13.1 Introduction to graphs€¦ · 3 13.1 Introduction to graphs A vertex is usually represented by a dot with a label. Each edge in E is a set of two vertices from V and is drawn

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13.1 Introduction to graphs

Graphs can model relations between pairs of objects and are fundamental in Discrete Mathematics.

In the previous lectures we covered directed graphs, now let’s spend some time with undirected graphs!

[Def] A graph G = ( V , E ) consists of a non-empty set of vertices (nodes) V and a set of edges E.

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b c

da

b c

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A vertex is usually represented by a dot with a label.

Each edge in E is a set of two vertices from V and is drawn as a line connecting the two vertices.

each edge has either 1 or 2 vertices associated with it, called endpoints

edge connects its endpoints

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b c

d

a

b c

d

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A vertex is usually represented by a dot with a label.

Each edge in E is a set of two vertices from V and is drawn as a line connecting the two vertices.

each edge has either 1 or 2 vertices associated with it, called endpoints

edge connects its endpoints

An infinite graph is a graph withinfinite number of vertices or edges.

A finite graph is a graph with a finite vertex set and a finite edge set.

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b c

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b c

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In directed graph (digraph) all edges have direction.edge (a,b) starts at a and ends at b.

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b c

da

b c

d

directed graphundirected graph

a b

undirected edge {a,b}

a b

directed edge (a,b)

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[Def] A simple (directed) graph is a graph in which each edge connects two different vertices (no loops) and where no two edges connect the same pair of vertices

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b c

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b c

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b c

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b c

d

simple graphs

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[Def] A multigraph is an undirected graph in which multiple edges connect the same vertices, but no loops

[Def] A directed multigraph is a directed graph in which multiple edges connect the same vertices, and loops are allowed.

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b c

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b c

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b c

dSome authors allow multigraphs to have loops, while others call these pseudographs, reserving the term multigraph for the case with no loops.

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a

b c

da

b c

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b c

d

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b c

d[Def] A directed multigraph is a directed graph in which multiple edges connect the same vertices, and loops are allowed.

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edge of multiplicity m: m different undirected edges are associated to the same pair of different vertices

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b c

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edge {b,d} has multiplicity 3

edge {a,c} has multiplicity 2

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[Def] A pseudograph is an undirected graph that may include loops, and possibly multiple edges connecting the same pair of vertices or a vertex to itself.

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b

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[Def] A mixed graph is a graph with both undirected and directed edges.

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b

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Type Edges Multiple edges

allowed?

Loops allowed?

Simple graph undirected No No

Multigraph undirected Yes No

Pseudograph undirected Yes Yes

Simple directed graph

directed No No

Directed multigraph

directed Yes Yes

Mixed graph both types Yes Yes

Graph terminology:

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Graph are used in a wide variety of models. When we build a graph model we need to answer the following three key questions:

● Are the edges directed, undirected, or both?

● If the graph is undirected: are multiple edges or loops present?If the graph is directed: are multiple edges present?

● Are loops present?

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Social Networks

Graphs are uses to model social structures based on different kinds of relationships between people or groups of people.

vertices: individuals (organizations)edges: relationships between individuals(organizations)

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Social Networks: Acquaintanceship / friendship graphs

vertices: individualsedges: friendship/acquantanceship between individuals

Alex

Samantha

Maria

Luis

Julia

Tom

Juan Olivia

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Social Networks: Acquaintanceship / friendship graphs

● no loops (not including self-knowledge) / loops (including)● no multiple edges

vertices: individualsedges: friendship/acquantanceship between individuals

Alex

Samantha

Maria

Luis

Julia

Tom

Juan Olivia

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Social Networks: Collaboration graphs

vertices: individualsedges: between individuals who work together in a particular way

MariaFred

Tom

Juan Olivia

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● no loops ● no multiple edgesi.e. simple graph

vertices: individualsedges: between individuals who work together in a particular way

MariaFred

Tom

Juan Olivia

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The Hollywood graph is a collaboration graph that represents actors by vertices and connects two actors if they the have worked together on a movie or TV show.

It is a huge graph with > 1.5 million vertices (as of 2011)

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Communication Networks

Different communications networks can be modeled by graphs using vertices to represent devices and edges for particular type of communications links of interest.

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vertices: data centersedges: communication links

Los Angeles

San FranciscoChicago

Detroit

Denver

New York

Washington

Computer network with diagnostic links

feedback for diagnostic purposes

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vertices: data centersedges: communication links

Los Angeles

San FranciscoChicago

Detroit

Denver

New York

Washington● loops ● multiple edgesi.e. pseudograph

Computer network with diagnostic links

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Transportation Networks

Graphs can be used to model road, air, rail, shipping ...etc networks.

