About this Course 97, recent views We invite you to a fascinating journey into Graph Theory — an area which connects the elegance of painting and the rigor of mathematics; is simple, but not unsophisticated. Graph Theory gives us, both an easy way to pictorially represent many major mathematical results, and insights into the deep theories behind them. In this course, among other intriguing applications, we will see how GPS systems find shortest routes, how engineers design integrated circuits, how biologists assemble genomes, why a political map can always be colored using a few colors. We will study Ramsey Theory which proves that in a large system, complete disorder is impossible!

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Definitions[ edit ] Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph[ edit ] A graph with three vertices and three edges. To avoid ambiguity, this type of object may be called precisely an undirected simple graph. The edge is said to join x and y and to be incident on x and on y.

A vertex may exist in a graph and not belong to an edge. Multiple edges are two or more edges that join the same two vertices. To avoid ambiguity, this type of object may be called precisely an undirected multigraph. A loop is an edge that joins a vertex to itself. So to allow loops the definitions must be expanded.

To avoid ambiguity, these types of objects may be called precisely an undirected simple graph permitting loops and an undirected multigraph permitting loops respectively. V and E are usually taken to be finite, and many of the well-known results are not true or are rather different for infinite graphs because many of the arguments fail in the infinite case.

Moreover, V is often assumed to be non-empty, but E is allowed to be the empty set. The order of a graph is V , its number of vertices. The size of a graph is E , its number of edges. The degree or valency of a vertex is the number of edges that are incident to it, where a loop is counted twice.

Main article: Directed graph A directed graph with three vertices and four directed edges the double arrow represents an edge in each direction. A directed graph or digraph is a graph in which edges have orientations. To avoid ambiguity, this type of object may be called precisely a directed simple graph. In the edge x, y directed from x to y, the vertices x and y are called the endpoints of the edge, x the tail of the edge and y the head of the edge.

The edge y, x is called the inverted edge of x, y. To avoid ambiguity, this type of object may be called precisely a directed multigraph. To avoid ambiguity, these types of objects may be called precisely a directed simple graph permitting loops and a directed multigraph permitting loops or a quiver respectively.

Applications[ edit ] The network graph formed by Wikipedia editors edges contributing to different Wikipedia language versions vertices during one month in summer Many practical problems can be represented by graphs. Emphasizing their application to real-world systems, the term network is sometimes defined to mean a graph in which attributes e.

Computer science[ edit ] In computer science , graphs are used to represent networks of communication, data organization, computational devices, the flow of computation, etc. For instance, the link structure of a website can be represented by a directed graph, in which the vertices represent web pages and directed edges represent links from one page to another. A similar approach can be taken to problems in social media, [9] travel, biology, computer chip design, mapping the progression of neuro-degenerative diseases, [10] [11] and many other fields.

The development of algorithms to handle graphs is therefore of major interest in computer science. The transformation of graphs is often formalized and represented by graph rewrite systems. Complementary to graph transformation systems focusing on rule-based in-memory manipulation of graphs are graph databases geared towards transaction -safe, persistent storing and querying of graph-structured data.

Linguistics[ edit ] Graph-theoretic methods, in various forms, have proven particularly useful in linguistics , since natural language often lends itself well to discrete structure. Traditionally, syntax and compositional semantics follow tree-based structures, whose expressive power lies in the principle of compositionality , modeled in a hierarchical graph.

More contemporary approaches such as head-driven phrase structure grammar model the syntax of natural language using typed feature structures , which are directed acyclic graphs. Within lexical semantics , especially as applied to computers, modeling word meaning is easier when a given word is understood in terms of related words; semantic networks are therefore important in computational linguistics.

Still, other methods in phonology e. Physics and chemistry[ edit ] Graph theory is also used to study molecules in chemistry and physics.

In condensed matter physics , the three-dimensional structure of complicated simulated atomic structures can be studied quantitatively by gathering statistics on graph-theoretic properties related to the topology of the atoms. Also, "the Feynman graphs and rules of calculation summarize quantum field theory in a form in close contact with the experimental numbers one wants to understand. This approach is especially used in computer processing of molecular structures, ranging from chemical editors to database searching.

In statistical physics , graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems. Similarly, in computational neuroscience graphs can be used to represent functional connections between brain areas that interact to give rise to various cognitive processes, where the vertices represent different areas of the brain and the edges represent the connections between those areas. Graph theory plays an important role in electrical modeling of electrical networks, here, weights are associated with resistance of the wire segments to obtain electrical properties of network structures.

Chemical graph theory uses the molecular graph as a means to model molecules. Graphs and networks are excellent models to study and understand phase transitions and critical phenomena. Removal of nodes or edges lead to a critical transition where the network breaks into small clusters which is studied as a phase transition. This breakdown is studied via percolation theory.

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.

Biology[ edit ] 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 ; [18] nervous systems can be seen as a graph, where the nodes are neurons and the edges are the connections between them.

Mathematics[ edit ] 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. Other topics[ edit ] 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. 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. 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. Another important factor of common development of graph theory and topology came from the use of the techniques of modern algebra.

Main article: Graph drawing 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 arrow. 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. 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. Drawings on surfaces other than the plane are also studied.

Graph-theoretic data structures[ edit ] Main article: Graph abstract data type 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. Implementations of sparse matrix structures that are efficient on modern parallel computer architectures are an object of current investigation.

In both cases a 1 indicates two adjacent objects and a 0 indicates two non-adjacent objects. The degree matrix indicates the degree of vertices. 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.

Enumeration[ edit ] 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 Subgraphs, induced subgraphs, and minors[ edit ] 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: Finding the largest complete subgraph is called the clique problem NP-complete. One special case of subgraph isomorphism is the graph isomorphism problem.

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## Introduction To Graph Theory

Definitions[ edit ] Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph[ edit ] A graph with three vertices and three edges. To avoid ambiguity, this type of object may be called precisely an undirected simple graph. The edge is said to join x and y and to be incident on x and on y. A vertex may exist in a graph and not belong to an edge. Multiple edges are two or more edges that join the same two vertices.

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