Imagine that 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 certain purposes, especially those that involve relating two variables that related by some kind of formula, a line graph is better because it shows the definitive relationship between the two variables, a relationship that, due to the formulaic nature of the relationship, will not change. Scatter plots are best used for other relationships.
Figure 1. Cost of an object (y) and the Year (x) (http://www.westegg.com/inflation/)
Figure 2. Test Scores (y) and Income (x) (Hypothetical)
Figure 3. Distance traveled by a car in meters (y) and Time in minutes (x) (Hypothetical)
...a connected graph G, a spanning graph of G that is a tree is called a spanning tree. A spanning tree for an undirected graph G = (V,E) is a graph G’ = (V,E’) such that G’ is a tree. In other words, G’ has the same set of vertices, but edges have been removed from E so that the resulting graph is a tree. This amounts to saying that G’ is acyclic. If G is directed, it means that cycles have been removed. Since a tree with V vertices has V1 edges, to generate a spanning tree of a connected graph G having V vertices and E edges we must delete all but (V1) edges from the G. We cannot do that randomly because it has to be a tree which is acyclic and connected. We must delete E(V1) = EV+1 edges, none of which is a bridge. A graph G can have several spanning tree.
Removal of any single edge from a spanning tree causes the graph to be unconnected.
For any spanning tree T of graph G, if an edge e that is not in T is added, a cycle is created. And also see one thing if we add any edge from ~G, we will also create a cycle.
Minimum Spanning Trees
A spanning tree is minimum if there is no other spanning tree with smaller cost. If the graph is unweighted, then the cost is just the number of edges. If it has weighted edges, then the cost is the sum of the edge weights of the edges in the spanning tree.
An example of...
...V. Adamchik
1
Graph Theory
Victor Adamchik Fall of 2005
Plan
1. Basic Vocabulary 2. Regular graph 3. Connectivity 4. Representing Graphs
Introduction
A.Aho and J.Ulman acknowledge that “Fundamentally, computer science is a science of abstraction.” Computer scientists must create abstractions of realworld problems that can be represented and manipulated in a computer. Sometimes the process of abstraction is simple. For example, we use a logic to design a computer circuits. Another example  scheduling final exams. For successful scheduling we have to take into account associations between courses, students and rooms. Such set of connections between items is modeled by graphs. Let me reiterate, in our model the set of items (courses, students and rooms) won't be much helpful. We also have to have a set of connections between pairs of items, because we need to study the relationships between connections. The basic idea of graphs were introduced in 18th century by the great Swiss mathematician Leonhard Euler. He used graphs to solve the famous Königsberg bridge problem. Here is a picture (taken from the internet)
V. Adamchik
21127: Concepts of Mathematics
German city of Königsberg (now it is Russian Kaliningrad) was situated on the river Pregel. It had a park situated on the banks of the river and two islands. Mainland and islands were joined by seven bridges. A problem was whether it...
..."Cartesian" is named after the French mathematician and philosopher Rene Descartes, who lived from 1596 until 1650. Cartesian diver lab is used commonly in scientific experiments to illustrate principle of buoyancy. The objective of this Cartesian diver lab is to demonstrate Pascal's law and Archimedes' principles.
Observation is the key to conduct this experimental study of the Cartesian diver. First a 2liter bottle is filled with water to almost all the way to the top, then prepare the diver which is a test tube, fill the test tube about 5060% with water, place the diver inside the bottle the diver should float near the water surface then secure the cap on the bottle. When the container is squeezed, the diver should sink to the bottom of the container. Release the bottle slowly, the diver should come up in reverse order.
The Cartesian diver shows that air is compressible and water is incompressible. When the container is squeeze, the pressure from squeeze is distributed equal throughout the container and the volume of air in the diver decreases because of the increased pressure of the water surrounding the diver. Since the volume of air inside the diver decreased, and water filled up where the air use to be, the diver becomes denser and will begin to sink if enough pressure is applied. It begins to sink because it becomes denser so the upward force of the water is not great enough to keep the...
...Paul Erdos and Alfred Renyi. Their work suggested that systems such as communications could be effectively modelled by connecting nodes with randomly placed links. Their simple approach revitalised graph theory and led to the emergence of the field of random networks.