Airline routes:

vertices: airportsedge: direct flight (and back) – usually both ways

undirected multigraph

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Delta Airlines routes

Source: https://blogs.cornell.edu/info2040/2015/09/11/an-analysis-of-delta-route-maps/

https://blogs.cornell.edu/info2040/2015/09/11/an-analysis-of-delta-route-maps/

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Connected World: Untangling the Air Traffic Network

Source: http://www.martingrandjean.ch/connected-world-air-traffic-network/

http://www.martingrandjean.ch/connected-world-air-traffic-network/

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Transportation Networks

Graphs can be used to model road, air, rail, shipping ...etc networks.

Road networks:

vertices: intersectionsedges: roads

One-way roads: directed edgesTwo-way roads: undirected edges

mixed graph

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Topology of a Network Data Model

Source: https://people.hofstra.edu/geotrans/eng/methods/nettopology.html

https://people.hofstra.edu/geotrans/eng/methods/nettopology.html

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Tournaments

Graphs can be used to model different kinds of tournaments

vertices: players / teamsedges: one round of play / tournament

directed / undirected edges: depends

If directed: shows who won

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Tournaments: single-elimination tournament

vertices: players / teamsedges: one roundundirected: each contestant is eliminated after one loss

Team 1

Team 2

Team 3

Team 4

Team 1

Team 4

Team 1

Team 5

Team 6

Team 7

Team 8

Team 6

Team 7

Team 7

Team 7

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Tournaments: single-elimination tournament

vertices: players / teamsedges: one roundundirected: each contestant is eliminated after one loss

● no loops● no multiple edges i.e. simple graph

Team 1

Team 2

Team 3

Team 4

Team 1

Team 4

Team 1

Team 5

Team 6

Team 7

Team 8

Team 6

Team 7

Team 7

Team 7

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[Def] Two vertices u and v in an undirected graph G are called adjacent (neighbors) in G if u and v are endpoints of an edge e of G.Such an edge is called incident with vertices u and v, and is said to connect u and v.

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b c

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adjacent vertices

incident with vertices a and c

connects b and d


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[Def] neighborhood of v, N(v), is the set of all neighbors of v.

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b c

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adjacent vertices


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[Def] neighborhood of v, N(v), is the set of all neighbors of v.

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b c

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adjacent vertices

neighborhood of a: N(a) = {b,c} ,neighborhood of b: N(b) = {a,d} ,neighborhood of c: N(c) = {a} ,neighborhood of d: N(d) = {b} ,


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[Def] the degree of a vertex in 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.

denotaion: deg(v)

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b c

d


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denotaion: deg(v)

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b c

d

deg(a) = 2 deg(c) = 1deg(b) = 2+2 = 4 deg(d) = 1


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denotaion: deg(v)

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b c

d

A vertex of degree 0 is called isolated.A vertex of degree 1 is called pendant.

pendant

pendant


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[Theorem] The Handshaking TheoremLet G = (V,E) be an undirected graph with m edges.

Then or2m=∑v∈V

deg(v)


∑v∈V

deg(v)=2|E|

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b c

dm = 4deg(a) = 2, deg(b) = 4, deg(c) = 1, deg(d) = 12 4 = 2 + 4 + 1 + 1 8 = 8



Then or2m=∑v∈V

deg(v) ∑v∈V

deg(v)=2|E|

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Why is it so?



Then or2m=∑v∈V

deg(v) ∑v∈V

deg(v)=2|E|

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Then or

Why is it so?

● each non-loop edge contributes 2 to the sum of the degrees ( 1 for each of adjacent vertices)

● each loop contributes 2 to the sum of the degrees (only for one vertex)

2m=∑v∈V

deg(v) ∑v∈V

deg(v)=2|E|


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Then or

[Theorem] An undirected graph has an even number of vertices of odd degree.

2m=∑v∈V

deg(v) ∑v∈V

deg(v)=2|E|


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Then or


Proof: Let V1 and V

2 be the set of vertices of even

degree and odd degree respectively, in a undirected graph G = (V,E) with m edges. |E| = m


2m=∑v∈V

deg(v) ∑v∈V

deg(v)=2|E|

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Then or


Proof: Let V1 and V



2m=∑v∈V

deg(v)=∑v∈V 1

deg(v)+∑v∈V 2

deg(v)


2m=∑v∈V

deg(v) ∑v∈V

deg(v)=2|E|

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Then or


Proof: Let V1 and V



2m=∑v∈V

deg(v)=∑v∈V 1

deg(v)+∑v∈V 2

deg(v)even

even


2m=∑v∈V

deg(v) ∑v∈V

deg(v)=2|E|

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Then or


Proof: Let V1 and V



2m=∑v∈V

deg(v)=∑v∈V 1

deg(v)+∑v∈V 2

deg(v)even

must be even q.e.d.

even


2m=∑v∈V

deg(v) ∑v∈V

deg(v)=2|E|

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We can take a directed graph and convert to to undirected graph by ignoring the directions.Such a graph is called underlying undirected graph.

