Graph theory introduction - Samy

Prescribed Books:
Graph Theory by Narsingh Deo
Recommended Books for Reference:
• Graph Theory by Hararay.
• Discrete Mathematics by Sahaum Outline Series.
 Discrete Mathematics and its applications by
Kenneth H. Rosen.

Topics Covered
 History of Graph Theory
 Why “Graphs”
 Definitions
 Types
 Terminology
 Representation
 Sub-graphs
 Connectivity
 Hamilton and Euler definitions
 Shortest Path
 Planar Graphs
 Graph Coloring

The origin of graph theory can be
traced back to Euler's work on the
Konigsberg bridges problem (1735),
which subsequently led to the concept
of an Eulerian graph.
The study of cycles on polyhedra by
the Thomas P. Kirkman (1806 - 95)
and William R. Hamilton (1805-65) led
to the concept of a Hamiltonian graph.
History of Graph Theory

The concept of a tree, a connected
graph without cycles, appeared implicitly
in the work of Gustav Kirchhoff (1824-
87), who employed graph-theoretical
ideas in the calculation of currents in
electrical networks or circuits.
Later, Arthur Cayley (1821-95), James J.
Sylvester(1806-97), George Polya(1887-
1985), and others use 'tree' to
enumerate chemical molecules.

The study of planar graphs originated in two
recreational problems involving the
complete graph K5 and the complete
bipartite graph K3,3. These graphs proved to
be planarity, as was subsequently
demonstrated by Kuratowski.

In particular the term “graph” was
introduced by Sylvester in a paper
published in 1878 in Nature, where
draws an analogy between “quantic
invariants” and “co-variants” of
algebra and molecular diagrams:

The first example of such use comes from work of
physicist Gustav Kirchhoff, who published in 1845 his
Kirchhoff’s circuit laws for calculating the voltage and
current in electric circuits.
The paper written by Leonhard Euler on the seven
bridges of konigsberg and published in 1736 is
regarded as the first paper in the history of graph
theory.
Euler’s fromula relating the number of edges, vertices
and faces of a convex polyhedron was studied and
generalised by Cauchy and L’Huillier, and is at the
origin of topology.

More than one century after Euler’s paper on the bridges of
konigsberg and while Listing introduced topology, Cayley was led
by the study of particular analytical forms arising from differential
calculus to study a particular class of graphs, the trees. This
study had many implications in theoretical chemistry. The
involved techniques mainly concerned the enumeration of graph
having particular properties. Enumerative graph theory then rose
from the results of Cayley and the fundamental results published
by polya between 1935 and 1937 and the generalisation of these
by De Bruijn in 1959. Cayley linked his results on trees with the
contemporary studies of chemical composition. The fusion of the
ideas coming from mathematics with those coming from chemistry
is at the origin of a part of the standard terminology of graph
theory

Once upon a time, there was a king with five sons.
In his will he stated that after his death the sons
should divide the kingdom into five provinces so
that the boundary of each province should have
a frontiers line in common with each of the other
four provinces.
Here the problem is whether one can draw five
mutually neighboring regions in the plane.
The king further stated that all five brothers should
join the provincial capital by roads so that no two
roads intersect.
Here the problem is that deciding whether the
graph K5 is planar.
First problem was presented by A. F. Mobius around
the year 1840 as follows

The origin of second problem is unknown but
it is first mentioned by H. Dudeney in 1913 in
its present form.
The puzzle is to lay a water, gas, and electricity
to each of the three houses without any pipe
crossing another.
This problem is that of deciding whether the
graph K3,3 is planar
Second problem-Is K3,3 planar ?

