Graph theory - dii.unisi.it · books Graph Theory by Reinhard Diestel and Introduction to Graph...

Graph TheoryAcknowledgementMuch of the material in these notes (only for private use) is from thebooks Graph Theory by Reinhard Diestel and Introduction to GraphTheory by Douglas West.

Giulia Simi (Universita di Siena) Graph theory Siena 2016-2017 1 / 78







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My name is: Giulia Simi

If you want to ask for an appointment you can write an email.

My email is: [email protected]

My office hours will be on Thursday afternoon from 4 p.m. to 5p.m.


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Introduction

The primary aim of this course is to present a coherentintroduction to graph theory, suitable for advancedundergraduate and beginning graduate students in mathematicsand computer science.


Introduction

The primary aim of this course is to present a coherentintroduction to graph theory, suitable for advancedundergraduate and beginning graduate students in mathematicsand computer science.


Programm

The arguments will be the following:

Graphs and simple graphs;Graph Isomorphism;Incidence and adjacency matrix;Subgraph;The degree of a vertex;Paths and connection, cycles, connected graph;Trees, equivalent definitions of trees, Cayley’s formula;Connectivity, vertex connectivity, edge connectivity, blocks,2-connected graphs, Menger’s theorem;Eulerian and Hamiltonian cycles;


Programm




Programm




Programm




Programm




Programm




Programm




Programm




Programm




Programm




Programm

Matchings, Hall’s theorem, Matchings in general graphs: Tutte’sTheorem;Planar graphs;The four-colour problem;Unsolved problems.


Programm



Programm



Programm



Programm



Reference textbooks

Lecture Notes distributed by the teacher;

Reinhard Diestel, Graph Theory, Springer.

Bondy, Adrian; Murty, U.S.R, Graph theory, Springer, 2008.

Additional material or information on line.


Reference textbooks






Reference textbooks






Reference textbooks






Reference textbooks






Exam session

It will be an oral examination;You must sign up online for taking an exam.


Exam session



Exam session



Graphs and simple graphs

Many real world situations can conveniently be described by means ofdiagram consisting of a set points together with lines joining certainpairs of these points.


Example

the points could represent people;

lines join pairs of friends;


Example




Example




Example

the points could represent communication centres;

lines represent communication links.


Example




Example




Definition

DefinitionA graph G is an ordered triple 〈V (G),E(G), ψG〉 consisting of anonempty set V (G) of vertices(or nodes, or points), e set of E(G) ofedges (or lines), such that V (G) ∩ E(G) = ∅, incidence function ψGthat associates with each edge of G an unordered pair, not necessarilydistinct, vertices of G.If e is an edge and u and v are vertices such that ψG(e) = uv, then eis said to join u and v;the vertices u and v are called the ends of e.


Definition



Definition



Definition



Observation

ObservationThe usual way to picture a graph is by drawing a dot for eachvertices and joining two of these dots by a line if thecorresponding two vertices form an edge.

Just how these dots and lines are drawing are consideredirrelevant: all that matters is the information which pairs of verticesform an edge and which do not.


Observation




Observation




Two following examples shouldserve to clarify the definition.


Example 1

G = 〈V (G),E(G), ψG〉where

V (G) = {v1, v2, v3, v4, v5}

E(G) = {e1,e2,e3,e4,e5,e6,e7,e8}

v1

v2

e1

v3

e2

e3

v4

e4

e5

v5

e8

e6

e7

Figure: Diagram graph G


Example 1


V (G) = {v1, v2, v3, v4, v5}

E(G) = {e1,e2,e3,e4,e5,e6,e7,e8}

v1

v2

e1

v3

e2

e3

v4

e4

e5

v5

e8

e6

e7



Example 1


V (G) = {v1, v2, v3, v4, v5}

E(G) = {e1,e2,e3,e4,e5,e6,e7,e8}

v1

v2

e1

v3

e2

e3

v4

e4

e5

v5

e8

e6

e7



Example 1


V (G) = {v1, v2, v3, v4, v5}

E(G) = {e1,e2,e3,e4,e5,e6,e7,e8}

v1

v2

e1

v3

e2

e3

v4

e4

e5

v5

e8

e6

e7



Example 1


V (G) = {v1, v2, v3, v4, v5}

E(G) = {e1,e2,e3,e4,e5,e6,e7,e8}

v1

v2

e1

v3

e2

e3

v4

e4

e5

v5

e8

e6

e7



Example 1

v1

v2

e1

v3

e2

e3

v4

e4

e5

v5

e8

e6

e7


G = 〈V (G),E(G), ψG〉

where ψ is defined by

ψG(e1) = v1v2, ψG(e2) = v2v3, ψG(e3) = v3v3, ψG(e4) = v3v4

ψG(e5) = v2v4, ψG(e6) = v4v5, ψG(e7) = v2v5, , ψG(e8) = v2v5.


