Baker Machine Company
Layout
Problem 3.4. (Summary)
Baker Machine is considering two alternative layouts. We will compare the Weighted-Distance Scores using rectilinear distance* of the two block plans to determine which alternative layout is better. Alternative Layout 1Alternative Layout 2

3| 6 | 4|
5| 1| 2|
3| 1 | 4|
5| 6| 2|

* rectilinear distance – the distance between two points with a series 90-degree 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|

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)

...V. Adamchik
1
GraphTheory
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 real-world 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
21-127: 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...

...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...

...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 |V|-1 edges, to generate a spanning tree of a connected graph G having |V| vertices and |E| edges we must delete all but (|V|-1) 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|-(|V|-1) = |E|-|V|+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...

...
Graphs
1 Introduction
We have studied one non-linear data structure so far i.e Trees. A graph is another non-linear data structure that is widely used to solve many real-life computing problems. For example, we need to use a graph to find out whether two places on a road-map are connected and what is the shortest distance between them. Graphs are used in simulating electrical circuits to find out current flows and voltage drops at various points in the circuit. Graphs are widely used in telephone and computer networks.
Graphs have great historical significance too. In 1736, the famous mathematician Euler used the concept of a graph to solve the “Koenigsberg problem”. In the small town of Koenisberg in Prussia, the river Pregal flows around the island of Kneiphof and then divides into two. The four land areas ( A, B, C, D) bordering the river are connected by seven bridges ( a,b,c,d,e,f,g). The problem is to find out whether it is possible to start walking from some area, cross each bridge exactly once and return to the starting land area. Euler used graphs to prove that this would not be possible. A walk which achieves this is called an “Eulerian Walk”.
{{{ Diagram }}}
In this chapter, we will study this data structure, its implementation and its applications. Before that, we will study some definitions and terminology.
Definitions and Terminology...

...scientists treated all complex networks as being completely random. This theory had its roots in the work of two mathematicians, 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 graphtheory and led to the emergence of the field of random networks.
An important prediction of random networktheory 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...

...Graphtheory - 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 graphtheory 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...

...Applications of GraphTheory in Real Life
Sharathkumar.A,
Final year, Dept of CSE,
Anna University, Villupuram
Email: kingsharath92@gmail.com
Ph. No: 9789045956
Abstract
Graphtheory is becoming increasingly significant as it is applied to other areas of
mathematics, science and technology. It is being actively used in fields as varied as
biochemistry (genomics), electrical engineering (communication networks and codingtheory), computer science (algorithms and computation) and operations research
(scheduling). The powerful combinatorial methods found in graphtheory have also been used
to prove fundamental results in other areas of pure mathematics. We discuss about computer
network security (worm propagation) using minimum vertex covers in graphs. We also show
how to apply edge coloring and matching in graphs for scheduling (the timetabling problem)
and vertex coloring in graphs for map coloring and the assignment of frequencies in GSM
mobile phone networks. Finally, we revisit the classical problem of finding re-entrant
knight’s tours on a chessboard using Hamiltonian circuits in graphs.
Introduction
Graphtheory is rapidly moving into the mainstream of mathematics mainl y because
of its applications in diverse fields which include biochemistry (genomics), electrical...

...MATH1081 Discrete Mathematics
T. Britz/D. Chan/D. Trenerry
§5 GraphTheory
Loosely speaking, a graph is a set of dots and dot-connecting lines. The dots are called vertices and the lines are called edges. Formally, a (ﬁnite) graph G consists of A ﬁnite set V whose elements are called the vertices of G; A ﬁnite set E whose elements are called the edges of G; A function that assigns to each edge e ∈ E an unordered pair of vertices called the endpoints of e. This function is called the edge-endpoint function. Note that these graphs are not related to graphs of functions. Graphs can be used as mathematical models for networks such roads, airline routes, electrical systems, social networks, biological systems and so on. Graphtheory is the study of graphs as mathematical objects.
1
Exercise. Consider the following graph G with vertices and edges V = {v1 , v2 , v3 , v4 , v5 } and E = {e1 , e2 , e3 , e4 , e5 , e6 , e7 } :
v3 e2 e5 v1 e 1 v2 e3 v4 e 6 v5 e7 e4
Edge e1 e2 e3 e4 e5 e6 e7
Endpoints {v1 , v2 } {v2 , v3 } {v2 , v4 } {v3 , v4 } {v3 , v4 } {v4 , v5 } {v5 }
2
Example. Below are 3 diﬀerent pictorial representations of the same graph. e1 v1 e2 v2 e1 e3 v3 e4 v1 e4 e2 e3 v3 v2 e4 v1 e1 e3 v3 e2 v2
The edge-endpoint function of this graph is as follows: Edge e1 e2 e3...

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