Why “graphs”?
Many problems can be stated in terms of a graph.
The properties of graphs are well-studied
Many algorithms exists to solve problems posed as graphs.
Many problems are already known to be intractable.
 By reducing an instance of a problem to a standard graph
problem, we may be able to use well-known graph
algorithms to provide an optimal solution.
Graphs are excellent structures for storing, searching, and
retrieving large amounts of data
 Graph theoretic techniques play an important role in
increasing the storage/search efficiency of computational
techniques.
Go To

Some applications of Graph Theory
 Models for communications and electrical networks
 Models for computer architectures
 Network optimization models for operations analysis,
including scheduling and job assignment
 Analysis of Finite State Machines
 Parsing and code optimization in compilers
 Graphs to printed circuit and microchip design. Graphs
seem an intuitively natural way to model many situations
in the Creation (connections of wires / leads, logistics /
transportation problems, pipelines between points with
known capacities, family trees, organizational charts,
among many more).

Graph theory has been applied in
• social science
• electrical engineering
• computer science
• engineering material
• kinematic chains
• mechanisms
• Graph engineering
• failure analysis
• Quality

• automobile vehicle design
• reinforced polymer composites
• electroplating
• mechatronic products
• thermal power plant
• manufacturing systems
• total quality management
• image processing and analysis
• modeling neighborhood relationships
• defining graph-theoretical algorithms

• filtering
• enhancement
• reliability
• restoration
• object extraction.,
• Quantum Graph Theory
• web search engines, ect

Draw a Triangle
0
0
Side
Vertex
Draw a Cube
Vertex
Edge

Loyola
Sidhartha
Stella
Besant
Road
1 Town
Benz Circle
Vertex
Vertex
Vertex
Vertex
Vertex
Vertex

ALC
Sidhartha
MSC
Besant
Road
Benz Circle
1 Town
Edge
Edge
Edge
Edge
Edge
Edge
Edge
Edge Edge

ALC
Sidhartha
MSC
Besant
Road
1 Town
Benz Circle
1 KM
4 KM
2.5 KM
6 KM
6 KM
6.5 KM
2.5 KM
2 KM
0.5 KM
Weight
Weight
Weight
?
?

Influence of graph theory is so great
that Department of Science and
Technology, Government of India,
announced 10th June of every year as
Graph Theory Day.

We celebrate June 10th as Graph
Theory Day marking the birth
anniversary of Professor E.
Sampathkumar, the pioneer in
introducing Graph Theory into the
postgraduate curriculum in India
during the academic year 1970-71 at
the famed Karnatak University,
Dharwad.
Statistic Day 29th June-2007
National Mathematics Day 22nd Dec -2012

Influence of graph theory is so great that
Department of Science and Technology,
Government of India, announced 10th June
of every year as Graph Theory Day.(2014)
Vertices
Recently we celebrated 14th Graph Theory
Day.

of every year as Graph Theory Day.
Edges

Vertices
Edges

A graph G=(V,E) consisting of a set of objects
V={v1,v2, … ,vn} called vertices and another
set E={e1,e2,…,em} called edges such that
each edge ek is identified with an unordered
pair (vi , vj) of vertices.
What is a Graph?

What is a Graph?
vi vjek

What is a Graph?
vi vjek
Nott
X
Y

What is a Graph?
vi vj
ek
Nott
a
c
b
hg i
d
k
j

What is a Graph?
vi vj
ek
Nott
Vertex V = { a, b, c, d}
Edge E = {g, h, i, j, k}
a
c
b
hg i
d
k
j

This graph consists of Vertices and Edges
Example
5 7

A vertex is also referred to as a
node or a junction or a point or
0-cell or an 0-simplex.
Other terms used for an edge
are a branch or a line or an
element, a 1-cell or a 1-simplex
or an arc.

• The magnitude of graph G is characterized by number
of vertices |V| (called the order of G) and number of
edges |E| (size of G)
• The running time of algorithms are measured in terms of
the order and size
Order and Size of a Graph
|V|=5 and |E | =6 |V|=6 and |E | =9 |V|=5 and |E | =10

Size – of a graph is the number of
vertices that the graph has
8
2
4
3 7
5
6

Graphs ↔ Networks
Graph
(Network)
Vertexes
(Nodes)
Edges
(Arcs)
Towns
Atoms
Roads
Chemical Bonds
Terminals Wire joining Terminals
People
Pair of people knowing
each other
Grammatical RelationsWords
LinksWeb Pages
Road maps
Chemical molecules
Electrical Networks
Social Networks
Linguistics
W.W.W