Example 1

v1

v2

e1

v3

e2

e3

v4

e4

e5

v5

e8

e6

e7


G = 〈V (G),E(G), ψG〉





Example 1

v1

v2

e1

v3

e2

e3

v4

e4

e5

v5

e8

e6

e7


G = 〈V (G),E(G), ψG〉





Example 1

v1

v2

e1

v3

e2

e3

v4

e4

e5

v5

e8

e6

e7


G = 〈V (G),E(G), ψG〉





Example 1

v1

v2

e1

v3

e2

e3

v4

e4

e5

v5

e8

e6

e7


G = 〈V (G),E(G), ψG〉





Example 2

u

v

w

xy

a

b

cd

e

f

g

h

Figure: Diagram graph H

H = 〈V (H),E(H), ψH〉where

V (H) = {u, v ,w , x , y}

E(H) = {a,b, c,d ,e, f ,g,h}


Example 2

u

v

w

xy

a

b

cd

e

f

g

h



V (H) = {u, v ,w , x , y}

E(H) = {a,b, c,d ,e, f ,g,h}


Example 2

u

v

w

xy

a

b

cd

e

f

g

h



V (H) = {u, v ,w , x , y}

E(H) = {a,b, c,d ,e, f ,g,h}


Example 2

u

v

w

xy

a

b

cd

e

f

g

h



V (H) = {u, v ,w , x , y}

E(H) = {a,b, c,d ,e, f ,g,h}


Example 2

u

v

w

xy

a

b

cd

e

f

g

h



V (H) = {u, v ,w , x , y}

E(H) = {a,b, c,d ,e, f ,g,h}


Example 2

u

v

w

xy

a

b

cd

e

f

g

h


H = 〈V (H), E(H), ψH〉

where

ψH is defined by

ψH(a) = uv , ψH(b) = uu, ψH(c) = vw , ψH(d) = wx

ψH(e) = vx , ψH(f ) = wx , ψH(g) = ux , ψH(h) = xy .


Example 2

u

v

w

xy

a

b

cd

e

f

g

h


H = 〈V (H), E(H), ψH〉

where

ψH is defined by




Example 2

u

v

w

xy

a

b

cd

e

f

g

h


H = 〈V (H), E(H), ψH〉

where

ψH is defined by




Example 2

u

v

w

xy

a

b

cd

e

f

g

h


H = 〈V (H), E(H), ψH〉

where

ψH is defined by




Observation

ObservationGraphes are so named because they can be representedgraphically, and it is representation which helps us understandmany of their properties.

Most of the definitions and concepts in graphic theory aresuggested by the graphical representation.


Observation




Observation




Incident,adjacent,loop, link

The ends of the edge are said to be incident with the edge, andan edge joins its ends.

Two vertices x , y of a graphic G are adjacent, or neighbours, ifxy is an edge of G, that is xy ∈ E(G). Two edges e 6= f areadjacent if they have an end in common, that is e ∩ f 6= ∅.

An edge with identical ends is called a loop, and a edge withdistinct ends a link. For example the edge b of G of Example 2, isa loop; all the other edges are link.

















Order

The number of vertices of a graph G is its order, written as |G|; itsnumber of edges is denoted by ||G|| or e(G), that is ||G|| = e(G).

A graph is finite if both of its vertices and edges are finite;

ObservationIn this course we only study finite graphs, so the term “graph” alwaysmeans “finite graph”.


Order





Order





Order





Order





Trivial graph, simple graph

A graph of order 1, that is with one just vertex, is trivial and allother graphs are nontrivial.

The graph is simple if it has no loops and no two of its links jointhe same pair of vertices.

Much of graph theory is concerned with the study of simple graphs.

















Any symmetric relation between objects givesgraph.

