The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition. Gary Chartrand continues to be prolific, even in retirement. The book is a fine introduction to the field and is rich with real world applications of this interdisciplinary subject. Three appendices provide some background in discrete mathematics, such as set theory, equivalence relations and proof techniques.
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By Gary Chartrand and Ping Zhang. Indeed, names of people were becoming part of the discussion. Mathematics has existed for many centuries. In the ancient past, certain cultures developed their own mathematics. In recent centuries, there has become only one international mathematics. It has become more organized and has been divided into more clearly defined areas even though there is significant overlap.
While this was occurring, explanations proofs as to why mathematical statements are true were becoming more structured and clearly written.
The goal of this book is to introduce undergraduates to the mathematical area called graph theory , which only came into existence during the first half of the 18th century. Since the second half of the 20th century, however, the subject has exploded. It is our intent to describe some of the major topics of this subject to you and to inform you of some of the people who helped develop and shape this area. In the beginning, most of these people were just like you — students who enjoyed mathematics but with a great sense of curiosity.
This will give you the chance to do some creative thinking of your own. In fact, maybe the next person who will have an influence on this subject is you.
Part of what makes graph theory interesting is that graphs can be used to model situations that occur within certain kinds of problems. These problems can then be studied and possibly solved with the aid of graphs. Because of this, graph models occur frequently throughout this textbook. However, graph theory is an area of mathematics and consequently concerns the study of mathematical ideas — of concepts and their connections with each other. As we said, this text has been written for undergraduates.
Keeping this in mind, we have included a proof of a theorem if we believe it is appropriate, the proof technique is informative and if the proof is not excessively long. We would like to think that the material in this text will be useful and interesting for mathematics students as well as for other students whose areas of interest include graphs. This text is also appropriate for self-study.
We have included three appendixes. In Appendix 1, we review some important facts about sets and logic. Appendix 2 is devoted to equivalence relations and functions while Appendix 3 describes methods of proof. We understand how frustrating it is for students or anyone! Consequently, we have endeavored to give clear, well-written proofs.
Although this can very well be said about any area of mathematics or indeed about any scholarly activity, we feel that appreciation of graph theory is enhanced by being familiar with many of the people, past and present, who were or are responsible for its development.
Consequently, we have included several remarks that we find interesting about some of the people of graph theory. Since we believe that these people are part of the story that is graph theory, we have discussed them within the text and not simply as footnotes.
We often fail to recognize that mathematics is a living subject. Graph theory was created by people and is a subject that is still evolving. There are several sections that have been designated as Excursion. These can be omitted with no negative effect if this text is being used for a course.
In other cases, an Excursion brings up a sidelight of graph theory that perhaps has little, if any, mathematical content but which we simply believe is interesting. There are also sections that we have designated as Exploration. These sections contain topics with which students can experiment and use their imagination.
These give students opportunities to practice asking questions. In any case, we believe that this might be fun for some students. As far as using this text for a course, we consider the first three chapters as introductory.
Much of this could be covered quite quickly. Students could read these chapters on their own. Sections 8. Solutions or hints for the odd-numbered exercises in the regular sections of the text, references, an index of mathematical terms, an index of people and a list of symbols are provided at the end of the text.
We thank him for this and for his encouragement. A major publishing company has ten editors referred to by 1, 2, …, 10 in the scientific, technical and computing areas. These ten editors have a standard meeting time during the first Friday of every month and have divided themselves into seven committees to meet later in the day to discuss specific topics of interest to the company, namely, advertising, securing reviewers, contacting new potential authors, finances, used and rented copies, electronic editions and competing textbooks.
This leads us to our first example. Example 1. They have set aside three time periods for the seven committees to meet on those Fridays when all ten editors are present. Some pairs of committees cannot meet during the same period because one or two of the editors are on both committees. This situation can be modeled visually as shown in Figure 1. In this figure, there are seven small circles, representing the seven committees and a straight line segment is drawn between two circles if the committees they represent have at least one committee member in common.
In other words, a straight line segment between two small circles committees tells us that these two committees should not be scheduled to meet at the same time. This gives us a picture or a model of the committees and the overlapping nature of their membership. What we have drawn in Figure 1.