Graphs ↔ Networks
Graph
(Network)
Vertexes
(Nodes)
Edges
(Arcs)
Flow
Telephones exchanges,
computers, satellites
Gates, registers,
processors
Cables, fiber optics,
microwave relays
Voice, video,
packets
Wires Current
Joints
Rods, beams,
springs Heat, energy
Reservoirs, pumping
stations, lakes
Pipelines Fluid, oil
MoneyTransactionsStocks, currency
Highways, rail beds,
airway routes
Airports, rail yards,
street intersections
Freight, vehicles,
passengers
Communications
Circuits
Mechanical
Hydraulic
Financial
Transportation
Flow

An edge where the two end vertices are
the same is called a loop, or a self-loop
Multiple Edges (or) Parallel Edges: Two or
more edges joining the same pair of vertices.
Multiple Edges
Loops and Multiple Edges
They are NOT multiple edges
Yes, Parallel or
Multiple EdgesAre they parallel Edges?

Undirected graph Directed graph
G=(V,E)

Multigraph: A graph that contains
u
v
we3
e2
e1
multiple edges but no loops is called a
multigraph
The edges e1 and e3 are called multiple or
parallel edges.

Multigraphs
If computers are connected via internet
instead of directly, there may be several
routes to choose from for each
connection. Depending on traffic, one
route could be better than another.
Makes sense to allow multiple edges,
but still no self-loops:

u
v
w
e3
e1
Pseudograph
A graph that may contain multiple edges
and loops is called a pseudograph.

Example of a Pseudograph
 A computer network may contain vertices with
loops, which are edges from a vertex to itself.

Undirected Graphs
pseudographs
simple graphs
multigraphs

Labeled graphs: Labels
are just the names we
give vertices and edges
so we can tell them apart.
Infinite graphs: A
graph with infinite
number of vertices
and edges is called
an infinite graph.
Finite graphs: A graph with finite number of
vertices and edges is called a finite graph.
a b
cd
56
2
3

Which of the following statements
hold for this graph?
(a) nodes v and u are adjacent;
(b) nodes v and x are adjacent;
(c) node u is incident with edge 2;
(d)Edge 5 is incident with node x.
Ans: (a) Yes , connected by the edge 2
(b) No, No edge joins the vertices v and x
(c) Yes, node v is also incident with edge 2
(d) No, edge 5 is incident with nodes v and w

A Directed Graph
San Francisco
Denver
Los Angeles
New York
Chicago
Washington
Detroit
SOME TELEPHONE LINES IN THE NETWORK MAY OPERATE
IN ONLY ONE DIRECTION .
Those that operate in two directions are represented by pairs of
edges in opposite directions.

A Directed Multigraph
San Francisco
Denver
Los Angeles
New York
Chicago
Washington
Detroit
THERE MAY BE SEVERAL ONE-WAY LINES IN THE
SAME DIRECTION FROM ONE COMPUTER TO
ANOTHER IN THE NETWORK.

loop
node
multiple arc
or edges
Directed Graph (digraph)
Arc or edge
Recall

Que: For a set V with n elements, how
many possible digraphs are there?
Ans: The same as the number of
relations on V, which is the number
of subsets of V V so
2
2n

60
Types of Graphs
TYPE EDGES MULTIPLE EDGES LOOPS
ALLOWED? ALLOWED?
Simple graph
Multigraph
Pseudograph
Directed graph
Directed multigraph
Undirected
Undirected
Undirected
Directed
Directed NO
NO
NONO
YES
YESYES
YES
YESYES

2
n
C
2
( 1)
2
n n n
C


The maximum number of edges possible
in a simple graph with n vertices is
where
.
The maximum number of edges with n=3 vertices is
2
( 1) 3(3 1) 3(2)
3
2 2 2
n n n
C
 
    edges.

