Are number theory and graph theory linked

Graph theory

The Graph theory is a branch of mathematics that studies the properties of graphs and their relationships to one another.
Because, on the one hand, many algorithmic problems can be traced back to graphs and, on the other hand, the solution of graph-theoretical problems is often based on algorithms, graph theory is also of great importance in computer science, especially complexity theory. The study of graphs is also part of network theory.
At first glance, graph theory seems to be more of an abstract and unrealistic discipline of mathematics. In fact, many everyday problems can be modeled with the help of graphs.

Object viewed

In graph theory, a graph is a set of points (these are then called nodes or corners) that may be connected to one another by lines (so-called edges or arcs). The shape of the points and lines does not play a role in graph theory.
A distinction is made between:
More complex graph types are:
Depending on the problem, nodes and edges can also be given colors (formally with natural numbers) or weights (i.e. rational or real numbers). One then speaks of node- or edge-colored or -weighted graphs.

Basic concepts and problems

Graph theory defines a large number of fundamental terms, knowledge of which is essential to understand scientific treatises. Fortunately, the majority of the terms are very intuitive, so they can be learned quickly and only occasionally have to look up the exact definition. Before reading further articles on graph theory, we recommend reading the following articles in particular:
Further basic terms can be found in:
Graphs can have different properties. A graph can be connected, bipartite, planar, Eulerian or Hamiltonian. You can ask about the existence of special sub-graphs or examine certain parameters, such as number of nodes, number of edges, minimum degree, maximum degree, waist size, diameter, node connection number, edge connection number, chromatic number, stability number or number of cliques.
The various properties can be related to one another. Investigating the relationships is a task of graph theory. For example, the node connection number is always smaller than the edge connection number, which in turn is always smaller than the minimum degree of the graph under consideration. In plane graphs, the color number is always less than 5. This statement is also known as the four-color theorem.
Some of the graph properties listed are relatively easy to determine algorithmically, that is, depending on the size of the graph, the corresponding algorithms require only a little time to calculate the graph property. Other properties are practically unsolvable even with a computer.
The main problems and results of graph theory are presented in the following articles:

history

The beginnings of graph theory go back to the year 1736. At that time, Leonhard Euler published a solution to the Königsberg bridge problem. The question was whether there is a tour through the city of Königsberg - today Kaliningrad - that uses each of the seven bridges over the Pregel River exactly once. Euler was able to specify a necessary condition for this problem and thus deny the existence of such a tour. A sufficient condition, as well as an efficient algorithm that can find such a tour in a graph, was only given by Hierholzer in 1873. The term graph was first mentioned in the literature by the mathematician Sylvester in 1878 and was derived from the graphic notation of chemical structures.

Applications

As explained above, graphs can be used to model many problems.
The task of finding the shortest route between two locations is a classic. It can be solved with graphs in which the road network is suitably modeled as an edge-weighted graph and a shortest route is efficiently calculated in this with the help of Dijkstra's algorithm.
Similar, but algorithmically much more difficult, is the determination of a shortest round trip (see problem of the traveling salesman), in which all locations of a network must be visited exactly once. In practice, you will be able to visit places several times. Then the triangle inequality applies indirectly and in this case approximation algorithms can be used that find a round trip that is at most twice (MST heuristic) or at most 1.5 times (Christofides heuristic) as long as the shortest round trip.
Also prominent is the problem of coloring the countries of a map with as few colors as possible so that neighboring countries do not get the same color. Here the map is also translated into a graph and then an attempt is made to solve this problem with an algorithm. Similar to the problem of the traveling salesman, according to the current state of knowledge, this problem cannot be solved exactly in a reasonable time, even with computers above a certain size of the map. The problem of coloring general graphs optimally is considered to be one of the most difficult problems in the class of NP-complete problems. Assuming P = / NP (see P / NP problem) even an approximate solution is not possible except for a constant factor.

Visualization

In the field of computer graphics, the visualization of graphs is a challenge. Particularly complex networks only become clear through sophisticated auto-layout algorithms.

There is no silver bullet to mathematics.

Euclid

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