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.
A Perspective on Graph Theory and Network ScienceMarko Rodriguez
The graph/network domain has been driven by the creativity of numerous individuals from disparate areas of the academic and the commercial sector. Examples of contributing academic disciplines include mathematics, physics, sociology, and computer science. Given the interdisciplinary nature of the domain, it is difficult for any single individual to objectively realize and speak about the space as a whole. Any presentation of the ideas is ultimately biased by the formal training and expertise of the individual. For this reason, I will simply present on the domain from my perspective---from my personal experiences. More specifically, from my perspective biased by cognitive and computer science.
This is an autobiographical lecture on my life (so far) with graphs/networks.
FREQUENT SUBGRAPH MINING ALGORITHMS - A SURVEY AND FRAMEWORK FOR CLASSIFICATIONcscpconf
Data mining algorithms are facing the challenge to deal with an increasing number of complex
objects. Graph is a natural data structure used for modeling complex objects. Frequent subgraph
mining is another active research topic in data mining . A graph is a general model to represent
data and has been used in many domains like cheminformatics and bioinformatics. Mining
patterns from graph databases is challenging since graph related operations, such as subgraph
testing, generally have higher time complexity than the corresponding operations on itemsets,
sequences, and trees. Many frequent subgraph Mining algorithms have been proposed. SPIN,
SUBDUE, g_Span, FFSM, GREW are a few to mention. In this paper we present a detailed
survey on frequent subgraph mining algorithms, which are used for knowledge discovery in
complex objects and also propose a frame work for classification of these algorithms. The
purpose is to help user to apply the techniques in a task specific manner in various application domains and to pave wave for further research.
Map Coloring and Some of Its Applications MD SHAH ALAM
This is a research paper which I have conducted at the final year of undergrad study and got 4.00/4.00. It is mainly related to graph theory and has many applications in practical life.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
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
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.
Embracing GenAI - A Strategic ImperativePeter 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.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
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.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
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.
4. Introduction to Graphs
Graph theory may be said to have its beginning in 1736
EULER considered the Königsberg bridge problem
It took 200 years before the first book on graph theory
was written.
Since then graph theory has developed into an
extensive and popular branch of mathematics, which
has been applied to many problems in mathematics,
computer science, and other scientific and not-so-
scientific areas.
5. Graphs
Definition: A graph G = (V, E) consists of a nonempty set V of vertices (or
nodes) and a set E of edges. Each edge has either one or two vertices associated
with it, called its endpoints. An edge is said to connect its endpoints.
Remarks:
We have a lot of freedom when we draw a picture of a graph. All that matters is the connections made
by the edges, not the particular geometry depicted. For example, the lengths of edges, whether edges
cross, how vertices are depicted, and so on, do not matter
A graph with an infinite vertex set is called an infinite graph. A graph with a finite vertex set is called a
finite graph. We (following the text) restrict our attention to finite graphs.
a
c
b
d
Example:
This is a graph
with four
vertices and five
edges.
6. Some Terminology
In a simple graph each edge connects two different vertices and no two
edges connect the same pair of vertices.
Multigraphs may have multiple edges connecting the same two
vertices. When m different edges connect the vertices u and v, we say
that {u,v} is an edge of multiplicity m.
An edge that connects a vertex to itself is called a loop.
A pseudograph may include loops, as well as multiple edges connecting
the same pair of vertices.
Example:
This pseudograph
has both multiple
edges and a loop.
a b
c
7. Directed Graphs
Definition: An directed graph (or digraph) G = (V, E)
consists of a nonempty set V of vertices (or nodes) and
a set E of directed edges (or arcs). Each edge is
associated with an ordered pair of vertices. The
directed edge associated with the ordered pair (u,v) is
said to start at u and end at v.
Remark:
Graphs where the end points of an edge are not ordered
are said to be undirected graphs.
8. Some Terminology (continued)
A simple directed graph has no loops and no multiple edges.
A directed multigraph may have multiple directed edges. When there
are m directed edges from the vertex u to the vertex v, we say that (u,v)
is an edge of multiplicity m.
a b
c
c
a b
In this directed multigraph the
multiplicity of (a,b) is 1 and the
multiplicity of (b,c) is 2.
Example:
This is a directed graph with
three vertices and four edges.
Example:
9. Graph Models:
Computer Networks
When we build a graph model, we use the appropriate type of graph to
capture the important features of the application.
We illustrate this process using graph models of different types of
computer networks. In all these graph models, the vertices represent
data centers and the edges represent communication links.
To model a computer network where we are only concerned whether
two data centers are connected by a communications link, we use a
simple graph. This is the appropriate type of graph when we only care
whether two data centers are directly linked (and not how many links
there may be) and all communications links work in both directions.
