Discrete Structure Chapter 7 Graphs

8/10/2019 Discrete Structure Chapter 7 Graphs

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GRAPH

Learning Outcomes

Students should be able to:

Explain basic terminology of a graph

Identify Euler and Hamiltonian cycle

Represent graphs using adjacency matrices


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Contents

Introduction Paths and circuits

Matrix representations of graphs


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Introduction to Graphs

DEF: A simple graphG= (V,E) consistsof a non-empty set Vof vertices(ornodes) and a set E(possibly empty) of

edgeswhere each edge is associated witha set consisting of either one or twovertices called its endpoints.

Q: For a set Vwith n elements, how manypossible edges there?


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Terminology

Loop, parallel edges, isolated, adjacent Loop - an edge connects a vertex to itself

Two edges connect the same pair of

vertices are said to be parallel. Isolated vertexunconnected vertex.

Two vertices that are connected by an

edge are called adjacent.An edge is said to be incident on each of

its end points.


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Example of a graph

Vertex set = {u1, u2, u3} Edge set = {e1, e2, e3, e4}

e1, e2, e3 are incident on u1

u2 and u3 are adjacent to u1 e4is a loop

e2and e3are parallel


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618 November 2014 Graphs and Trees 6

Types of Graphs

Directedordercounts whendiscussing edges

Undirected(bidirectional)

Weightedeach

edge has a valueassociated with it

Unweighted


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Examples

http://richard.jones.name/google-hacks/google-cartography/google-cartography.htm


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Special Graphs

Simpledoes not have any loops or paralleledges Complete graphsthere is an edgebetween

every possible tuple of vertices

Bipartite graphV can be partitioned into V1and V2, such that: (x,y)E (xV1 yV2) (xV2 yV1)

Sub graphs G1 is a subset of G2 iff

Every vertex in G1 is in G2 Every edge in G1 is in G2

Connected graphcan get from any vertex toanother via edges in the graph


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Complete Graphs

there is an edgebetween every possibletuple of vertices. |e| = C(n,2) = n. (n-1)/2


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Bipartite graph

A graph is bipartiteif its vertices can bepartitioned into two disjoint subsets U and

V such that each edge connects a vertex

from U to one from V.


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Complete bipartite

A bipartite graph is a complete bipartitegraph if every vertex in U is connected to everyvertex in V. If U has melements and V has n,then we denote the resulting complete bipartite

graph by Km,n. The illustration shows K3,2


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Degree of Vertex

Defined as the number of edges attached(incident) to the vertex. A loop is counted twice.


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Handshake Theorem

If Gis any graph, then the sum of thedegrees of all the vertices of Gequalstwice the number of edges of G.

Specifically, if the vertices of Gare v1, v2,, v

n, where nis a nonnegative integer,

then:

The total degree of G= d(v1)+d(v2)++d(vn)= 2 (the number of edges of G)


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Total degree of a graph is even

Prove that the total of the degrees of allvertices in a graph is even.

Since the total degree equals 2 times of

edges, which is an integer, the sum of alldegree is even.


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Whether certain graphs exist

Draw a graph with the specified propertiesor show that no such graph exists.

(a) A graph with four vertices of degrees

1,1,2, and 3(b) A graph with four vertices of degrees

1,1,3 and 3

(c) A simple graph with four vertices ofdegrees 1,1,3 and 3


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Even no. of vertices with odd degree

In any graph, there are an even numberof vertices with odd degree

Is there a graph with ten vertices of

degrees 1,1,2,2,2,3,4,4,4, and 6?


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Learning Outcomes

Students should be able to:Explain basic terminology of a graph




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Seven Bridges of Knigsberg

Is it possible for a person to take a walk aroundtown, starting and ending at the same location andcrossing each of the seven bridges exactly once?


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Definitions

Terminology - Walk, path, simple path,circuit, simple circuit.

Walk from two vertices is a finite alternating

sequence of adjacent vertices and edges Trivial walk from v to v consists of single vertex

v0e1v1e2envn


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Path

Patha walk that does not contain arepeated edge (may have a repeatedvertex)

v0e1v1e2envn where all the ei are distinct Simple patha path that does not contain

a repeated vertex

v0e1v1e2envn where all the ei and vjaredistinct. e1is represented by {v0,v1}.


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Example - path

Path

v

Simple path

w


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Circuit

Closed walkstarts and ends at samevertex

Circuita closed walk without repeatededge

Simple circuita circuit with no repeatedvertex except first and last


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Examples

Cuircuit

Simple circuit


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Connectedness

Connectednessif there is a walk fromone to the other

Let G be a graph. Two vertices v and w of

G are connected if, and only if, there is awalk from v to w.

The graph G is connected if, and only if,

given any two vertices v and w in G, thereis a walk from v to w.


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Examples


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Euler Circuits

A circuit that contains every vertex andevery edge of G.

A sequence of adjacent vertices and edges

that starts and ends at the same vertex, uses every vertex of G at least once, and

uses every edge of G exactly once.


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If every vertex of a graph has even degree

then the graph has an Euler circuit.

Contrapositive: if the graph does not havean Euler circuit, then, some vertices havean odd degree.


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If every vertex of nonempty graphhas even degree and if graph is

connected, then the graph has anEuler circuit.


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Euler Circuit

A graph G has an Euler circuit if, and onlyif, G is connected and every vertex of Ghas even degree.


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Hamiltonian Path

A path in an undirected graph which visitseach vertex exactly once.


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Hamiltonian Circuit

A simple circuit that includes every vertexof G.

A sequence of adjacent vertices and

distinct edges in which every vertex of Gappears exactly once, except for the firstand last, which are the same.


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Hamiltonian circuit

An Euler circuit for a graph G may not bea Hamiltonian circuit.

An Hamiltonian circuit may not be an Euler

circuit.


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Hamiltonian Circuit

Proved simple criterion for determiningwhether a graph has an Euler circuit

No analogous criterion for determining

whether a graph has a Hamiltonian circuit Nor is there an efficient algorithm for

finding such an algorithm


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Hamiltonian Circuit

Finding Hamiltonian circuits


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Traveling Salesman Problem

http://en.wikipedia.org/wiki/Traveling_Salesman_Problem
http://en.wikipedia.org/wiki/Traveling_Salesman_Problemhttp://en.wikipedia.org/wiki/Traveling_Salesman_Problemhttp://en.wikipedia.org/wiki/Traveling_Salesman_Problemhttp://en.wikipedia.org/wiki/Traveling_Salesman_Problem


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Learning Outcomes

Students should be able to:

Explain basic terminology of a graph




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Matrix Representations of Graphs


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Matrices and Digraph

Let G be a directed graph with orderedvertices v1,v2,,vn. The adjacency matrixof G is the n x n matrix A =(aijover the

set of nonnegative integers such thataij= the numbers of arrows from vito vjfor all i,j = 1,2,,n.


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Matrices and Connected Components


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Counting Walks of Length n

Matrix multiplication


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How do these graphs relate?

=


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Summary

Documents

Discrete Structure Chapter 7 Graphs