The number of simple graphs possible with n
vertices = 2
( 1)
C 2
2 2
n
n n

The maximum number of simple graphs
with n=3 vertices is
2
( 1) 3(3 1) 3(2)
C 32 2 2
2 2 2 2 2 8
n
n n 
    

Definition: An empty graph on zero nodes is
called a null graph.
Definition: A graph with only one vertex is
called a trivial graph or singleton graph.
Definition: A graph having no edges is called an
empty graph.
Example: In the above graph, there are three
vertices named a, b, and c, but there are no edges
among them.

Example of a Directed Multigraph
 The algorithm uses a finite number of steps, since
it terminates after all the integers in the sequence
have been examined.

Weighted graphs
1 2 3
4 5 6
.5
1.2
.2
.5
1.5
.3
1
4 5 6
2 3
2
1
35
 is a graph for which each edge has an associated
weight, usually given by a weight function w: E
 R.
2
4
3
1
4
8
6
2
9
4
8 13
5

Acquaintanceship Graph
 To represent whether two people know
each other (whether they are acquainted)
Each person is
represented by a
vertex.
An undirected edge
connects two people
when they know
each other.

Degree of all the vertices of the graph is even.
deg (v1) =4
deg(v2) =6
deg(v3) =6
deg(v4) =6
Find the degrees of all the vertices:
deg(v2) =1+1+1+1+2(loop)=6
deg (v1) =1+
1
1
1
1
deg (v1) =1+1deg (v1) =1+1+1deg (v1) =1+1+1+1=4
deg(v4) =1+1+2(loop)+2(loop)=6
Degree of a self-loop is 2
The number of edges incident (connected) to a vertex

The degree of a vertex is the number of
edges incident (connected) to it, i.e. the
number of edges that have it as an endpoint.
The degree of a vertex is sometimes also refer
as its valency.
Degree 0 Degree 1 Degree2 Degree 3 Degree 4

Degree of a Vertex
Definition: The degree of a vertex in an
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.
In other words, you can determine the
degree of a vertex in a displayed graph by
counting the lines that touch it.
The degree of the vertex v is denoted by
deg(v).

vertex whose degree is one is called a
pendant vertex.
a
b
g f e
c d
Which is the pendant vertex ?
d is the pendant vertex ?
What about the vertex e ?

A vertex whose degree is zero called an
isolated vertex.
A loop counts as two degree.
A vertex is said to be even or odd according
as its degree is even or odd number.
a
b
g f e
c d
The vertex e an isolated vertex
a,
Odd vertices are
Even vertices are
d and f
and gb, c, e

Degree of a Vertex (deg (v)):
u
w
v
A loop contributes twice to the degree (why?).
Pendant Vertex: deg (v) =1 and deg (W) =1.
Isolated Vertex: deg (k) = 0.
4.
2.deg (U) =
deg (U) =
the number of edges incident on a vertex.

Example
 Find the degrees of all the vertices:
deg(a) = ?, deg(b) = ?, deg(c) = ?, deg(d) = ?,
deg(e) = ?, deg(f) =?, deg(g) = ?.
a
b
g e
c d
f

a
deg( b ) =
Degree of a vertex
The degree of a vertex in an 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. The
degree of an isolated vertex is zero.
b
g f e
c d

a
deg( b ) = 6
Degree of a vertex
Find the degree of all the other vertices.
deg( a ) deg( c ) deg( f ) deg( g )
b
g f e
c d
deg( d ) =
deg( e ) =
1
0

a
deg( b ) = 6
Degree of a vertex
deg( a ) = deg( c ) = deg( f ) = deg( g ) =
b
g f e
c d
deg( d ) = 1
deg( e ) = 0
2 4 3 4

a
deg( b ) =
Degree of a vertex
deg( a ) = 2 deg( c ) = 4 deg( f ) = 3 deg( g ) = 4
TOTAL of degrees = 2 + 4 + 3 + 4 + 6 + 1 + 0 = 20
b
g f e
c d
deg( d ) = 1
deg( e ) = 0
6

