The document provides an introduction to graph theory. It lists prescribed and recommended books, outlines topics that will be covered including history, definitions, types of graphs, terminology, representation, subgraphs, connectivity, and applications. It notes that the Government of India designated June 10th as Graph Theory Day in recognition of the influence and importance of graph theory.
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.Graph theory is also important in real life.
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.Graph theory is also important in real life.
what is Hamilton path and Euler path?
History of Euler path and Hamilton path
Vertex(node) and edge
Hamilton path and Hamilton circuit
Euler path and Euler circuit
Degree of vertex and comparison of Euler and Hamilton path
Solving a problem
what is Hamilton path and Euler path?
History of Euler path and Hamilton path
Vertex(node) and edge
Hamilton path and Hamilton circuit
Euler path and Euler circuit
Degree of vertex and comparison of Euler and Hamilton path
Solving a problem
The Age of EulerRarely has the world seen a mathematician as pro.docxmehek4
The Age of Euler
Rarely has the world seen a mathematician as prolific as the great Leonhard Euler1 (1707-1783). Born in Switzerland, he eventually obtained royal appointments in two European courts, namely Russia and Germany (under Frederick the Great). He published so many mathematics articles that his work fills seventy thick volumes. His publications account for one-third of all the technical articles of eighteenth-century Europe. The preceding century saw the rise of scientific and mathematical journals – the new media of the times and the quickest way of making innovations known to colleagues across the continent. This outgrowth of the printing revolution of the fifteenth century accelerated the pace of mathematical and scientific progress by transmitting new ideas in a timely manner – much like the present computer revolution has just begun to affect the dissemination of knowledge.
1Euler was the person who gave us the notation π for pi, i for , Δy for the change in y, f (x) for a function, and Σ for summation.
After Euler’s death, it took forty years for the backlog of his work to appear in print. Although he lost his sight in 1768, for the last fifteen years of his life he continued his research at his usual energetic pace while his students copied his pearls of wisdom. It is inconceivable to most how he did mathematics without pencil and paper – without being able to see the multitude of diagrams, equations, and graphs needed to do research.
What areas of math did he enrich and expand? The question is what field of math did he not enrich and expand! Not only did he contribute substantially to calculus, geometry, algebra, and number theory, he also invented several fields. Though a father to eleven children, Euler found time to become the father of an important branch of mathematics, known today as graph theory, which would be important in modern fields such as computer science and operations research, as well as traditional areas such as physics and chemistry.
Euler became the father of graph theory as well as topology after solving the notorious “Seven Bridges of Königsberg” problem. The diagram of Figure 10-1 shows the four landmasses of the city of Königsberg and the seven bridges interconnecting them.
Figure 10-1
The problem was to devise a route that traverses each bridge exactly once and to end where one starts. Euler observed that the task could not be done!! He noticed that each landmass has an odd number of bridges connecting it with the rest of the city. Hence a traveler departing, returning, departing, and so forth, an odd number of times would wind up departing on the last bridge, rendering impossible his return to his point of origin.
Let’s consider this gem of thinking one more time. Number the bridges contiguous with landmassA, 1, 2, and 3. Then if one starts the trip by departing A on bridge number one, he must return on bridge number two or number three, leaving only one more bridge. Clearly he must depart on that ...
Characteristics of Fuzzy Wheel Graph and Hamilton Graph with Fuzzy Ruleijtsrd
Graph theory is the concepts used to study and model various application in different areas. We proposed the wheel graph with n vertices can be defined as 1 skeleton of on n 1 gonal pyramid it is denoted by w nwith n 1 vertex n=3 . A wheel graph is hamiltonion, self dual and planar. In the mathematical field of graph theory, and a Hamilton path or traceable graph is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle is a hamiltonian path that is a cycle. In this paper, we consider the wheel graph and also the hamilton graph using if then rules fuzzy numbers. The results are related to the find the degree of odd vertices and even vertices are same by applying if then rules through the paths described by fuzzy numbers. Nisha. D | Srividhya. B "Characteristics of Fuzzy Wheel Graph and Hamilton Graph with Fuzzy Rule" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-6 , October 2019, URL: https://www.ijtsrd.com/papers/ijtsrd29319.pdf Paper URL: https://www.ijtsrd.com/mathemetics/other/29319/characteristics-of-fuzzy-wheel-graph-and-hamilton-graph-with-fuzzy-rule/nisha-d
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2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
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Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
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The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
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June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
3. 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.
4. 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
7. 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
8. 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.
History of Graph Theory
9. 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.
History of Graph Theory
10. 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:
11. 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.
12. 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
13. 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
14. 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 ?
15. 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
18. 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).
19. Graph theory has been applied in
• social science
• electrical engineering
• computer science
• engineering material
• kinematic chains
• mechanisms
• Graph engineering
• failure analysis
• Quality
20. • 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
28. 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.
29. 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
30. 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.
31. 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.
Edges
32. 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.
Vertices
Edges
33. 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.
34. 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.
Vertices
Edges
35. 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?
36. 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?
vi vjek
37. 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?
vi vjek
Nott
X
Y
38. 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?
vi vj
ek
Nott
a
c
b
hg i
d
k
j
39. 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?
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
42. 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.
43. • 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
44. Size – of a graph is the number of
vertices that the graph has
8
2
4
3 7
5
6
47. 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?
49. 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.
50. 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:
54. 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
55. 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
56. 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.
57. 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.
59. 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. 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
61. 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.
62. 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
63. 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.
64. 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.
65. 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
66. 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.
68. 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
69. 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
70. 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).
71. 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 ?
72. 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
73. 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.
74. 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
75. 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
76. 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
77. 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 ) = 1
deg( e ) = 0
2 4 3 4
78. a
deg( b ) =
Degree of a vertex
Find the degree of all the other vertices.
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
80. 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
• Out-degree : Number of edges leaving
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.
81. 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
• Out-degree : Number of edges leaving
82. 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.
83. 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)
84. 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
85. Graphs in chemistry
• molecular (structural) graphs (often: hydrogen-
supressed)
• degree of a vertex = valence of atom
86. • 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
87. 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},
88. 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, }
89. a
deg( b ) = 6
Degree of a vertex
Find the degree of all the other vertices.
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
90. 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