The directed graph and its underlying undirected graph have the same number of edges.

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Example: recall Hollywood graph.a) What does the degree of a vertex represent in the Hollywood graph?

b) What does the neighborhood of a vertex represent?

c) What do isolated and pendant vertices represent?


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the degree of a vertex represents the number of times the actor worked together with other actors on a movie or a TV show




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N(a) represents the list of all actors actor a worked with on a movie or a TV show.



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Isolated vertices represent actors who didn't work with any other actor (present in the graph) on a movie or a TV show.Pendant vertices represent only actors with one collaboration only.


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Example: show that in a simple graph with at least two vertices there must be two vertices that have the same degree.


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Solution: recall that simple graphs are undirected graphs that have no loops, no multiple edges.


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deg(v1) = 0, deg(v

2) = 0

v1 v

2

2 vertices:


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deg(v1) = 1, deg(v

2) = 1

v1 v

2

2 vertices:


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deg(v1) = 0, deg(v

2) = 0, deg(v

3) = 0,

v1 v

2

3 vertices:

v3


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deg(v1) = 1, deg(v

2) = 1, deg(v

3) = 0,

v1 v

2

3 vertices:

v3


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deg(v1) = 2, deg(v

2) = 1, deg(v

3) = 1,

v1 v

2

3 vertices:

v3


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deg(v1) = 2, deg(v

2) = 2, deg(v

3) = 2,

v1 v

2

3 vertices:

v3


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v1 v

2

v1 v

2

v1 v

2

v3

v1 v

2

v3

v1 v

2

v3

v1 v

2

v3

...

...

...


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v1 v

n+1

...


vn

v1 v

n+1v

n

v1 v

n+1v

n

v1 v

n+1v

n

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[Def]: A graph with all the vertices of the same degree is called a regular graph.

In a d-regular graph, all the vertices have degree d.


3-regular graphs

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[Def]: A graph H = (VH, E

H) is a subgraph of a graph G =

(VG, E

G) if V

H V

G and E

H E⊆

G.

Note: graph G is a subgraph of itself


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b c

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H

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[Def]: A graph H = (VH, E

H) is a subgraph of a graph G =

(VG, E

G) if V

H V

G and E

H E⊆

G.

Note: graph G is a subgraph of itself

13.1 Introduction to graphs: common graphs

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Special Simple Graphs

- complete graphs ( Kn, for n Z+ )

- cycles ( Cn, for n Z+ and n 3)

- wheels ( Wn, for n Z+ and n 3)

- n-cubes ( Qn, for n Z+ )


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Special Simple Graphs: Complete graphs

A complete graph Kn on n vertices is a simple graph

that contains exactly one edge between each pair of distinct vertices.n = 1, 2, 3, 4, …

Sometimes Kn is called clique of size n or n-clique.

K1 K

2 K3 K

4K

5


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Special Simple Graphs: Cycles

A cycle Cn on n vertices consists on n vertices v

1, v

2, …,

vn and edges {v

1,v

2}, {v

2,v

3},..., and {v

n-1,v

n}.

n = 3, 4, 5, ...

C3 C

4C

5 C6


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Special Simple Graphs: Wheels

A wheel Wn can be obtained from cycle C

n by adding

an additional vertex and connecting this new vertex with each of the n vertices in C

n.

n = 3, 4, 5, ...

W3 W

4W

5W

6


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Special Simple Graphs: n-Cubes

Graph of Qn has 2n vertices

Q1 Q

2 Q3


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n-Cubes can represent bit strings of length nn = 1, 2, 3, 4, ...

Q1 Q

2 Q3


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Q1 Q

2

10Q

1

Q3


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Q1 Q

2

Q2

10Q

1 00 01

10 11

2nd place

1st place

Q3


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Q1 Q

2

Q2

10Q

1 00 01

10 11

2nd place

1st place

111110

000 001

011100

3rd place

1st place

2nd place

Q3

Q3


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Q1 Q

2 Q3

Q2

10Q

1 00 01

10 11

2nd place

1st place

111110

000 001

011100

3rd place

1st place

2nd place

010101

Q3


Documents

13.1 Introduction to graphs€¦ · 3 13.1 Introduction to graphs A vertex is usually represented by a dot with a label. Each edge in E is a set of two vertices from V and is drawn