Simple Graphs and Graph Structures in Discrete Mathematics, Lecture notes of Mathematics

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Discrete Mathematics 5. Graphs & Trees

What are Graphs? •^ General meaning in everyday math:^ A plot or chart of numerical data using a coordinatesystem.

Not

Discrete Mathematics, Spring 2009

-^ Technical meaning in discrete mathematics:^ A particular class of discrete structures (to bedefined) that is useful for representing relations andhas a convenient webby-looking graphicalrepresentation.

Simple Graphs •^ Definition:^ A

simple graph G

=(V

,E)

consists of:^ −^

a set

V^ of

vertices

or^

nodes

(V^

corresponds to the

Visual Representationof a Simple Graph

Discrete Mathematics, Spring 2009

−^ a set

V^ of

vertices

or^

nodes

(V^

corresponds to the

universe of the relation

R), and

−^ a set

E^ of

edges

/^ arcs

/^ links

: unordered pairs of

[distinct?]

elements

u,v

∈^ V

, such that

uRv

-^ Let

V^ be the set of states in the far-southeastern U.S.:

V={FL, GA, AL, MS, LA, SC, TN, NC}

-^ Let

E={{

u,v

}|u

adjoins

v}

Example of a ={{FL,GA},{FL,AL},{FL,MS},

Simple Graph Discrete Mathematics, Spring 2009

={{FL,GA},{FL,AL},{FL,MS},^ {FL,LA},{GA,AL},{AL,MS},{MS,LA},{GA,SC},{GA,TN},{SC,NC},{NC,TN},{MS,TN},{MS,AL}}

TN AL

MS LA

NC SC GA FL

Pseudographs •^ Like a multigraph, but edges connecting anode to itself are allowed. •^ Definition

Discrete Mathematics, Spring 2009

-^ Definition

A^ pseudograph G

=(V

,^ E,

f^ ) where

f:E→

{{u,

v}| u,v

∈V

}. Edge

e∈ E^ is a

loop

if

f(e)={

u,u

}={u

-^ Example:

nodes are campsitesin a state park and edges arehiking trails through the woods.

loop

Directed Graphs •^ Correspond to arbitrary binary relations

R,

which need not be symmetric. • Definition

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-^ Definition

A^ directed graph

(V,

E) consists of a set of vertices

V^ and a binary relation

E^ on

V.

-^ Example:

V^ = people,

E={(

x,y) |

x^ loves

y}

Directed Multigraphs

-^ Like directed graphs, but there may be more than onearc from a node to another. •^ Definition

A^ directed

multigraph

G=(

V,^ E

,^ f^ ) consists of a set

V^ of

Discrete Mathematics, Spring 2009

A^ directed

multigraph

G=(

V,^ E

,^ f^ ) consists of a set

V^ of

vertices, a set

E^ of edges, and a function

f:E

→V

×V.

-^ Example

−^ The WWW is a directed multigraph. −^ V

=web pages,

E=hyperlinks.

Types of Graphs: Summary

-^ Keep in mind this terminology is not fully standardized...

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T er m

E d g ety p e

M u ltip leed g es o k?

S elf-lo o p s o k?

S im p le g rap h

U n d ir.

N o

N o

M u ltig rap h

U n d ir.

Y es

N o

P seu d o g rap h

U n d ir.

Y es

Y es

D irected g rap h

D irected

N o

Y es

D irected m u ltig rap h

D irected

Y es

Y es

Adjacency Let

G^

be an undirected graph with edge set

E. Let

e∈E

be (or map to) the pair {

u,v

}. Then we say:

-^ u

,^ v^

are

adjacent

/^ neighbors

/^ connected

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-^ Edge

e^ is

incident with

vertices

u^ and

v.

-^ Edge

e connects u

and

v.

-^ Vertices

u^ and

v^ are

endpoints

of edge

e.

Degree of a Vertex •^ Let

G^

be an undirected graph,

v∈

V^ a vertex.

-^ The

degree

of^

v, deg(

v), is its number of incident

edges. (Except that any self-loops are counted twice.)

Discrete Mathematics, Spring 2009

twice.) • A vertex with degree 0 is

isolated

-^ A vertex of degree 1 is

pendant

Directed Adjacency •^ Let

G^

be a directed (possibly multi-) graph, and

let^

e^ be an edge of

G^

that is (or maps to) (

u,v

Then we say:^ −^

u^ is

adjacent to

v,^ v

is^ adjacent from

u

Discrete Mathematics, Spring 2009

−^ u

is^ adjacent to

v,^ v

is^ adjacent from

u

−^ e comes from

u, e

goes to

v.

−^ e connects u to v

,^ e goes from u to v

−^ the

initial vertex

of^

e^ is

u

−^ the

terminal vertex

of^

e^ is

v

Directed Degree •^ Definition

Let

G^ be a directed graph,

v^ a vertex of

G.

−^ The

in-degree

of^

v, deg

−(v), is the number of

edges going to

v. Discrete Mathematics, Spring 2009

edges going to

v.

−^ The

out-degree

of^

v, deg

+(v), is the number of

edges coming from

v.

−^ The

degree

of^

v, deg(

v)≡

deg

−(v)+deg

+(v), is the

sum of

v’s in-degree and out-degree.

Special Graph Structures Special cases of undirected graph structures: •^ Complete Graphs

Kn

-^ Cycles

Cn

Discrete Mathematics, Spring 2009

-^ Cycles

Cn

-^ Wheels

W

n

-^ n

-Cubes

Qn

-^ Bipartite Graphs •^ Complete Bipartite Graphs

Km

,n

Complete Graphs •^ Definition:

−^ For any

n∈

N, a

complete graph

on

n^ vertices,

K,n

is a simple graph with

n^ nodes in which every

node is adjacent to every other node:

∀u

,v∈

V:

Discrete Mathematics, Spring 2009

node is adjacent to every other node:

∀u

,v∈

V:

u≠v

u,v

}∈E

K^1

K^2

K^3

K^4

K^5

K^6

Note that K

hasn

edges.

) (^1) ( 2 (^11)

− − n^ nn = i ∑= i