Graph theory - the study of graphs and networks, is often considered part of combinatorics, but has grown large enough and distinct enough, with its own kind of problems, to be regarded as a subject in its own right.[12] Graphs are one of the prime objects of study in discrete mathematics. They are among the most ubiquitous models of both natural and human-made structures. They can model many types of relations and process dynamics in physical, biological and social systems. In computer science, they can represent networks of communication, data organization, computational devices, the flow of computation, etc. In mathematics, they are useful in geometry and certain parts of topology, e.g. theory. Algebraic has close links with group theory. There are also continuous graphs, however for the most part research in graph theory falls within the domain of discrete mathematics. Combinatorics -studies the way in which discrete structures can be combined or arranged. Enumerative combinatorics concentrates on counting the number of certain combinatorial objects - e.g. the twelvefold way provides a unified framework for counting permutations, combinations and partitions. Analytic combinatorics concerns the enumeration (i.e., determining the number) of combinatorial structures using tools from complex analysis and probability theory. In contrast with enumerative combinatorics which uses explicit combinatorial formulae and generating functions to describe the results, analytic combinatorics aims at obtaining asymptotic formulae. Design theory is a study of combinatorial designs, which are collections of subsets with certain intersection properties. Theory studies various enumeration and asymptotic problems related to integer partitions, and is closely related to q-series, functions and orthogonal polynomials. Originally a part of number theory and analysis, partition theory is now considered a part of combinatorics or an independent field. Order theory is the study...

...LAB # 1
Graph Matching
Principles of Physics I Laboratory
Breanna Wilhite
Introduction
In this lab motion will be represented by graphs that plots distance and velocity vs. time. A motion detector will be used to measure the time it takes for a high frequency sound pulse to travel from the detector to an object and back. By using this method sound can determine the distance to the object, or its position. This device will determine in what direction the woman in the video was walking and how fast she was walking. This information will be plotted on a graph and show the motion as the woman moves, whether she speed up or slowed down. Logger Pro will use the change in position to calculate the object’s velocity and acceleration. All of this information is in graph form. A qualitative analysis of the graphs of motion will help you develop an understanding of the concepts of kinematics.
Theory
The motion of an object can be measured using a motion detector. The detector helps in knowing where an object is according to an indication point. How fast and in what direction an object is moving, and how an object is accelerating is necessary in understanding the kinematics graphs.
The Motion detector uses pulses of ultrasound that bounces off of an object to determine the position of the person/object. As the person moves, the change in its position is measured many times...

...Introduction in GraphTheory
(BASIC CONCEPTS)
BASIC CONCEPTS
We used decision trees in Unit DT and used them to study decision making. However,
we did not look at their structure as trees. In fact, we didn’t even define a tree precisely.
What is a tree? It is a particular type of graph, which brings us to the subject of this unit.
What is a Graph?
There are various types of graphs, each with its own definition. Unfortunately, some
people apply the term “graph” rather loosely, so you can’t be sure what type of graph
they’re talking about unless you ask them. After you have finished this chapter, we expect you to use the terminology carefully, not loosely. To motivate the various definitions, we’ll begin with some examples.
Example 1 (A computer network) Computers are often linked with one another so
that they can interchange information. Given a collection of computers, we would like to describe this linkage in fairly clean terms so that we can answer questions such as “How can we send a message from computer A to computer B using the fewest possible intermediate computers?”
We could do this by making a list that consists of pairs of computers that are connected. Note that these pairs are unordered since, if computer C can communicate with computer D, then the reverse is also true. (There are sometimes exceptions to this, but they are rare and we will...

...objective of line graphs is to define raw data, making it easily understandable with a visual representation. By plotting data on a line graph, you assign it a vertical and horizontal value that corresponds to the raw data determining the graph. For instance, if tracking annual sales at a retail store, the data would be defined by the amount of sales in dollars and the months during which these sales took place.
Interpreting Data
The visual representation of data in line graphs allows users to easily interpret information over a given period of time. Once figures have been plotted on the graph and a line has connected all the points, users can visually analyze data that occurs over time without having to compare specific figures. For instance, monthly temperatures over the period of a year can be easily visualized by looking at the points on a line graph. The user can quickly ascertain the warmer and colder months based on viewing the graph, without having to compare specific temperatures. This also allows users to quickly and visually identify trends in data.
Uses of Line Graph
Line graphs show the rate of change to a specific data set over a period of time. For instance, line graphs are commonly used to check the changes in various weather patterns, including temperature, humidity and rainfall, over the course of a month or a...

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1. Name three different kinds of graphs that are often used to plot information and discuss the value of each.
Answer:
Three types of graphs are line graph, histogram or bar graph, and pie chart graph.
The line graph is used to describe how an object moves explaining relationship between time and distance traveled.
A histogram or bar graph is used to compare quantities using a series of vertical bars.
A pie chart graph represents data in a chart that resembles a pie cut into pieces. This is valuable when comparison to the whole is important.
Each type of graph presents information in a visual manner, which often makes interpretation more interesting.
(7 points)
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2. Explain what chart junk is and how it differs from the kind of items you should include in your graphs. Provide four examples.
Answer:
Chart junk consists of decorative and distracting elements added to a graph that do not supply useful information on the graph such as texture or designs in the bars of a bar graph.
These may include strange highlighting or coloration, unusual formatting, cartoon drawings or pictures placed into the graph purely for visual effects such as gradation of color inside the...

... So let's start with the graph where all variables, a, b and c, are equal to 1.
The graph opens upward because a is postive. If a was negative the graph would open in the negative direcction. It is not symmetrical around the y-axis because c = 1. Because c = 1, when x = 0, the parabola passes through the point (0, 1). If c equaled 0, then the parabola would be symmetrical around the y-axis.
Now let's look at what happens when we change b, while a and c remain 1.
For these positive values of b, the graph always intersects the y-axis at the point (0, 1). The vertex is always to the left of the graph when b > 0. When b = 0, the vertex is on the y-axis. We know that where the parabola intersects or hits the x-axis is where the real roots of that particular equation occur. For those graphs that do not intersect or hit the x-axis, then they do not have any real roots. For example, when b = 1, the graph does not hit the x-axis, therefore it does not have any real roots. Notice that when tb is positive, the real roots occur on the negative side of the y-axis.
Now let's see what happens when b is negative....

...a line on a Cartesian graph is approximately the distance y in feet a person walks in x hours. What does the slope of this line represent? How is this graph useful? Provide another example for your colleagues to explain.
The slope of the line represents the speed of the person in feet per hour. This graph is useful because it provides a visual representation of the continuous motion of the person walking, something that could not provided by something like a bar graph. In a bar graph, the sheer number of columns and their shape makes representation of a temporal action cumbersome, whereas in a line graph, the information is represented fluidly, as it is reality. (See Figure 1 for another example.)
What is the difference between a scatterplot and a line graph? Provide an example of each. Does one seem better than the other? In what ways is it better?
A scatter plot consists of several data points placed on a Cartesian graph that are not connected to each other, where the data is shown as a collection of points. A line graph is an extension of the scatter plot in which the data is connected by straight segments of lines. A scatter plot might be used to analyze the relationship between achievement test scores and income (Figure 2), while a line graph might be used to analyze the distance traveled by a car in minutes (Figure 3).
For...

...Graphs - Introduction
Terminology
Graph ADT
Data Structures
Reading: 12.1-12.2
COSC 2011, Summer 2004
Definition
• A graph is a pair (V, E), where
– V is a set of nodes, called vertices
– E is a collection of pairs of vertices, called edges
• Both are objects (i.e. store data)
G
E
B
F
A
Vertex
city
computer
web page
airport
C
D
COSC 2011, Summer 2004
H
Edge
road
cable
hyperlink
flight
Example Applications
•
•
•
•
•
Electronic circuits
– Printed circuit board
– Integrated circuit
Transportation networks
– Highway network
– Flight network
Computer networks
– Local area network
– Internet
– Web
Databases
– Entity-relationship diagram
Questions
– Are two points connected?
– What is the shortest path between
them?
cslab1a
cslab1b
math.brown.edu
cs.brown.edu
brown.edu
qwest.net
att.net
cox.net
John
Paul
COSC 2011, Summer 2004
David
Types of edges
• Undirected edge {u,v}
– Does not indicate direction
– We can “travel” in either direction
A
B
• Directed edge (u,v)
– Has a direction
A
– We can only “travel” in one direction along the edge
B
• Self-loop
– Edge that originates and ends at the same vertex
• Parallel edges
A
– Two undirected edges with same end vertices
– Two directed edges with same origin and destination
COSC 2011, Summer 2004
A
B
Types of graphs
• Undirected graph
– All edges are undirected
• Directed graph
– All edges are directed
• Mixed...

...GraphTheoryGraphsGraph:- A graph consists of a nonempty set of points or vertices, and a set of edges that link together the vertices. A simple real world example of a graph would be your house and the corner store. Where the house and the store are the vertices and the road between them is the edge connecting the two vertices.
Or a graph is a network consisting of vertices (or nodes) and edges (V,E)
Simple Graph
A graph can take on many forms: directed or undirected.
Directed Graph:- A directed graph is one in which the direction of any given edge is defined.
Or A graph with directed edges = directed graph (digraph)
Directed edges = arcs
Directed Graph
Undirected Graph:- An undirected graph is one in which the direction of any given edge is not defined. Conversely, in an undirected graph you can move in both directions between vertices. Or a graph with undirected edges is called undirected graph.
Undirected graph
Mixed Graph:- A graph is one in which contains...