connected by exactly one path.

5.21 Alternative definitions of the tree

Let G be a connected graph. A spanning tree in G is a subgraph of G that includes all the vertices of G and is also a tree. The edges of the tree are called branches.

5.22 Spanning [ 包括 , 延伸 ] Trees

A graph G

Spanning Trees

Example 5.22-1 (Spanning Trees)

Let T be a spanning tree of minimum total weight in a connected weighted graph G. Then T is a minimum spanning tree of G.

5.23 Minimum Spanning Tree

Illustrated Example

5.24 Greedy Algorithm

Iteration 1 (Choose an edge of minimum weight)

5.24 Greedy Algorithm (cont.)

Iteration 2 (Select the vertex from unconnected set {A,B,C} that is closest to connected set {D,E})

Iteration 3 (Select the vertex from unconnected set {A,C} that is closest to connected set {B,D,E,})

Iteration 4 (Select the vertex from unconnected set {C} that is closest to connected set {A,B,D,E})

unit 5 graphs & trees 1 it discipline itd1111 discrete mathematics & statistics stdtlp unit...

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