Examplelet V be a set of people in a room, and E the set of pairs of peoplewho met for the first time today;

let V be a set of cities in a country, and let the edges in Ecorrespond to the road connecting them;

the internet: let V be the set of computers, and let the edges in Ecorrespond to the links connecting them.






















Graph Isomorphism

Two graphs G and H are identical, written G = H, ifV (G) = V (H), E(G) = E(H) and ψG = ψH .

If two graphs are identical then they can clearly be represented byidentical diagrams.

However, it is also possible for graphs that are not identical tohave the same graphs.


Graph Isomorphism





Graph Isomorphism





Graph Isomorphism





Graph Isomorphism

There are different ways to draw the same graph. Consider thefollowing two graphs.

You probably feel that these graphs do not differ from each other.


Graph Isomorphism




Graph Isomorphism




Graph Isomorphism

What about this pair?

Another paira a

b

b

e

e

c

c

d

d

Are the same?


Graph Isomorphism


Another paira a

b

b

e

e

c

c

d

d

Are the same?


Graph Isomorphism


Another paira a

b

b

e

e

c

c

d

d

Are the same?


Graph Isomorphism


Another paira a

b

b

e

e

c

c

d

d

Are the same?


Graph Isomorphism


Another paira a

b

b

e

e

c

c

d

d

Are the same?


Graph Isomorphism

We have the following definition:Two graphs G and H are said to be isomorphic, written G ∼= H, ifthere are bijections θ : V (G)→ V (H) and φ : E(G)→ E(H) suchthat

ψG(e) = uv ⇔ ψH(φ(e)) = θ(u)θ(v);

such a pair (θ, φ) of mappings is called an isomorphism between

G and H.

If G = H, it is called an automorphism.


Graph Isomorphism


ψG(e) = uv ⇔ ψH(φ(e)) = θ(u)θ(v);


G and H.



Graph Isomorphism


ψG(e) = uv ⇔ ψH(φ(e)) = θ(u)θ(v);


G and H.



Graph Isomorphism


ψG(e) = uv ⇔ ψH(φ(e)) = θ(u)θ(v);


G and H.



Graph Isomorphism


ψG(e) = uv ⇔ ψH(φ(e)) = θ(u)θ(v);


G and H.



Graph Isomorphism


ψG(e) = uv ⇔ ψH(φ(e)) = θ(u)θ(v);


G and H.



Invariant

A map taking graphs as arguments is called a graph invariant ifit assigns equal value to isomorphisms graphs.

The number of vertices and the number of edges of a graph aretwo simple graph invariants;the gratest number of pairwise adjacent vertices is another;there are some other.

In other words: an invariant is a property such that if a graph has it allisomorphic graphs have it.


Invariant





Invariant





Invariant





Invariant





Invariant





Notation

Will be often use the following simple definition of graph and hence thesimple definition of isomorphism of graphs.

A graph is a pair G = 〈V ,E〉 of sets satisfying E ⊆ [V ]2;thus, the elements of E are 2-element subset of V .We shall always assume tacitly that V ∩ E = ∅.Let G = 〈V ,E〉 and G′ = 〈V ′,E ′〉 be two graphs.We call G and G′ isomorphic, and write G ' G′, if there exists abijection φ : V → V ′ with

xy = (x , y) ∈ E ⇔ φ(x)φ(y) = (φ(x), φ(y)) ∈ E ′ ∀x , y ∈ V .


Notation





Notation





Notation





Notation





Notation





Notation





Notation





Question

Are these graphics in some sense the same?

1

2

3

4

a b

cdG G′

Yes!


Question


1

2

3

4

a b

cdG G′

Yes!


Question


1

2

3

4

a b

cdG G′

Yes!


Why?

1

2

3

4

a b

cdG G′

The function φ : G→ G′ given by

φ(1) = a, φ(2) = c, φ(3) = b, φ(4) = d

such that (u, v) ∈ E(G) if and only if (φ(u), φ(v)) ∈ E(G′), is anisomorphism.


Why?

1

2

3

4

a b

cdG G′


φ(1) = a, φ(2) = c, φ(3) = b, φ(4) = d



Why?

1

2

3

4

a b

cdG G′


φ(1) = a, φ(2) = c, φ(3) = b, φ(4) = d



Why?

1

2

3

4

a b

cdG G′


φ(1) = a, φ(2) = c, φ(3) = b, φ(4) = d



Why?

