basics of graph theory

8/14/2019 basics of graph theory

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Graphs

Nitin Upadhyay

February 24, 2006


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Discussion

What is a Graph?

Applications of Graphs

Categorization

Terminology


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What is a Graph?

A graph G=(V,E) consists of a finitenonempty set V, the elements of which are

the verticesof G, and a finite set Eof

unordered pairs of distinct elements ofV

called edges.

G = ( V, E )

Where, V- set of vertices

E - set of edges

1

2

3

4

5

Vertex

(Node)Edge

V = {1, 2, 3, 4, 5}

E = { (1,2), (1,3), (1,4), (2,3), (3,5), (4,5) }


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Graph applications

Graphs can be used for: Finding a route to drive from one city to another Finding connecting flights from one city to another Determining least-cost highway connections

Designing optimal connections on a computer chip Implementing automata Implementing compilers Doing garbage collection Representing family histories Doing similarity testing (e.g. for a dating service) Pert charts Playing games


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Graph applications

Computer Networks

Electrical Circuits

Road Map

Computer

Resistor/Inductor/

City


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Graph Models

Acquaintanceship Graphs showing whether two people know eachother, that is, whether they are acquainted.

Influence Graphs showing whether a person can influence another(e.g., ab) by using a directed graph.

The Bollywood Graph showing whether the actors have workedtogether in a movie.

Round-Robin Tournaments showing if team a beats team b (usingan edge (a, b)) via a directed graph.

Collaboration Graphs modeling joint authorship of academic papers(an edge links two people if they have jointly written a paper).

Call Graphs modeling telephone calls made in a network (can useeitherdirected multigraphs orundirected ones, depending on the

interest) The Web Graph representing hyperlinks between Web pages using a

directed graph. E.g., Web page a links to Web page b via an edgepointing from a to b.


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Structure of the Internet

Europe

Japan

Backbone 1

Backbone 2

Backbone 3

Backbone 4, 5, N

Australia

Regional A

Regional B

NAP

NAP

NAP

NAP

MAPS UUNET MAP

SOURCE: CISCO SYSTEMS
http://www.caida.org/tools/visualization/mapnet/Backbones/http://www.uunet.com/network/maps/http://www.uunet.com/network/maps/http://www.caida.org/tools/visualization/mapnet/Backbones/


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Graph Models Where are we right now?


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Graph categorization

Simple Graphs

Multigraphs

Pseudographs

Directed Graphs

Directed Multigraphs

Weighted Graph


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Simple Graphs

A simple graphG = (V,E)

consists of: a set Vofvertices ornodes

(Vcorresponds to the

universe of the relation R),

a set Eofedges / arcs / links: unordered pairs of

distinct elements u, v

V, such that uRv.


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Multigraphs

Like simple graphs, but there may be more than one

edge connecting two given nodes.

A multigraphG=(V, E, f) consists of a set Vof

vertices, a set Eof edges (as primitive objects), anda function ffrom Eto

{{u,v}| u,vVuv}.

The edges e1 and e2 are called multiple

or parallel edges if f(e1) = f(e2).

E.g., nodes are cities, edges

are segments of major highways

Parallel edges


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Pseudographs

Like a multigraph, but edges connecting a

node to itself are allowed.

ApseudographG=(V, E, f) where

f:E{{u,v}|u,vV}. Edge eEis

a loop iff(e)={u,u}={u}.

E.g., nodes are campsites

in a state park, edges arehiking trails through the woods.

Self loops


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Directed Graphs

Correspond to arbitrary binary relations R,

which need not be symmetric.

A directed graph (V,E) consists of a set of

vertices Vand a binary relation Eon V.

E.g.: V= people,

E={(x,y) |xloves y}


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Directed Multigraphs

Like directed graphs, but there may be more

than one arc from a node to another.

A directed multigraphG=(V, E, f) consists of

a set Vof vertices, a set Eof edges, and afunction f:EVV.

E.g., V=web pages,

E=hyperlinks. The WWW isa directed multigraph...


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Weighted Graph

A Weighted Graph is a graph where all theedges are assigned weights

1

2

3

4

5

50

4020

30

40

30


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Graph Terminology

Adjacent, connects, endpoints, degree, initial,

terminal, in-degree, out-degree, complete,

cycles, wheels, n-cubes, bipartite, subgraph,

union.


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Adjacency

Let G be an undirected graph with vertices u

and v, and a linking edge e = {u,v}. Then we

say:

u, vare adjacent/ neighbors / connected. Edge e is incident with vertices uand v.

Edge econnectsuand v.

Vertices uand vare endpoints of edge e.


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Adjacency

Adjacent vertices: If (i,j) is an edge of the

graph, then the nodes i and j are adjacent

An edge (i,j) is Incidenttovertices i and j

1

2

3

4

5

Vertices 2 and 5 are not

adjacent


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Degree of a Vertex

Let G be an undirected graph with vertex v.

The degree ofv, deg(v), is its number of

incident edges. (Except that any self-loops

are counted twice.) A vertex with degree 0 is isolated.

A vertex of degree 1 ispendant.


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A Degree Example

What are the degrees of the vertices in

the graphs G and Has shown below?In G, deg(a) = 2, deg(b) = deg(c) = deg(f) = 4,

deg(d) = 1, deg(e) = 3 and deg(g) = 0

In H, deg(a) = 4, deg(b) = deg(e) = 6, deg(c) = 1,

deg(d) = 5

a

b c d

ef g

a b c

de


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Handshaking Theorem

Let G = (V, E) be an undirected (simple,

multi-, or pseudo-) graph with e edges. Then

Corollary: Any undirected graph has an even

number of vertices of odd degree.


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Handshaking TheoremExample How many edges are there in a graph with

ten vertices each of degree six?

The sum of the degrees of the vertices is 6x10 = 60,

it follows that 2e = 60. Therefore, e = 30.


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Directed Adjacency

Let G be a directed (possibly multi-) graph,

and let e be an edge ofG that is (or maps to)

(u,v). Then we say:

uis adjacent tov, vis adjacent fromu ecomes from u, e goes to v.

e connects u to v, e goes from u to v

the initial vertexofe is u the terminal vertexofe is v


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Directed Degree

Let G be a directed graph, va vertex ofG. The in-degree ofv, deg(v), is the number of

edges going to v.

The out-degree ofv, deg+(v), is the number ofedges coming from v.

The degree ofv, deg(v)deg(v)+deg+(v), is the

sum ofvs in-degree and out-degree.


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Directed HandshakingTheorem Let G = (V, E) be a directed (possibly multi-)

graph with vertex set Vand edge set E.

Then:

Note that the degree of a node is unchanged

by whether we consider its edges to bedirected or undirected.


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Path

Path: A sequence of edges in the graph There can be more than one path between

two vertices

Vertex A is reachable from B if there is a pathfrom A to B

A

D

C

F

E

B

G

Paths from B to D

- B, A, D- B, C, D

-


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Path

Simple Path:A path whereall the vertices are

distinct

1

2

3

45

1,4,5,3 is a simplepath.

But 1,4,5,4 is not asimple path.


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Length

Length :Sum of the lengths of the edges on

the path.

Length of the path1,4,5,3 is 3

1

2

3

45


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Circuit

Circuit: A path whose first and last vertices

are the same

The path 3,2,1,4,5,3

is a circuit.

1

2

3

45


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Cycle

Cycle: A circuit where all the vertices are

distinct except for the first (and the last)

vertex

1,4,5,3,1 is a cycle

1,4,5,4,1 is not a cycle

1

2

3

45


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basics of graph theory