Formally, a graph G consists of a finite nonempty set V of objects called vertices the singular is vertex and a set E of 2-element subsets of V called edges. The sets V and E are the vertex set and edge set of G , respectively.
So a graph G is a pair actually an ordered pair of two sets V and E. At times, it is useful to write V G and E G rather than V and E to emphasize that these are the vertex and edge sets of a particular graph G. Vertices are sometimes called points or nodes and edges are sometimes called lines.
Indeed, there are some who use the term simple graph for what we call a graph. It is common to represent a graph by a diagram in the plane as we did in Figure 1.
The diagram itself is then also referred to as a graph. For the graph G of Figure 1. Every integer in the sequence is the sum of the two integers immediately preceding it except for the first two integers of course.
These numbers are well known in mathematics and are called the Fibonacci numbers. In fact, these integers occur so often that there is a journal The Fibonacci Quarterly , frequently published five times a year! Our second example concerns these numbers. There is a more visual way of identifying these pairs, namely by the graph H of Figure 1. If uv is an edge of G , then u and v are said to be adjacent in G.
The number of vertices in G is often called the order of G , while the number of edges is its size. Since the vertex set of every graph is nonempty, the order of every graph is at least 1. A graph with exactly one vertex is called a trivial graph , implying that the order of a nontrivial graph is at least 2. The graph G of Figure 1.
We often use n and m for the order and size, respectively, of a graph. So, for the graph G of Figure 1. There are occasions when we are interested in the structure of a graph and not in what the vertices are called. In this case, a graph is drawn without labeling its vertices.
For this reason, the graph G of Figure 1. There are twelve such configurations, shown in Figure 1. A configuration can be transformed into other configurations according to certain rules. Specifically, we say that the configuration ci if cj can be obtained from ci by performing exactly one of the following two steps:. Necessarily, if ci can be transformed into cj , then cj can be transformed into ci. For example, c 2 can be transformed i into c 1 by shifting the silver coin in c 2 to the right, ii into c 4 by shifting the gold coin to the right or iii into c 8 by interchanging the two coins see Figure 1.
Now consider the twelve configurations shown in Figure 1. This graph F is shown in Figure 1. We say that a word W 1 can be transformed into a word W 2 if W 2 can be obtained from W 1 by performing exactly one of the following two steps:.
Therefore, if W 1 can be transformed into W 2, then W 2 can be transformed into W 1. This situation can be modeled by a graph G , where the given words are the vertices of G and two vertices are adjacent in G if the corresponding words can be transformed into each other.
This graph is called the word graph of the set of words. For the 11 words above, its word graph G is shown in Figure 1. In this case, a graph G is called a word graph if G is the word graph of some set S of 3-letter words.
For example, the unlabeled graph G of Figure 1. This idea is related to the concept of isomorphic graphs, which will be discussed in Chapter 3. When a vehicle approaches this intersection, it could be in one of the nine lanes: L1, L2, …, L9.
This intersection has a traffic light that informs drivers in vehicles in the various lanes when they are permitted to proceed through the intersection. To be sure, there are pairs of lanes containing vehicles that should not enter the intersection at the same time, such as L1 and L7.
However, there would be no difficulty for vehicles in L1 and L5 to drive through this intersection at the same time. This situation can be represented by the graph G of Figure 1.
The Math Citadel
By Gary Chartrand and Ping Zhang. Indeed, names of people were becoming part of the discussion. Mathematics has existed for many centuries. In the ancient past, certain cultures developed their own mathematics.
A First Course in Graph Theory
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A first course in graph theory
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A First Course in Graph Theory. Gary Chartrand , Ping Zhang. This comprehensive text offers undergraduates a remarkably student-friendly introduction to graph theory. Written by two of the field's most prominent experts, it takes an engaging approach that emphasizes graph theory's history. Unique examples and lucid proofs provide a sound yet accessible treatment that stimulates interest in an evolving subject and its many applications. Optional sections designated as "excursion" and "exploration" present interesting sidelights of graph theory and touch upon topics that allow students the opportunity to experiment and use their imaginations. Three appendixes review important facts about sets and logic, equivalence relations and functions, and the methods of proof.