# Application graph theory real life

The number of colors given to color the graph is equal to number of registers. This technique is used in order to approximate the data which has to be stored while minimizing the loses of precision. For example a data which represents taken temperatures during the time and builds a nice graph. However if this data was taken at high frequency, there might be too many records.

The idea is to minimize the number of records, while keeping most of the information about the evolvement of the temperature. The shortest path algorithm can be used to solve this problem. For instance the blue line in the following graphics represents the data to be stored. It is 10 values: The green and red line represent possible approximations, when we are leaving out one or two nodes. Of course there are several nodes which can be left out and the shortest path algorithm can help us to find which ones can be left out.

We can construct a full graph, which will contain 5 nodes, representing the 5 data points the times x. The following picture represents the partial graph — the skipping edges start only in the first node A. The edge with value x1 corresponds to the red line in the graph etc. The graph should be also filled with other edges starting in B and C the only edge going from D to E is already present and there are no edges starting in E , but I have left them out for simplicity.

The values of the edges in the standard path should be the lowest here all of them have value 1. On the other hand values of edges which will make us loose more precision should be the greatest. Then of course we can introduce some bonus to motivate the algorithm to take less edges better compression.

All this constraints can be modeled using heuristics. One possible heuristics to evaluate the value of the edge is to measure the distance between the real data and the estimated date. For instance the value of the second point is 5. If we estimate the value using the red line leaving out the second point the corresponding value on the red line is 3.

The distance between these two values is: On the other hand the green line also estimates the second point 3 point. And we see that the distance for the second point will be more or less 1.

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So we should add these distance together. So we get:. This is just a proposition. We could also multiply it by some coefficient to obtain some reasonable results. In this problem we have a group of workers and group of tasks.

### Obvious applications

Each task can be processed by each worker. However the workers do not have the same performance on all tasks — the time for the processing of each task differs for each worker. The values in the table represent the processing time, that the worker needs for the given task. Not that each edge has two values: Finding the cheapest flow in this graph from S to F will give us the best assignment of workers to tasks. Applications of Graph Theory Graph Theory is just a beautiful part of mathematics. For my personal clasification I have separated the tasks, which you can solve using Graph Theory into two groups: Obvious applications — I mean, that the analogy between the graph and the problem is quite easy to imagine maps, cities, relations etc.

In this post you can find the following: Than you see one of them, and than you think: I will provide the following ones in this post: Notifications View Subscribe. Font Size. Article Tools Print this article. Indexing metadata. How to cite item. Email this article Login required. Then in , Thomas. Kirkman and William R. Hamilton studied cycles on polyhydra and invented the concept called Hamiltonian graph by studying trips that visited certain sites exactly once. In , H. Dudeney mentioned a puzzle problem. Eventhough the four color problem was invented it was solved only after a century by Kenneth Appel and Wolfgang Haken.

This time is considered as the birth of Graph Theory. Caley studied particular analytical forms from differential calculus to study the trees. This had many implications in theoretical chemistry. This lead to the invention of enumerative graph theory. In , Ramsey worked on colorations which lead to the identification of another branch of graph theory called extremel graph theory. In , the four color problem was solved using computers by Heinrich. The study of asymptotic graph connectivity gave rise to random graph theory.

The number of vertices in a graph is usually denoted n while the number of edges is usually denoted m. Vertices are also known as nodes, points and in social networks as actors, agents or players. Edges are also known as lines and in social networks as ties or links. Two vertices u and v are adjacent if there exists an edge u,v that connects them. An edge u,v is said to be incident upon nodes u and v. The natural graphical representation of an adjacency matrix is a table, such as shown below. Examining either Figure 1 or given adjacency Matrix, we can see that not every vertex is adjacent to every other.

### Hidden applications

A graph in which all vertices are adjacent to all others is said to be complete. A subgraph of a graph G is a graph whose points and lines are contained in G. A complete subgraph of G is a section of G that is complete 1. While not every vertex in the graph in Figure 1 is adjacent, one can construct a sequence of adjacent vertices from any vertex to any other.

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## Applications of Graph Theory

Graphs with this property are called connected. Similarly, any pair of vertices in which one vertex can reach the other via a sequence of adjacent vertices is called reachable. If we determine reachability for every pair of vertices, we can construct a reachability matrix R such as depicted in Figure 2. The matrix R can be thought of as the result of applying transitive closure to the adjacency matrix A. A component of a graph is defined as a maximal subgraph in which a path exists from every node to every other i.

## Applications of Graph Theory – Jan Fajfr's wall – Software engineer @ ITG RFQ-hub

The size of a component is defined as the number of nodes it contains. A connected graph has only one component. A sequence of adjacent vertices v0,v1,…,vn is known as a walk. In Figure 3, the sequence a,b,c,b,c,g is a walk.

A walk can also be seen as a sequence of incident edges, where two edges are said to be incident if they share exactly one vertex. A walk in which no vertex occurs more than once is known as a path. In Figure 3, the sequence a,b,c,d,e,f is a path. A walk in which no edge occurs more than once is known as a trail.

In Figure 3, the sequence a,b,c,e,d,c,g is a trail but not a path. Every path is a trail, and every trail is a walk. The sequence c,e,d in Figure 3 is a cycle. A tree is a connected graph that contains no cycles. In a tree, every pair of points is connected by a unique path. That is, there is only one way to get from A to B. Figure 3: A labeled tree with 6 vertices and 5 edges 1. A spanning tree for a graph G is a sub-graph of G which is a tree that includes every vertex of G.

Applications of Graphs to real life problems. Introduction