Chemical graph theory uses the molecular graph as a means to model molecules. Graph theory is also widely used in sociology as a way, for example, to measure actors' prestige or to explore rumor spreading , notably through the use of social network analysis software. Under the umbrella of social networks are many different types of graphs. Influence graphs model whether certain people can influence the behavior of others. Finally, collaboration graphs model whether two people work together in a particular way, such as acting in a movie together.

Likewise, graph theory is useful in biology and conservation efforts where a vertex can represent regions where certain species exist or inhabit and the edges represent migration paths or movement between the regions. This information is important when looking at breeding patterns or tracking the spread of disease, parasites or how changes to the movement can affect other species. Graph theory is also used in connectomics ; [16] nervous systems can be seen as a graph, where the nodes are neurons and the edges are the connections between them.

In mathematics, graphs are useful in geometry and certain parts of topology such as knot theory. Algebraic graph theory has close links with group theory. Algebraic graph theory has been applied to many areas including dynamic systems and complexity. A graph structure can be extended by assigning a weight to each edge of the graph. Graphs with weights, or weighted graphs , are used to represent structures in which pairwise connections have some numerical values.

For example, if a graph represents a road network, the weights could represent the length of each road. There may be several weights associated with each edge, including distance as in the previous example , travel time, or monetary cost. Such weighted graphs are commonly used to program GPS's, and travel-planning search engines that compare flight times and costs. Euler's formula relating the number of edges, vertices, and faces of a convex polyhedron was studied and generalized by Cauchy [18] and L'Huilier , [19] and represents the beginning of the branch of mathematics known as topology.

The techniques he used mainly concern the enumeration of graphs with particular properties. These were generalized by De Bruijn in Cayley linked his results on trees with contemporary studies of chemical composition. In particular, the term "graph" was introduced by Sylvester in a paper published in in Nature , where he draws an analogy between "quantic invariants" and "co-variants" of algebra and molecular diagrams: [22]. One of the most famous and stimulating problems in graph theory is the four color problem : "Is it true that any map drawn in the plane may have its regions colored with four colors, in such a way that any two regions having a common border have different colors?

Many incorrect proofs have been proposed, including those by Cayley, Kempe , and others. The study and the generalization of this problem by Tait , Heawood , Ramsey and Hadwiger led to the study of the colorings of the graphs embedded on surfaces with arbitrary genus. The four color problem remained unsolved for more than a century.

In Heinrich Heesch published a method for solving the problem using computers. A simpler proof considering only configurations was given twenty years later by Robertson , Seymour , Sanders and Thomas. The autonomous development of topology from and fertilized graph theory back through the works of Jordan , Kuratowski and Whitney. Another important factor of common development of graph theory and topology came from the use of the techniques of modern algebra. 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. 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.

### 1st Edition

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.

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. 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. 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. 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.

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:. 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. This article is about sets of vertices connected by edges. For graphs of mathematical functions, see Graph of a function. For other uses, see Graph disambiguation.

## Handbook of graph theory, combinatorial optimization, and algorithms

Area of discrete mathematics. Main article: Directed graph. Main article: Graph drawing. Method of Instruction: Each minute class consists of two, roughly equal parts. In the first part of the class some new theory is introduced while the second part is dedicated to solving problems.

## Handbook of graph theory, combinatorial optimization, and algorithms - Semantic Scholar

Problems constitute an inherent part of the learning process: they give the students the opportunity to better understand and deepen their knowledge of the notions, theorems and algorithms covered in the first part of the class. Students are given homeworks at the end of each class.

These can be of two types: the first type is to be solved individually and then handed in. These problems are intended to ensure that the student has acquired the necessary command of the material to be able to further follow the course. Their solution requires nothing but the adaptation of methods already covered in class. The second type of homework might require some original ideas and insights and students are encouraged to work together on these if they wish.

The solutions of these problems often provide the grounds for getting acquainted with the new material next class. In he was awarded as "Excellent Teacher of the Department", based on student feedback surveys. He obtained his Ph. Combinatorial Optimization. View PDF Instructor s :. Contact hours. Learning Objectives: By completing the course the students will learn to use linear and integer programming to formulate various problems arising in real-life circumstances; will learn to identify the range and the limits of applicability of linear and integer programming; will learn to use certain efficient algorithms for some bipartite matching and network flow problems; will get a chance to further develop their mathematical skills.

Detailed Program and Class Schedule: Matchings in bipartite graphs, the augmenting path algorithm. The maximum network flow problem, the Ford-Fulkerson algorithm. The basic problem of linear programming. Graphical solution for two-variable problems. Modeling practical problems as multivariable LP instances. Solving LP problems with Microsoft Excel.

The notion of integer programming. Modeling practical problems as IP instances. Using decision variables, incorporating logical constraints. Solving IP problems with Microsoft Excel. Linear algebra revision: fundamental operations on matrices, determinants.

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The matrix form of LP problems. Solving systems of linear inequalities with the Fourier-Motzkin elimination. Midterm test. A necessary and sufficient condition for the solvability of systems of linear inequalities: the Farkas-lemma.