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An introduction to graph theory Arash Rafiey 13 October, 2015 Arash Rafiey An introduction to graph theory

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An introduction to graph theory - Indiana State University

An introduction to graph theory Arash Rafiey 13 October, 2015 Arash Rafiey An introduction to graph theory

An introduction

Arash
13 Octob

Arash Rafiey

to graph theory

Rafiey
ber, 2015

An introduction to graph theory

Definition of Graph

Let V be a finite nonempty set an
(V , E ) is called a digraph where V
called a set of (directed) edges or

Arash Rafiey

nd let E ⊆ V × V .
V is a set of vertices and E is

arcs.

An introduction to graph theory

Definition of Graph

Let V be a finite nonempty set an
(V , E ) is called a digraph where V
called a set of (directed) edges or
When the order does not matter (
graph G = (V , E ) and E (G ) ⊆ {{

a

bc

de
V={a,b,c,d,e}
E={{a,b},{a,c},{b,c},{b,d},{d,c},{c,e

Arash Rafiey

nd let E ⊆ V × V .
V is a set of vertices and E is

arcs.
(relation is symmetric) we have a
{u, v }|u, v ∈ V (G )}

a

bc

de
V={a,b,c,d,e}
e}} E={(a,b),(b,d),(d,b),(c,b),(c,e)}

An introduction to graph theory

For simplicity instead of edge {u,

Arash Rafiey

v } we write edge uv .

An introduction to graph theory

For simplicity instead of edge {u,
Two vertices u and v are called ad

Arash Rafiey

v } we write edge uv .
djacent if uv is an edge of G .

An introduction to graph theory

For simplicity instead of edge {u,
Two vertices u and v are called ad
We say v is a neighbor of u if uv
Let x, y be two vertices of graph G

Arash Rafiey

v } we write edge uv .
djacent if uv is an edge of G .
is an edge of G .
G.

An introduction to graph theory

For simplicity instead of edge {u,
Two vertices u and v are called ad
We say v is a neighbor of u if uv
Let x, y be two vertices of graph G
An x − y walk is an alternating se
starting at x and ending at y .

Arash Rafiey

v } we write edge uv .
djacent if uv is an edge of G .
is an edge of G .
G.
equence of vertices and edges,

An introduction to graph theory

For simplicity instead of edge {u,
Two vertices u and v are called ad
We say v is a neighbor of u if uv
Let x, y be two vertices of graph G
An x − y walk is an alternating se
starting at x and ending at y .
If x = y then the walk is called cl

Arash Rafiey

v } we write edge uv .
djacent if uv is an edge of G .
is an edge of G .
G.
equence of vertices and edges,
losed.

An introduction to graph theory

For simplicity instead of edge {u,
Two vertices u and v are called ad
We say v is a neighbor of u if uv
Let x, y be two vertices of graph G
An x − y walk is an alternating se
starting at x and ending at y .
If x = y then the walk is called cl
A trail is a walk in which all the e

Arash Rafiey

v } we write edge uv .
djacent if uv is an edge of G .
is an edge of G .
G.
equence of vertices and edges,
losed.
edges are distinct.

An introduction to graph theory

For simplicity instead of edge {u,
Two vertices u and v are called ad
We say v is a neighbor of u if uv
Let x, y be two vertices of graph G
An x − y walk is an alternating se
starting at x and ending at y .
If x = y then the walk is called cl
A trail is a walk in which all the e
A path is a simple walk (no verte

Arash Rafiey

v } we write edge uv .
djacent if uv is an edge of G .
is an edge of G .
G.
equence of vertices and edges,

losed.
edges are distinct.
ex repeated).

An introduction to graph theory

For simplicity instead of edge {u,
Two vertices u and v are called ad
We say v is a neighbor of u if uv
Let x, y be two vertices of graph G
An x − y walk is an alternating se
starting at x and ending at y .
If x = y then the walk is called cl
A trail is a walk in which all the e
A path is a simple walk (no verte
A cycle is a simple closed walk (n
beginning ).

Arash Rafiey

v } we write edge uv .
djacent if uv is an edge of G .
is an edge of G .
G.
equence of vertices and edges,

losed.
edges are distinct.
ex repeated).
no vertex repeated except the

An introduction to graph theory

Theorem
Let G = (V , E ) be undirected gra
exists a trail from a to b, then the

Arash Rafiey

aph with a, b ∈ V , a = b. If there
ere exists a path from a to b.

An introduction to graph theory

Theorem
Let G = (V , E ) be undirected gra
exists a trail from a to b, then the

Proof.
Consider the shortens trail a, x1, x
(ax1, x1x2, x2x3, . . . , xn−1xn, xnb ar
path then there exist k, m where x
k < m ). But then we can contra
a, x1, . . . , xk , xm+1, . . . , xn, b from

Arash Rafiey

aph with a, b ∈ V , a = b. If there
ere exists a path from a to b.

x2, . . . , xn, b in G
re edges). If this trail is not a
xk = xm (x0 = a, xm+1 = b,
act and get a shorter trail
a to b.

