A graph G = (V, E) consists of a set of vertices V = { V1, V2, . . . } and set of edges E = { E1, E2, . . . }. The set of unordered pairs of distinct vertices whose elements are called edges of graph G such that each edge is identified with an unordered pair (Vi, Vj) of vertices.The vertices (Vi, Vj) are said to be adjacent if there is an edge Ek which is associated to Vi and Vj. In such a case Vi and Vj are called end points and the edge Ek is said to be connect/joint of Vi and Vj.
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.
Graphs are propular to visualize a problem . Matrix representation is use to convert the graph in a form that used by the computer . This will help to get the efficent solution also provide a lots of mathematical equation .
A graph G = (V, E) consists of a set of vertices V = { V1, V2, . . . } and set of edges E = { E1, E2, . . . }. The set of unordered pairs of distinct vertices whose elements are called edges of graph G such that each edge is identified with an unordered pair (Vi, Vj) of vertices.The vertices (Vi, Vj) are said to be adjacent if there is an edge Ek which is associated to Vi and Vj. In such a case Vi and Vj are called end points and the edge Ek is said to be connect/joint of Vi and Vj.
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.
Graphs are propular to visualize a problem . Matrix representation is use to convert the graph in a form that used by the computer . This will help to get the efficent solution also provide a lots of mathematical equation .
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
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.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
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.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
2. Contents
Basic terminology,
Multi graphs and weighted graphs
Paths and circuits
Shortest path in weighted graph
Hamiltonian and Euler paths and circuits
Planer graph
Travelling salesman problem.
2
3. Introduction to Graphs
Definition: A graph is collection of points called vertices &
collection of lines called edges each of which joins either a pair of
points or single points to itself.
Mathematically graph G is an ordered pair of (V, E)
Each edge eij is associated with an ordered pair of vertices (Vi,Vj).
3
4. Introduction to Graphs
In Fig. G has graph 4 vertices namely
v1, v2, v3, v4& 7 edges
Namely e1, e2, e3, e4, e5, e6, e7 Then e1=(v1, v2)
Similarly for other edges.
In short, we can represent G=(V,E) where V=(v1, v2, v3, v4) &
E=(e1, e2, e3, e4, e5, e6,e7 )
4
v1
v2
v3
v4
e1
e2
e4
e5
e3
e6
e7
Graph G
5. Self Loops & Parallel Edges
Definition: If the end vertices Vi & Vj of any edge eij are same, then
edge eij called as Self Loop.
For Example, In graph G, the edge e6 =(v3, v3) is self loop.
Definition: If there are more than one edge is associated with given
pair of vertices then those edge called as Parallel or Multiple edge.
For Example, In graph G, e4 & e7 has (v3, v4) are called as Parallel
edge.
5
6. Simple & Multiple Graphs
Definition: A graph that has neither self loops or
parallel edge is called as Simple Graph
otherwise it is called as Multiple Graph.
For Example,
G1 (Simple Graph ) G2 (Multiple Graph)
6
7. Weighted Graph
Definition: If each edge or each vertex or both are associated with
some +ve no. then the graph is called as Weighted Graph
For Example,
7
V1 V2
V4 V3
1.4
1.5
1.7
6
8. Finite & Infinite Graph
Definition: A graph is Finite no. of vertices as well as finite no. of
edges called as Finite Graph otherwise it is Infinite Graph.
For Example, The graph G1 & G2 is Finite Graph.
Definition: A graph G=(V,E) is called as Labeled Graph if its
edges are labeled with some names or data.
For Example, Graph G is labeled graph.
8
9. Adjacency & Incidence
Definition: Two vertices v1 & v2 vertices of G joins directly by at
least one edge then there vertices called Adjacent Vertices.
For Example, In Graph G, v1 & v2 are adjacent vertices.
Definition: If Vi is end vertex of edge eij=(vi,vj) then edge eij is said
to be Incident on vi. Similarly eij is said to be Incident on vj.
For Example, In Graph G, e1 is incident on v1 & v2.
9
10. Degree of a Vertex
Definition: The no. of edges incident on a vertex vi with self loop
counted twice is called as degree of vertex vi.
For Example, Consider the Graph G, d(v1)=3 ,d(v2)=2 ,d(v3)=5
,d(v4)= 4
Definition :
A vertex with degree zero is called as Isolated Vertex & A vertex
with degree one is called as Pendant Vertex.
10
v1
v2
v3
v4
e1
e2
e4
e5
e3
e6
e7
11. Handshaking Lemma
Theorem: The graph G with e no. of edges & n no. of vertices, since
each edge contributes two degree, the sum of the degrees of all
vertices in G is twice no. of edges in G.
i.e. d(vi)=2e is called as Handshaking Lemma.
Example: How many edges are there in a graph with 10 vertices,
each of degree 6? Solution: The sum of the degrees of the vertices is
6*10 = 60. According to the Handshaking Theorem, it follows that
2e = 60, so there are 30 edges.
11
n
i=1
12. Matrix Representation of Graphs
A graph can also be represented by matrix.