1

2

3

4

a b

cdG G′


φ(1) = a, φ(2) = c, φ(3) = b, φ(4) = d



Remark

Isomorphism is an equivalence relation of graphs. This means that:Any graphic is isomorphism to itself;

If G1 is isomorphic to G2 then G2 is ismorphic to G1.

If G1 is isomorphic to G2 and G2 is isomorphic to G3, then G1 isisomorphic to G3.


Remark





Remark





Remark





Remark





Observation

ObservationIf two graphs G and H are isomorphic, then G and H have thesame structure, and they differ only in the names of vertices andedges.

Since it is in structural properties that we shall be interested, weshall often omit labels when drawing graphs;

an unlabelled graph can be thought of as a representative of anequivalence class of isomorphic graphs.


Observation





Observation





Observation





Observation

ObservationWe assign labels to vertices and edges in a graph mainly for apurpose of referring to them.

For instance, when dealing with simple graphic, it is often useful torefer to the edge e with ends u and v as “ the edge uv” .

This is not ambiguous since, in a simple graph, at most one edgejoins any pair of vertices.


Observation





Observation





Observation





Example of non-isomorphic graphs

a 1

b 2

c

3

d

e

4 5


Example of non-isomorphic graphs

a 1

b 2

c

3

d

e

4 5


Invariant property

In practice, it is not a simple task to prove that two graphs areisomorphic.It is much simpler to show that two graphs are not isomorphic byshowing an invariant property that one has and other does not.Note, a complete set of such invariants is unknown.


Invariant property



Invariant property



Invariant property



Complete graph

A simple graph in which each pair of distinct vertices is joined byan edge is called a complete graph.

Up to isomorphism there is just one complete graph on n vertices;

it is denoted by Kn.

An empty graph 〈∅, ∅〉 is one with no edges, order 0.


Complete graph






Complete graph






Complete graph






Complete graph






Example: complete graph

Examplea K3 is called a triangle and it is shown in the following figure:

K3

Figure: K3

A drawing of K5 is shown in the following figure:

K5

Figure: K5




K3

Figure: K3


K5

Figure: K5




K3

Figure: K3


K5

Figure: K5




K3

Figure: K3


K5

Figure: K5




K3

Figure: K3


K5

Figure: K5


Bipartite graph

A bipartite graph G is graph such that V (G) = X ∪ Y withX ∩ Y = ∅ and each edge of G has one end in X and one end inY ;such a partition (X ,Y ) is called a bipartition of the graph G.

A complete bipartite graph is a simple bipartite graph withpartition (X ,Y ) in which each vertex of X is joined to each vertexof Y ;if |X | = m and |Y | = n such a graph is denoted by Km,n.


Bipartite graph




Bipartite graph




Bipartite graph




Bipartite graph




r - Partite

Let r ≥ 2 be an integer. A graph G = 〈V ,E〉 is called r -partite if Vadmits a partition into r classes such that every edge has its endsin different classes:vertices in the same partition class must not be adjacent.

An r -partite graph in which every two vertices from differentpartition classes are adjacent is called complete.


r - Partite

Let r ≥ 2 be an integer. A graph G = 〈V ,E〉 is called r -partite if Vadmits a partition into r classes such that every edge has its endsin different classes:vertices in the same partition class must not be adjacent.

An r -partite graph in which every two vertices from differentpartition classes are adjacent is called complete.


Example

ExampleThe graph in the following figure is the complete bipartite graph K3,3.

K3,3

Figure: K3,3


Incidence graph

Let |G| = ν be the number of vertices of a graph G, and ||G|| = ε thenumber of its edges

To any graph G there corresponds a ν × ε matrix called theincidence matrix of G.Let us denote the vertices of G by v1, v2, . . . vν and the edges bye1,e2, . . .eε.Then the incidence matrix of G is the matrix M(G) = [mij ],where mij is the number of time, 0, 1, 2, that vi and ej are incident.The incidence matrix of a graph is just a different way specifyingthe graph.


Incidence graph




Incidence graph




Example: Incidence MatrixThe graph G is shown in the following figure:

v1 v2

v3v4

e1

e2

e3

e4

e5 e6e7

G

its incidence matrix, M(G):

e1 e2 e3 e4 e5 e6 e7

v1 1 1 0 0 1 0 1v2 1 1 1 0 0 0 0v3 0 0 1 1 0 0 1v4 0 0 0 1 1 2 0


Adjacency matrix


Another matrix associated with G is the adjacency matrix;this is the ν × ν matrix A(G) = [ai,j ], in which aij is the number ofedges joining vi and vj .