a
b
cd
in-degrees and out-degrees
• In-degree : Number of edges entering
• Out-degree : Number of edges leaving

a c
b
e d f
Find the in-degrees and out-degrees of this digraph.
In-degrees : deg-(a) = , deg-(b) = , deg-(c) = ,
deg-(d) = , deg-(e) = , deg-(f) =
Out-degrees: deg+(a) = , deg+(b) = , deg+(c) = ,
deg+(d) = , deg+(e) = , deg+(f) =
Example• In-degree : Number of edges entering
2 2
2
2
2
3
3
3
0
0
4 1
Question: How does adding a loop to a vertex
change the in-degree and out-degree of that
vertex?
Answer: It increases both the in-degree and the
out-degree by one.

in-degrees and out-degrees
Example: What are the in-degree (deg-(v) ) and
out-degrees (deg+(v)) of the vertices a, b, c, d in
this graph:
a b
cd
deg-(a) = 1
deg+(a) = 2
deg-(b) = 4
deg+(b) = 2
deg-(d) = 2
deg+(d) = 1
deg-(c) = 0
deg+(c) = 2
• In-degree : Number of edges entering

in-degree & out-degree of a vertex v
Definition: In a graph with directed edges, the in-
degree of a vertex v, denoted by deg-(v), is the
number of edges with v as their terminal vertex.
The out-degree of v, denoted by deg+(v), is the
number of edges with v as their initial vertex.
Question: How does adding a loop to a vertex
change the in-degree and out-degree of that
vertex?
Answer: It increases both the in-degree and the
out-degree by one.

Degree (Directed Graphs)
 In-degree : Number of edges entering
 Out-degree: Number of edges leaving
outdeg(1)=2
indeg(1)=0
outdeg(2)=2
indeg(2)=2
outdeg(5)=1
indeg(5)=2
outdeg(3)=1
indeg(3)=4
outdeg(4)=2
indeg(4)=0
In-degree=0+2+4+0+2=8 Out-degree=2+2+1+2+1=8
Degree=2+4+5+2+3=16
Degree(16) = indeg(8) + outdeg(8)

Degree: Simple Facts
 The sum of the in-degrees of all vertices in a
digraph = the sum of the out-degrees = the
number of edges.
 If G is a digraph, then
 indeg(v )= outdeg(v ) = |E |
 i.e.,    deg deg
v V v V
v v E 
 
  

Graphs in chemistry
• molecular (structural) graphs (often: hydrogen-
supressed)
• degree of a vertex = valence of atom

• reaction graphs – union of the molecular graphs of
the supstrate and the product
C C
C C
CC
2 : 1
2 : 1
2 : 1
0 : 1
0 : 11 : 2
Diels-Alder reaction

Definition: If the degrees of all vertices in a graph are
arranged in descending or ascending order, then the
sequence obtained is known as the degree sequence
of the graph.
Example (DSG1):
Vertex a b c d e
Connecting to
Degree
In the above graph, for the vertices
the degree sequence is
2 2 2 3 1
b,c a,d a,d c,b,e d
{3, 2, 2, 2, 1}.
{d, a, b, c, e},

Example (DSG 2):
Vertex a b c d e f
In the above graph, for the vertices {a, b, c, d, e, f}, the
degree sequence is
Degree
Connecting to b,e a,c b,d c,e a,d -
2 2 2 2 2 0
{2, 2, 2, 2, 2, 0}.
Conti…
{2, 2, 0, 2, 2, 2, }

a
deg( b ) = 6
Degree of a vertex
deg( a ) = 2 deg( c ) = 4 deg( f ) = 3 deg( g ) = 4
b
g f e
c d
deg( d ) = 1
deg( e ) = 0
TOTAL - degrees = 2 + 4 + 3 + 4 + 6 + 1 + 0 = 20
TOTAL NUMBER OF EDGES = 10

Count the sum of the degrees and
the number of edges
a
c
b
hg i
d
k
j
What is the relation between sum of the
degrees and the number of edges

Graph theory introduction - Samy

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Graph theory introduction - Samy