An introduction to graph theory - Indiana State University

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 ).

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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.

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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 .

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

An introduction to graph theory