Adjacency matrix




Adjacency matrix




Example: Adjacency matrixThe graph G is shown in the following figure:

v1 v2

v3v4

e1

e2

e3

e4

e5 e6e7

G

its adjacency matrix, M(G):

v1 v2 v3 v4

v1 0 2 1 1v2 2 0 1 0v3 1 1 0 1v4 1 0 1 1


Example: Adjacency matrixThe graph G is shown in the following figure:

v1 v2

v3v4

e1

e2

e3

e4

e5 e6e7

G

its adjacency matrix, M(G):

v1 v2 v3 v4

v1 0 2 1 1v2 2 0 1 0v3 1 1 0 1v4 1 0 1 1


Observation

ObservationThe adjacency matrix of a graph is generally considerable smallerthan its incidence matrix, and it is in this form that graphs arecommonly stored in computers.


Adjacency Matrix of a simple graph

Let [n] = {1, . . .n}.

DefinitionLet G = 〈V ,E〉 with V = [n]. The adjacency matrix A = A(G) of G isthe n × n symmetric matrix defined by

aij =

{1 if (i , j) ∈ E ,0 otherwise.



Let [n] = {1, . . .n}.


aij =




Let [n] = {1, . . .n}.


aij =



ExampleLet be the following graph:

1

2 3

4 5G

Figure: The simple graph G

This is the adjacency matrix of the simple graph G.

A =

0 1 0 0 01 0 1 1 00 1 0 0 10 1 0 0 00 0 1 1 0


Spectral theorem

ObservationAny adjacency matrix A is real and symmetric, hence the spectraltheorem proves that A has an orthogonal basis of eigenvalues withreal eigenvectors.This important fact allows us to use spectral methods in graph theory.Indeed, there is a large subfield of graph theory called spectral graphtheory.


Spectral theorem



Spectral theorem



Incidence matrix of a simple graph

DefinitionLet G = 〈V ,E〉 be a graph with V = {v1, . . . , vn} and E = {e1, . . . ,em}.Then the incidence matrix B = B(G) of G is the n ×m matrix definedby

bij =

{1, if vi ∈ ej ,0, otherwise.




bij =





bij =



ExampleGiven the following simple graph G:

1

2

3

4e1

e2

e3e4


This is the incidence matrix of G

B =

1 1 1 00 0 1 11 1 0 11 0 0 0

Every column of B has |e| = 2 entries 1.



1

2

3

4e1

e2

e3e4



B =

1 1 1 00 0 1 11 1 0 11 0 0 0




1

2

3

4e1

e2

e3e4



B =

1 1 1 00 0 1 11 1 0 11 0 0 0




1

2

3

4e1

e2

e3e4



B =

1 1 1 00 0 1 11 1 0 11 0 0 0



Subgraph

A graph H is a subgraph of G, written as H ⊆ G, if V (H) ⊆ V (G),E(G) ⊆ E(G) and ψH is the restriction of ψG to E(H).

When H ⊆ G but H 6= G, we write H ⊂ G and call H a propersubgraph of G.

If H is a subgraph of G, G is supergraph of H


Subgraph





Subgraph





Spanning subgraph

A spanning subgraph of G is a subgraph H with V (H) = V (G),that is if V (H) spans all of G.

By deleting from G all loops and, for every of adjacent vertices, allbut one edge joining them, we obtain a simple spanning subgraphof G called underlying simple graph of G.


Spanning subgraph

A spanning subgraph of G is a subgraph H with V (H) = V (G),that is if V (H) spans all of G.

By deleting from G all loops and, for every of adjacent vertices, allbut one edge joining them, we obtain a simple spanning subgraphof G called underlying simple graph of G.


Example

ExampleThe following figure shows a graph G and its underlying simplesubgraph.

Figure: A graph G and in red its underlying simple subgraph


Induced subgraph G[V ′]

Let G be a graph. Let V ′ ⊆ V be any set of vertices of G such thatV ′ 6= ∅.The subgraph of G whose vertex set is V ′ and whose edge is theset of those edges of G that have both ends in V ′ is called thesubgraph of G induced by V ′ and it is denoted by G[V ′].

We say that G[V ′] is an induced subgraph of G.










