Graph Theory (draft)

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An extended bibliography includes not only the author, title, and publishing information, but also a two-to-three sentence explanation of the content of the source. I expect this bibliography to include a vast majority of the works you will use to write your paper.

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Final Draft, due November 24 : You need to have a final draft of your report. Bring in two printed copies of your draft to class on November 24 in order to discuss it in depth with your classmates. They will give feedback which will enable you to revise it to turn it in the following week. All previous students have found this peer review session extremely helpful in preparing the project for its submission. Project Due, due December 9 : The project is due for submission via Blackboard. Just as with the other homework assignments, if you are running into trouble or you would like my input on your project, I suggest coming to see me earlier rather than later.

Content : Based on your project type, you have different goals for the content of your project.

Spinrad. Graph Theory (Book Draft)

For the report on a mathematician : Include some background history of the mathematician, where your mathematician worked, and who were some of his or her colleagues. Explain why this person is important in graph theory. Spend one to two pages explaining one or two major results and how they relate to graph theory in general. Make sure to state the result and elaborate upon it; if the math is unclear, I suggest coming to talk with me!

Your paper must include a proper bibliography. For the lesson plan : A lesson plan is based on a well-researched understanding of the background topic with an eye towards your target audience.

Your lesson plan will conform to the specifications that you learn in teaching classes. In particular, you should map out the expected time to be spent on each part of the lesson plan and outline the concepts to be learned in each segment. The lesson plan should explain all parts of the lesson including the setting and the necessary props.

If you will be using worksheets, they must be prepared and stapled to the final product. The first example of such a use comes from the work of the physicist Gustav Kirchhoff , who published in his Kirchhoff's circuit laws for calculating the voltage and current in electric circuits.

Graphs are represented visually by drawing a point or circle for every vertex, and drawing a line between two vertices if they are connected by an edge. If the graph is directed, the direction is indicated by drawing an edge. A graph drawing should not be confused with the graph itself the abstract, non-visual structure as there are several ways to structure the graph drawing. All that matters is which vertices are connected to which others by how many edges and not the exact layout.

In practice, it is often difficult to decide if two drawings represent the same graph. Depending on the problem domain some layouts may be better suited and easier to understand than others.

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The pioneering work of W. Tutte was very influential on the subject of graph drawing. Among other achievements, he introduced the use of linear algebraic methods to obtain graph drawings. Graph drawing also can be said to encompass problems that deal with the crossing number and its various generalizations. The crossing number of a graph is the minimum number of intersections between edges that a drawing of the graph in the plane must contain. For a planar graph , the crossing number is zero by definition. There are different ways to store graphs in a computer system.

The data structure used depends on both the graph structure and the algorithm used for manipulating the graph. Theoretically one can distinguish between list and matrix structures but in concrete applications the best structure is often a combination of both. List structures are often preferred for sparse graphs as they have smaller memory requirements.

Matrix structures on the other hand provide faster access for some applications but can consume huge amounts of memory. List structures include the incidence list , an array of pairs of vertices, and the adjacency list , which separately lists the neighbors of each vertex: Much like the incidence list, each vertex has a list of which vertices it is adjacent to.

Matrix structures include the incidence matrix , a matrix of 0's and 1's whose rows represent vertices and whose columns represent edges, and the adjacency matrix , in which both the rows and columns are indexed by vertices.

Sébastien Bubeck – Publications by Year

In both cases a 1 indicates two adjacent objects and a 0 indicates two non-adjacent objects. The Laplacian matrix is a modified form of the adjacency matrix that incorporates information about the degrees of the vertices, and is useful in some calculations such as Kirchhoff's theorem on the number of spanning trees of a graph. The distance matrix , like the adjacency matrix, has both its rows and columns indexed by vertices, but rather than containing a 0 or a 1 in each cell it contains the length of a shortest path between two vertices.

There is a large literature on graphical enumeration : the problem of counting graphs meeting specified conditions. Some of this work is found in Harary and Palmer A common problem, called the subgraph isomorphism problem , is finding a fixed graph as a subgraph in a given graph.

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One reason to be interested in such a question is that many graph properties are hereditary for subgraphs, which means that a graph has the property if and only if all subgraphs have it too. Unfortunately, finding maximal subgraphs of a certain kind is often an NP-complete problem. For example:. One special case of subgraph isomorphism is the graph isomorphism problem. It asks whether two graphs are isomorphic. It is not known whether this problem is NP-complete, nor whether it can be solved in polynomial time.

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A similar problem is finding induced subgraphs in a given graph. Again, some important graph properties are hereditary with respect to induced subgraphs, which means that a graph has a property if and only if all induced subgraphs also have it. Finding maximal induced subgraphs of a certain kind is also often NP-complete.

Project Reflection (Graph Theory) - Thomas Petrou's DP

Still another such problem, the minor containment problem, is to find a fixed graph as a minor of a given graph. A minor or subcontraction of a graph is any graph obtained by taking a subgraph and contracting some or no edges. Many graph properties are hereditary for minors, which means that a graph has a property if and only if all minors have it too. For example, Wagner's Theorem states:. A similar problem, the subdivision containment problem, is to find a fixed graph as a subdivision of a given graph.

A subdivision or homeomorphism of a graph is any graph obtained by subdividing some or no edges. Subdivision containment is related to graph properties such as planarity. For example, Kuratowski's Theorem states:. Another problem in subdivision containment is the Kelmans—Seymour conjecture :.

Another class of problems has to do with the extent to which various species and generalizations of graphs are determined by their point-deleted subgraphs. Many problems and theorems in graph theory have to do with various ways of coloring graphs.

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Typically, one is interested in coloring a graph so that no two adjacent vertices have the same color, or with other similar restrictions. One may also consider coloring edges possibly so that no two coincident edges are the same color , or other variations. Among the famous results and conjectures concerning graph coloring are the following:. Constraint modeling theories concern families of directed graphs related by a partial order. In these applications, graphs are ordered by specificity, meaning that more constrained graphs—which are more specific and thus contain a greater amount of information—are subsumed by those that are more general.

Operations between graphs include evaluating the direction of a subsumption relationship between two graphs, if any, and computing graph unification. The unification of two argument graphs is defined as the most general graph or the computation thereof that is consistent with i. For constraint frameworks which are strictly compositional , graph unification is the sufficient satisfiability and combination function.

Well-known applications include automatic theorem proving and modeling the elaboration of linguistic structure. There are numerous problems arising especially from applications that have to do with various notions of flows in networks , for example:.

Various math topics (combinatorics, geometric probability, information theory)

Covering problems in graphs are specific instances of subgraph-finding problems, and they tend to be closely related to the clique problem or the independent set problem. Decomposition, defined as partitioning the edge set of a graph with as many vertices as necessary accompanying the edges of each part of the partition , has a wide variety of question. Often, it is required to decompose a graph into subgraphs isomorphic to a fixed graph; for instance, decomposing a complete graph into Hamiltonian cycles.

Many problems involve characterizing the members of various classes of graphs. Some examples of such questions are below:. From Wikipedia, the free encyclopedia.