An introduction to graph theory

Definition

A graph G is called connected if
distinct vertices of G .

a c
b

de
G (connected)

Definition

If G is not connected then it can
each piece is a connected graph a
component .
The number of connected compon

Arash Rafiey

f there is a path between any two

a
bc
de
H (not connected)

be partitioned into pieces where
and is called a connected
nents of G is denoted by κ(G ).

An introduction to graph theory

Example :
Let G = (V , E ) be an undirected
n-sequences and the two vertices
exactly two positions.
Find κ(G ).

Arash Rafiey

graph whose vertices are binary
x, y are adjacent if they differ in

An introduction to graph theory

Definition
A graph G is simple if there are n
self-loop.

Arash Rafiey

no parallel edges and there is no

An introduction to graph theory

Definition
A graph G is simple if there are n
self-loop.

Definition
If v is a vertex of graph G , then t
(dG (v ), or dv ) is the number of ed
of neighbors of v . The self-loop is
If G is a simple graph and each ve
called a k-regular graph.

Arash Rafiey

no parallel edges and there is no

the degree of v , denoted deg (v )
dges incident to v . Is the number
s counted twice.
ertex has degree k then G is

An introduction to graph theory

Theorem (Handshaking)
For any graph G = (V , E ) we hav

deg (v )

v ∈V

Arash Rafiey

ve
= 2|E (G )|.

An introduction to graph theory

Theorem (Handshaking)
For any graph G = (V , E ) we hav

deg (v )

v ∈V

Proof.
If an edge e has two end points u
to each of deg (u), deg (v ). If e is
then it will contribute two to deg (
contributes two in the sum and id

Arash Rafiey

ve
= 2|E (G )|.

u and v then it will contribute one
s a self loop incident to vertex u
(u). In any case each edge
dentity follows.

An introduction to graph theory

Theorem (Handshaking)
For any graph G = (V , E ) we hav

deg (v )

v ∈V

Proof.
If an edge e has two end points u
to each of deg (u), deg (v ). If e is
then it will contribute two to deg (
contributes two in the sum and id

Corollary
In any graph G the number of ver

Arash Rafiey

ve
= 2|E (G )|.

u and v then it will contribute one
s a self loop incident to vertex u
(u). In any case each edge
dentity follows.

rtices of odd degree must be even.

An introduction to graph theory

Some families of Graphs

1) A graph G is called complete
every other vertex.

Arash Rafiey

if every vertex is adjacent to

An introduction to graph theory

Some families of Graphs

1) A graph G is called complete
every other vertex.
2) For every n ≥ 2, the n-path-gra
on n vertices.
V (Pn) = {v1, v2, . . . , vn} and E (P

Arash Rafiey

if every vertex is adjacent to
aph, denoted by Pn is just a path
Pn) = {v1v2, v2v3, . . . , vn−1vn}.

An introduction to graph theory

Some families of Graphs

1) A graph G is called complete
every other vertex.
2) For every n ≥ 2, the n-path-gra
on n vertices.
V (Pn) = {v1, v2, . . . , vn} and E (P
3) For every n ≥ 2, the n-cycle-gr
on n vertices.
V (Cn) = {v1, v2, . . . , vn}, E (Cn) =

Arash Rafiey

if every vertex is adjacent to
aph, denoted by Pn is just a path
Pn) = {v1v2, v2v3, . . . , vn−1vn}.
raph, denoted by Cn is just a cycle
= {v1v2, v2v3, . . . , vn−1vn, vnv1}.

An introduction to graph theory

Some families of Graphs

1) A graph G is called complete
every other vertex.
2) For every n ≥ 2, the n-path-gra
on n vertices.
V (Pn) = {v1, v2, . . . , vn} and E (P
3) For every n ≥ 2, the n-cycle-gr
on n vertices.
V (Cn) = {v1, v2, . . . , vn}, E (Cn) =
4) For every n ≥ 2, the n-hypercu
V (Qn) = {length n bit string }
their bit strings differ in exactly on

Arash Rafiey

if every vertex is adjacent to

aph, denoted by Pn is just a path
Pn) = {v1v2, v2v3, . . . , vn−1vn}.
raph, denoted by Cn is just a cycle
= {v1v2, v2v3, . . . , vn−1vn, vnv1}.
ube, denoted by Qn has vertex set

and two vertices are adjacent if
ne position.

An introduction to graph theory

Bipartite Graphs

Definition
A graph G = (V , E ) is bipartite if
such that each edge of G has one
point in W .

Arash Rafiey

f V can be partitioned into U, W
e end point in U and one end

An introduction to graph theory


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