Induced subgraph

The induced subgraph G[V \ V ′] where V ′ is nonempty subset ofV , is denoted by G − V ′;it is the subgraph obtained from G by deleting the vertices in V ′

together with the incident edge.

If V ′ = {v} we write G − v for G − {v}.


Induced subgraph





Induced subgraph





Example

ExampleSome types of subgraphs of G

x

y

u

v

w

a

b

c

d

e

f

g

G

h

x

y

u

v

w

a

b

c

d

e

f

g

G

h

x

y

u

v

w

a

b

c

d

e

f

g

G

h

x

y

u

v

w

a

b

c

d

e

f

g

G

h

x

y

u

v

w

b

c

d

e

f

x

y

u

v

w

b

c

d

e

f

A spanning subgraph of G

x

y

df

g

x

y

df

g

G− {u,w}

v

Figure: Subgraphs of G


Example

ExampleThe subgraph of G induced by V ′ = {u, v , x}

x

y

u

v

w

a

b

c

d

e

f

g

G

h

x

y

u

v

w

a

b

c

d

e

f

g

G

h

x

y

u

v

w

a

b

c

d

e

f

g

G

h

x

y

u

v

w

a

b

c

d

e

f

g

G

h

v

a

x

u

v

a

x

u

v

a

x

u

v

a

The induced subgraph G[{u, v, x, }]

Figure: Subgraph G[{u, v , x}]


Edge-induced subgraph

Let E ′ be a nonempty set of E .

The subgraph of G whose vertex set is the set of ends of edges inE ′ and whose edge set is E ′ is called the subgraph of G inducedby E ′ and is denoted by G[E ′];

G[E ′] is an edge-induced subgraph of G.












Spanning subgraph of G

The spanning subgraph of G with edge set E \ E ′ is written simplyas G − E ′;

it is the subgraph obtained from G by deleting the edges in E ′.

Similarly, the graph obtained from G by adding a set of edges E ′ isdenoted by G + E ′.

If E ′ = {e} we write G − e and G + e instead of G − {e} andG + {e}.




















Example

ExampleOthers examples of subgraphs of G.

x

y

u

v

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G[{a, c, e, g}]

Figure: Subgraphs of G


UnionLet G = 〈V ,E〉 and G′ = 〈V ′,E ′〉 be graphs.

We set G ∪G′ = 〈V ∪ V ′,E ∪ E ′〉The following figure shows the union of two graphs G and G′.

1

2

3

4

5

G

1

2

3

4

5

3

4 6

5G′

6

G ∪G′

Figure: G ∪G′


UnionLet G = 〈V ,E〉 and G′ = 〈V ′,E ′〉 be graphs.

We set G ∪G′ = 〈V ∪ V ′,E ∪ E ′〉The following figure shows the union of two graphs G and G′.

1

2

3

4

5

G

1

2

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5

3

4 6

5G′

6

G ∪G′

Figure: G ∪G′


Observation

1

2

3

4

5

G

1

2

3

4

5

3

4 6

5G′

6

G ∪G′

Figure: G ∪G′

ObservationThe vertices 2,3,4 induce (or span) a triangle in G ∪G′ but not in G.


IntersectionLet G = 〈V ,E〉 and G′ = 〈V ′,E ′〉 be graphs.

We set G ∩G′ = 〈V ∩ V ′,E ∩ E ′〉.

The following figure shows the intersection of two graphs G and G′.

1

2

3

4

5

G3

4 6

5G′

3 5

4

G ∩G′

Figure: G ∩G′


IntersectionLet G = 〈V ,E〉 and G′ = 〈V ′,E ′〉 be graphs.

We set G ∩G′ = 〈V ∩ V ′,E ∩ E ′〉.

The following figure shows the intersection of two graphs G and G′.

1

2

3

4

5

G3

4 6

5G′

3 5

4

G ∩G′

Figure: G ∩G′


Observation

ObservationIf G ∩G′ = ∅, that is G and G′ have no vertex in common, then Gand G′ are disjoint.

We say that G and G′ are edge-disjoint if they have no edge incommon.

If G and G′ are disjoint, we sometimes denote their union byG + G′.


Observation





Observation





The graph G ∗G′

Let G and G′ be disjoint.

We denote by G ∗G′ the graph obtained from G ∪G′ by joining allthe vertices of G to all vertices of G′.

For example, K2 ∗ K3 = K5.