Two ways are used for matrix representation of graph are given as
follows,
1. Adjacent Matrix
2. Incident Matrix
Lets see one by one…
12
13. 1. Adjacent Matrix
The A.M. of Graph G with n vertices & no parallel edges is a
symmetric binary matrix A(G)=[aij] or order n*n where,
aij=1, if there is as edge between vi &vj.
aij=0, if vi & vj are not adjacent.
A self loop at vertex vi corresponds to aij=1.
For Example,
A(G)=
13
14. 1. Adjacent Matrix
The A.M. of multigraph G with n vertices is an n*n
matrix A(G)=[aij] where,
aij=N, if there one or more edge are there
between vi &vj & N is no. of edges between vi &
vj.
aij=0, otherwise.
For Example,
A(G)=
14
15. 2. Incident Matrix
Given a graph G with n vertices , e edges & no self loops. The
incidence matrix x(G)=[Xij] of the other graph G is an n*e matrix
where,
Xij=1, if jth edge ej is incident on ith vertex vi,
Xij=0, otherwise.
Here n vertices are rows & e edges are columns.
X(G)=
15
16. Directed Graph or Diagraph
Definition: If each edge of the graph G has a direction then graph
called as diagraph.
In a graph with directed edges, the in-degree of a vertex v, denoted
by deg-(v) & out-degree of v, denoted by deg+(v).
See the example in Next page….
16
17. Directed Graph or Diagraph
Example: What are the in-degrees and out-degrees of the vertices a,
b, c, d in this graph:
17
a
b
c
d
deg-(a) = 1
deg+(a) = 2
deg-(b) = 4
deg+(b) = 2
deg-(d) = 2
deg+(d) = 1
deg-(c) = 0
deg+(c) = 2
18. Adjacency Matrix of a diagraph
It is defined in similar fashion as it defined for undirected graph.
For Example,
A(D)=
18
19. Incident matrix of diagraph
Given a graph G with n, e & no self loops is matrix x(G)=[Xij] or
order n*e where n vertices are rows & e edges are columns such
that,Xij=1, if jth edge ej is incident out ith vertex vi
Xij=-1, if jth edge ej is incident into ith vertex vi
Xij=0, if jth edge ej not incident on ith vertex vi.
19
20. Null Graph
Definition: If the edge set of any graph with n vertices is an empty
set, then the graph is known as null graph.
It is denoted by Nn For Example,
N3 N4
20
21. Complete Graph
Definition: Let G be simple graph on n vertices. If the degree of
each vertex is (n-1) then the graph is called as complete graph.
Complete graph on n vertices, it is denoted by Kn.
In complete graph Kn, the number of edges are
n(n-1)/2,For example,
21
K1 K2 K3 K4 K5
22. Regular Graph
Definition: If the degree of each vertex is same say ‘r’ in any graph
G then the graph is said to be a regular graph of degree r.
For example,
22
K3 K4 K5
23. Bipartite Graph
Definition: The graph is called as bipartite graph , if its vertex set
V can be partitioned into two distinct subset say V1 & V2. such that
V1 U V2=V & V1 V2 = & also each edge of G joins a vertex of
V1 to vertex of V2.
A graph can not have self loop.
23
24. Bipartite Graphs
Example I: Is G1 bipartite?
24
v1
v2 v3
No, because there is no way to partition the
vertices into two sets so that there are no edges
with both endpoints in the same set.
Example II: Is G2 bipartite?
v5
v1
v2
v3 v4
v6
v1 v6
v2
v5
v3
v4
Yes, because we
can display G2 like
this:
25. Isomorphism
Definition: Two graphs are thought of as equivalent (called
isomorphic) if they have identical behavior in terms of graph
theoretic properties.
Two graphs G(V, E) & G’(V’,E’) are said to be isomorphic to each
other if there is one-one correspondence between their vertices &
between their edges such that incidence relationship in preserved.
It is denoted by G1=G2
25
26. Isomorphism
For Example,
1 2 a b
4 3 d c
It is immediately apparent by definition of isomorphism that two
isomorphic graphs must have,
the same number of vertices,
the same number of edges, and
the same degrees of vertices.
26
b
2
d
3
c
4
a
1
27. Sub Graph
Definition: A sub graph of a graph G = (V, E) is a graph G’ = (V’,
E’) where V’V and E’E.
For Example:
G G1 G2
27
28. Spanning Graph
Definition: Let G=(V, E) be any graph. Then G’ is said to be the
spanning subgraph of the graph G if its vertex set V’ is equal to
vertex set V of G.
For Example:
G G1 G2
28
29. Complement of a Graph
Definition: Let G is a simple graph. Then complement of G
denoted by ~G is graph whose vertex set is same as vertex set of G
& in which two vertices are adjacent if & only if they are not
adjacent in G.For Example:
G ~G H ~H
29
30. Operations on Graphs
Definition: The union of two simple graphs G1 =
(V1, E1) and G2 = (V2, E2) is the simple graph with vertex set V1 V2
and edge set E1 E2.
The union of G1 and G2 is denoted by G1 G2.
30
G1 G2 G1 G2
31. Operations on Graphs
Definition: The Intersection of two simple graphs G1 =(V1, E1) and
G2 = (V2, E2) is the simple graph with vertex set V1 V2 and edge set
E1 E2.
The Intersection of G1 and G2 is denoted by G1 G2.
31
G1 G2 G1 G2