10. Graph Models:
Computer Networks (continued)
• To model a computer network
where we care about the number
of links between data centers, we
use a multigraph.
• To model a computer network
with diagnostic links at data
centers, we use a pseudograph,
as loops are needed.
• To model a network with multiple one-
way links, we use a directed multigraph.
Note that we could use a directed graph
without multiple edges if we only care
whether there is at least one link from a
data center to another data center.
11. Graph Terminology: Summary
To understand the structure of a graph and to build a graph
model, we ask these questions:
• Are the edges of the graph undirected or directed (or both)?
• If the edges are undirected, are multiple edges present that
connect the same pair of vertices? If the edges are directed,
are multiple directed edges present?
• Are loops present?
12. Other Applications of Graphs
We will illustrate how graph theory can be used in
models of:
Social networks
Communications networks
Information networks
Software design
Transportation networks
13. Graph Models: Social Networks
Graphs can be used to model social structures based on
different kinds of relationships between people or groups.
In a social network, vertices represent individuals or
organizations and edges represent relationships between
them.
Useful graph models of social networks include:
friendship graphs - undirected graphs where two people
are connected if they are friends (in the real world, on
Facebook, or in a particular virtual world, and so on.)
collaboration graphs - undirected graphs where two
people are connected if they collaborate in a specific way
influence graphs - directed graphs where there is an edge
from one person to another if the first person can
influence the second person
14. Graph Models: Social Networks
(continued)
Example: A friendship
graph where two people
are connected if they are
Facebook friends.
15. Applications to Information Networks
Graphs can be used to model different types of
networks that link different types of information.
In a web graph, web pages are represented by vertices
and links are represented by directed edges.
A web graph models the web at a particular time.
In a citation network:
Research papers in a particular discipline are
represented by vertices.
When a paper cites a second paper as a reference, there
is an edge from the vertex representing this paper to the
vertex representing the second paper.
16. Transportation Graphs
Graph models are extensively used in the study of
transportation networks.
Airline networks can be modeled using directed
multigraphs where
airports are represented by vertices
each flight is represented by a directed edge from the vertex
representing the departure airport to the vertex representing
the destination airport
Road networks can be modeled using graphs where
vertices represent intersections and edges represent roads.
undirected edges represent two-way roads and directed edges
represent one-way roads.
17. We can use a directed graph called a precedence graph to
represent which statements must have already been
executed before we execute each statement.
Vertices represent statements in a computer program
There is a directed edge from a vertex to a second vertex if the
second vertex cannot be executed before the first
Software Design Applications
Example: This precedence
graph shows which statements
must already have been executed
before we can execute each of
the six statements in the
program.
19. Section Summary
Basic Terminology
Some Special Types of Graphs
Bipartite Graphs
Bipartite Graphs and Matchings (not currently
included in overheads)
Some Applications of Special Types of Graphs (not
currently included in overheads)
New Graphs from Old
20. Basic Terminology
Definition 1. Two vertices u, v in an undirected graph G
are called adjacent (or neighbors) in G if there is an edge e
between u and v. Such an edge e is called incident with the
vertices u and v and e is said to connect u and v.
Definition 2. The set of all neighbors of a vertex v of G =
(V, E), denoted by N(v), is called the neighborhood of v.
Definition 3. The degree of a vertex in a undirected graph
is the number of edges incident with it, except that a loop
at a vertex contributes two to the degree of that vertex. The
degree of the vertex v is denoted by deg(v).
21. Degrees and Neighborhoods of
Vertices
Example: What are the degrees and neighborhoods of the
vertices in the graphs G and H?
Solution:
G: deg(a) = 2, deg(b) = deg(c) = deg(f ) = 4, deg(d ) = 1,
deg(e) = 3, deg(g) = 0.
N(a) = {b, f }, N(b) = {a, c, e, f }, N(c) = {b, d, e, f }, N(d) = {c},
N(e) = {b, c , f }, N(f) = {a, b, c, e}, N(g) = .
H: deg(a) = 4, deg(b) = deg(e) = 6, deg(c) = 1, deg(d) = 5.
N(a) = {b, d, e}, N(b) = {a, b, c, d, e}, N(c) = {b},
N(d) = {a, b, e}, N(e) = {a, b ,d}.
23. Handshaking Theorem
Example: How many edges are there in a graph with 10
vertices of degree six?
Solution: Because the sum of the degrees of the vertices is
6 10 = 60, the handshaking theorem tells us that 2m = 60.
So the number of edges m = 30.
Example: If a graph has 5 vertices, can each vertex have
degree 3?