The graph G ∗G′





The graph G ∗G′





The complement G of G

The complement G of G is the graph on V with edge set [V ]2 \ E .

We show in the following picture the graph G and its complementG.

G G

Figure: The graphs G and G′


The complement G of G

The complement G of G is the graph on V with edge set [V ]2 \ E .

We show in the following picture the graph G and its complementG.

G G

Figure: The graphs G and G′


The degree of a vertex


Degree of a vertex

Let G = 〈V ,E〉 be a graph.If (u, v) ∈ E(G) then v is a neighbour of u.Given v ∈ V we define the neighbourhood N(v) of v to be theset of neighbours of v .

The degree dG(v) = d(v) of a vertex v in G is the number |E(v)|of edges at v , that is, the number of edges of G incident with v ,each loop counting as two edges.Hence the degree of v is |N(v)|, the numbers of neighbours of v .

A vertex of degree 0 is isolated.

ObservationHere, as elsewhere, we drop the index referring to the underlyinggraph if the reference is clear.


Degree of a vertex






Degree of a vertex






Degree of a vertex






Degree of a vertex






Degree of a vertex






Degree of a vertex






ExampleGiven the following simple graph G

1

2

3

4

5


Then we have

d(1) = 3, d(2) = 2, d(3) = 2, d(4) = 1, d(5) = 0

5 is a isolated vertex.



1

2

3

4

5


Then we have

d(1) = 3, d(2) = 2, d(3) = 2, d(4) = 1, d(5) = 0




1

2

3

4

5


Then we have

d(1) = 3, d(2) = 2, d(3) = 2, d(4) = 1, d(5) = 0




1

2

3

4

5


Then we have

d(1) = 3, d(2) = 2, d(3) = 2, d(4) = 1, d(5) = 0



Fact

For any graph G on the vertex set [n] with adjacency andincidence matrix A and B respectively, we have

BBT = A + D,

where

D =

d(1) . . . 0

0. . . 0

0 0 d(n)


Fact


BBT = A + D,

where

D =

d(1) . . . 0

0. . . 0

0 0 d(n)


Fact


BBT = A + D,

where

D =

d(1) . . . 0

0. . . 0

0 0 d(n)


ExampleGiven the following simple graph:

1

2 3

e1e2

e3

Figure: The graph G

We have B =

1 1 00 1 11 0 1

, A =

0 1 11 0 11 1 0

and

D =

2 0 00 2 00 0 2


ExampleGiven the following simple graph:

1

2 3

e1e2

e3

Figure: The graph G

We have B =

1 1 00 1 11 0 1

, A =

0 1 11 0 11 1 0

and

D =

2 0 00 2 00 0 2


ExampleWe have

BBT = D + A

1 1 00 1 11 0 1

· 1 0 1

1 1 00 1 1

=

2 1 11 2 11 1 2

= 2 0 00 2 00 0 2

+

0 1 11 0 11 1 0


ExampleWe have

BBT = D + A

1 1 00 1 11 0 1

· 1 0 1

1 1 00 1 1

=

2 1 11 2 11 1 2

= 2 0 00 2 00 0 2

+

0 1 11 0 11 1 0


ExampleWe have

BBT = D + A

1 1 00 1 11 0 1

· 1 0 1

1 1 00 1 1

=

2 1 11 2 11 1 2

= 2 0 00 2 00 0 2

+

0 1 11 0 11 1 0


δ(G), ∆(G)

The numberδ(G) := min{d(v) : v ∈ V}

is the minimum degree of G;

The number∆(G) := max{d(v) : v ∈ V}

is the maximum degree of G;


δ(G), ∆(G)






δ(G), ∆(G)






δ(G), ∆(G)






δ(G), ∆(G)






δ(G), ∆(G)






Average degree

Let G be a graph.The number

d(G) =1|V |

∑v∈V

d(v)

is the average degree of G.Clearly

δ(G) ≤ d(G) ≤ ∆(G).

The average degree quantifies globally what is measured locallyby the vertex degrees: the number of edges of G per vertex.


Average degree


d(G) =1|V |

∑v∈V

d(v)


δ(G) ≤ d(G) ≤ ∆(G).



Average degree


d(G) =1|V |

∑v∈V

d(v)


δ(G) ≤ d(G) ≤ ∆(G).



Average degree


d(G) =1|V |

∑v∈V

d(v)


δ(G) ≤ d(G) ≤ ∆(G).



Average degree


d(G) =1|V |

∑v∈V

d(v)


δ(G) ≤ d(G) ≤ ∆(G).



Average degree


d(G) =1|V |

∑v∈V

d(v)


δ(G) ≤ d(G) ≤ ∆(G).



Theorem1.23 Let G be a graph ∑

v∈V

d(v) = 2ε,

where ε is the number of edges of G.

ProofConsider the incidence matrix M = [mij ], where mij is the number oftime that vi and ej are incident.So the sum Σjmij of the entries in the row i corresponding to vertex viis precisely d(vi), and therefore

∑v∈V d(v) =

∑i,j mij .

Since every column sums of M is 2, then∑v∈V

d(v) =∑i,j

mij = 2ε.

♠Giulia Simi (Universita di Siena) Graph theory Siena 2016-2017 75 / 78


v∈V

d(v) = 2ε,



∑v∈V d(v) =

∑i,j mij .


d(v) =∑i,j

mij = 2ε.



v∈V

d(v) = 2ε,



∑v∈V d(v) =

∑i,j mij .


d(v) =∑i,j

mij = 2ε.



v∈V

d(v) = 2ε,



∑v∈V d(v) =

∑i,j mij .


d(v) =∑i,j

mij = 2ε.



v∈V

d(v) = 2ε,



∑v∈V d(v) =

∑i,j mij .


d(v) =∑i,j

mij = 2ε.


Another proof

ProofIn the sum

∑v∈V (G) d(v) every edge is counted twice: once from u

and once from v .

♠

|E | = 12∑

v∈V d(v) = 12d(G) · |V |.


Corollary1.24

In any graph G, the number of vertices of odd degree is even.

ProofLet V1 and V2 be the sets of vertices of odd and even degree in G,respectively.Then ∑

v∈V1

d(v) +∑v∈V2

d(v) =∑v∈V

d(v)

is even, by Theorem 2.23. Since∑

v∈V2d(v) is even,

it follows that∑

v∈V1d(v) is even.

Thus |V1| is even.

♠


Corollary1.24



v∈V1

d(v) +∑v∈V2

d(v) =∑v∈V

d(v)


v∈V2d(v) is even,

it follows that∑

v∈V1d(v) is even.

Thus |V1| is even.

♠


Corollary1.24



v∈V1

d(v) +∑v∈V2

d(v) =∑v∈V

d(v)


v∈V2d(v) is even,

it follows that∑

v∈V1d(v) is even.

Thus |V1| is even.

♠


Corollary1.24



v∈V1

d(v) +∑v∈V2

d(v) =∑v∈V

d(v)


v∈V2d(v) is even,

it follows that∑

v∈V1d(v) is even.

Thus |V1| is even.

♠


Corollary1.24



v∈V1

d(v) +∑v∈V2

d(v) =∑v∈V

d(v)


v∈V2d(v) is even,

it follows that∑

v∈V1d(v) is even.

Thus |V1| is even.

♠


Corollary1.24



v∈V1

d(v) +∑v∈V2

d(v) =∑v∈V

d(v)


v∈V2d(v) is even,

it follows that∑

v∈V1d(v) is even.

Thus |V1| is even.

♠


Corollary1.24



v∈V1

d(v) +∑v∈V2

d(v) =∑v∈V

d(v)


v∈V2d(v) is even,

it follows that∑

v∈V1d(v) is even.

Thus |V1| is even.

♠


Corollary1.24



v∈V1

d(v) +∑v∈V2

d(v) =∑v∈V

d(v)


v∈V2d(v) is even,

it follows that∑

v∈V1d(v) is even.

Thus |V1| is even.

♠


Regular graph

A graph G is k - regular, or simply regular, if all the vertices of Ghave the same degree k ,

d(v) = k ∀v ∈ V .

Complete graphs are regular.

Complete bipartite graphs Kn,n are regular.

Is there a 3 - regular on 9 vertices?No!For Corollary 2.24


Regular graph


d(v) = k ∀v ∈ V .





Regular graph


d(v) = k ∀v ∈ V .





Regular graph


d(v) = k ∀v ∈ V .





Regular graph


d(v) = k ∀v ∈ V .





Regular graph


d(v) = k ∀v ∈ V .





Regular graph


d(v) = k ∀v ∈ V .





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