An important prediction of random network theory is regardless of the random placement of links most nodes will still have approximately the same number of links. In fact, in a random network the nodes follow a Poisson distribution with a bell shape (see Fig.1). Random networks are also called exponential, because the probability that a node is connected to k other sites decreases exponentially for large k. This is better described by the famous small world networks. It was Watts and Strogatz in 1998 that recognised that a class of random graphs could be categorised as small world networks. They noted that graphs could be classified according to their clustering coefficient and their diameter. Many random graphs show a small diameter and also have a small clustering coefficient. What Strogatz and Watts found was that in real world networks the diameter is still small but has a clustering coefficient significantly higher than expected by random chance. Watts and Strogatz thus proposed a simple model of random graphs with (a) a small diameter and (b) a large clustering coefficient.
I wasn't until 1998 when AlbertLászlǒ Barabási...
...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 each second....
...Graphs
Data Structures and Algorithms
Prepared by: Engr. Martinez
Graph Concepts
Graph Concepts
Graphs are of 2 types
Undirected Graph
Undirected Graph examples
Directed Graph
Directed Graph example
Directed Graph
Directed GraphGraph Relationships
Graph RelationshipsGraph Relationships
Basic terms involved in graphs:
Basic terms involved in graphs:
Basic terms involved in graphs:
Degree of vertex
The number of edges incident onto the vertex For an undirected graph The degree of a vertex u is the number of edges connected to u. For a directed graph The outdegree of a vertex u is the number edges leaving u, and its indegree is the number of edges ending at u
Degree of vertex
Edges are of 2 types
Directed edge: A directed edge between the vertices vi and vj is an ordered pair. It is denoted by . Undirected edge: An undirected edge between the vertices vi and vj is an unordered pair. It is denoted by (vi,vj).
Different Types of Graphs
Subgraph Connected graph Completely connected graph
1. Subgraphs
A subgraph of a graph G = (V,E) is a...
...Graphs and Function
What is the relation between the graphs and function and how was it applied in the real world?
Graphs are frequently used in national magazines and newspaper to present information about things such as the world’s busiest airports (O’Hare in China is first, Heathrow in London is sixth), about the advertisingdollar receivers in the United States (newspaper are first, radio is fourth) and about NCAA men’s golf team title winner (Yael is first, Houston is second). The function concept is very closely connected to graphs, and functions are the heart of mathematics.
I gathered my information from books especially algebra books and some are from the internet. I went to the library to look for some books and I borrowed some so I have many resources of information.
Many reallife relations between two quantities expressed in the form of equation are functions. To visualize these relationships, geometric figures called graphs are used. Modern technology provides us with graphing utilities needed to draw these graphs as well as enhance man’s knowledge of graphing techniques. Scientist and astronomers identify, visualize, and explore graphical patterns useful in analyzing data about the universe. Economist and businessmen draw mathematical models to find curves of best fit. Generally, the use of function and graphs is found in every...
...Baker Machine Company
Layout
Problem 3.4. (Summary)
Baker Machine is considering two alternative layouts. We will compare the WeightedDistance Scores using rectilinear distance* of the two block plans to determine which alternative layout is better.
Alternative Layout 1 Alternative Layout 2
3  6  4 
5  1  2 
3  1  4 
5  6  2 
* rectilinear distance – the distance between two points with a series 90degree turns, as along city blocks
Data
Baker Machine Company is a job shop that specialized in precision parts for firms in the aerospace industry. The current block plan is as follows:
3  4  2 
1  5  6 
The weighteddistance score for the current layout is 115.
 Closeness Matrix       
       
  Trips Between Departments  
 Department  1  2  3  4  5  6 
1  Burr and grind    7  16   10  5 
2  Numerically controlled (NC) equipment      4   
3  Shipping and receiving       9  9 
4  Lathes and drills        3 
5  Tool crib        3 
6  Inspection        
Solution
To determine which alternative layout is better we calculate the weighted distance, wd, scores of the two block plans.
Layouts can be assessed using the Layout solver of OM Explorer.
Solution (continue)
Alternative Layout 1
Solver  Layout 
       

   
...