Solution: This is not possible by the handshaking thorem,
because the sum of the degrees of the vertices 3 5 = 15 is
odd.
24. Directed Graphs
Definition: An directed graph G = (V, E) consists of V,
a nonempty set of vertices (or nodes), and E, a set of
directed edges or arcs. Each edge is an ordered pair of
vertices. The directed edge (u,v) is said to start at u
and end at v.
Definition: Let (u,v) be an edge in G. Then u is the
initial vertex of this edge and is adjacent to v and v is
the terminal (or end) vertex of this edge and is adjacent
from u. The initial and terminal vertices of a loop are
the same.
Recall the definition of a directed graph.
25. Directed Graphs (continued)
Definition: The in-degree of a vertex v, denoted
deg−(v), is the number of edges which terminate at v.
The out-degree of v, denoted deg+(v), is the number of
edges with v as their initial vertex. Note that a loop at a
vertex contributes 1 to both the in-degree and the out-
degree of the vertex.
Example: In the graph G we have
deg−(a) = 2, deg−(b) = 2, deg−(c) = 3, deg−(d) = 2,
deg−(e) = 3, deg−(f) = 0.
deg+(a) = 4, deg+(b) = 1, deg+(c) = 2, deg+(d) = 2,
deg+ (e) = 3, deg+(f) = 0.
26. Directed Graphs (continued)
Theorem 3: Let G = (V, E) be a graph with directed edges.
Then:
Proof: The first sum counts the number of outgoing edges
over all vertices and the second sum counts the number of
incoming edges over all vertices. It follows that both sums
equal the number of edges in the graph.
27. Special Types of Simple Graphs:
Complete Graphs
A complete graph on n vertices, denoted by Kn, is the
simple graph that contains exactly one edge between
each pair of distinct vertices.
28. Special Types of Simple Graphs:
Cycles and Wheels
A cycle Cn for n ≥ 3 consists of n vertices v1, v2 ,⋯ , vn,
and edges {v1, v2}, {v2, v3} ,⋯ , {vn-1, vn}, {vn, v1}.
A wheel Wn is obtained by adding an additional vertex
to a cycle Cn for n ≥ 3 and connecting this new vertex
to each of the n vertices in Cn by new edges.
29. Special Types of Simple Graphs:
n-Cubes
An n-dimensional hypercube, or n-cube, Qn, is a graph
with 2n vertices representing all bit strings of length n,
where there is an edge between two vertices that differ
in exactly one bit position.
30. Special Types of Graphs and
Computer Network Architecture
Various special graphs play an important role in the design of
computer networks.
Some local area networks use a star topology, which is a complete
bipartite graph K1,n ,as shown in (a). All devices are connected to a
central control device.
Other local networks are based on a ring topology, where each device is
connected to exactly two others using Cn ,as illustrated in (b).
Messages may be sent around the ring.
Others, as illustrated in (c), use a Wn – based topology, combining the
features of a star topology and a ring topology.
31. Bipartite Graphs
Definition: A simple graph G is bipartite if V can be partitioned
into two disjoint subsets V1 and V2 such that every edge connects
a vertex in V1 and a vertex in V2. In other words, there are no
edges which connect two vertices in V1 or in V2.
It is not hard to show that an equivalent definition of a bipartite
graph is a graph where it is possible to color the vertices red or
blue so that no two adjacent vertices are the same color.
G is
bipartite
H is not bipartite
since if we color a
red, then the
adjacent vertices f
and b must both
be blue.
32. Bipartite Graphs (continued)
Example: Show that C6 is bipartite.
Solution: We can partition the vertex set into
V1 = {v1, v3, v5} and V2 = {v2, v4, v6} so that every edge of C6
connects a vertex in V1 and V2 .
Example: Show that C3 is not bipartite.
Solution: If we divide the vertex set of C3 into two
nonempty sets, one of the two must contain two vertices.
But in C3 every vertex is connected to every other vertex.
Therefore, the two vertices in the same partition are
connected. Hence, C3 is not bipartite.
33. Complete Bipartite Graphs
Definition: A complete bipartite graph Km,n is a graph
that has its vertex set partitioned into two subsets
V1 of size m and V2 of size n such that there is an edge
from every vertex in V1 to every vertex in V2.
Example: We display four complete bipartite graphs
here.
34. New Graphs from Old
Definition: A subgraph of a graph G = (V,E) is a graph
(W,F), where W ⊂ V and F ⊂ E. A subgraph H of G is a
proper subgraph of G if H ≠ G.
Example: Here we show K5 and
one of its subgraphs.
35. New Graphs from Old (continued)
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.
Example: