Applications of Graph Theory in Science, Technology, Engineering, and Mathematics

In mathematics, a graph is defined as a set of vertices (V) and edges (E) that is G=(V, E).

There are different types of graphs, and most importantly, the two broad groups of graphs are the directed graphs and the undirected graphs. All these graph types have sub-categories, and as argued by Singh (2014), all graphs have their significance both in science and in other fields of studies. Graph theory is defined as the mathematical theory that contains applications and properties of graphs. In the modern world, graphs are becoming increasingly crucial as graph theory is applied in many fields including other areas of mathematics, science, technology, and engineering.

Additionally, the powerful combinatorial methods found in graph theory are quite significant in roving other fundamental areas in re and applied mathematics.

For instance, in pure mathematics, graph theory is used to provide proofs for theories like Fermat’s Little Theorem and Nielson-Schreier Theorem. Minimum vertex covers have also been used to solve DNA sequencing problems and SNP assembly problems, neural networks and computer science in general. Additionally, Shu-Xi (2012) asserts that edge coloring can be applied in solving the timetabling problem and in explaining the assignment of frequencies in mobile phones’ networks. In this project, some of the primary applications of graph theory relating to the field of science, technology, engineering, and mathematics will be discussed.

Example of a graph

The two major classes of graphs that this paper will cover include the directed and the undirected graphs. In a directed graph, all the edges of the graph are directed from one vertex to the other. At some point, a directed graph is called a digraph or a directed network. On the other hand, undirected graphs are graphs whose edges have no particular directions. The figures below show directed and undirected graphs.

Figure 1: Directed graph and undirected graph respectively (Shu-Xi, 2012)

Application of graph theory in computer science

Apart from aiding in mathematical and scientific proofs, graphs are also quite essential in the field of computer science. In this field, graphs are used to indicate the flow of communication, the organization of data, different computational devices, and computational flow among others. For instance, graph theory has been used to study the link structure of a given website, like Wikipedia. Link websites can be represented by directed graphs, and in this case, the vertices are used to describe the web pages and the edges (weighted) are used to indicate the link from one page to the other.

Several algorithms have been designed in computer science to handle graphs, for instance, the Dijkstra’s algorithm which determines the shortest distance between two given points, and can be used in computer science to determine the webpage that has the highest number of viewers. Graph theory is also used to rank websites based on the number of viewers, and this is the idea that Google uses to perform page ranks. This is because using the damping factor may be too subtle especially when a web page has many viewers. The use of graph theory is then used as shown in the figure below.

Figure 2: Using graph theory to determine the page ranks of a website (Shu-Xi, 2012)

In Figure 2 above, it can be argued that Page C has the highest number of viewers (check the inward arrows). This implies that this website has the highest number of people visiting it than the other websites. Therefore, Google’s page rank algorithm will assign Page C to have the highest rank as shown in the figure below.

Figure 3: Google’s assigned page ranks using graph theory (Shu-Xi, 2012)

Dijkstra’s algorithm

Graph theory has become a vast field of study with applications in various areas. Dijkstra’s algorithm provides a solution of determining the shortest path between two points in a graph. Singh (2014) argues that it is possible to find the shortest distance from one point in a graph to all the other points. This algorithm is quite significant in supply chain and logistics as it can be used to find the shortest distance between different cities.

Additionally, this concept of graph theory can be applied in real life to determine the shortest distance that a car or even an individual can cover while traveling between given towns to save fuel and energy respectively. Such types of graphs are called weighted directed graphs and weights are assigned to each edge of the graph depending on the distances between given states or cities. The example below shows a weighted graph used to represent the distances between two given points/states.

Other important applications of graph theory

In epidemiology, graph theory is quite essential especially in determining the network models between groups of individuals. This is also useful in determining the number of contacts an infected person (with a contagious disease) may make with a susceptible individual. Additionally, the network designs are used to determine the spread of diseases and to come up with interventions that reduce the widespread of highly infectious illnesses and designing the response strategies. In this case, the graph’s vertices are considered as the individuals while the edges are the possible contacts. Graph theory is then applied to determine how an affected individual is connected to the others. Understanding and identifying the shortest path is relevant in indicating how susceptible an individual is and the potential of an infected individual to affect others.

Conclusion

Over the past few years, graph theory is one of the most significant theories that have been used to prove the existence of different phenomena in various fields. Most importantly, graphs are widely used in science, technology, engineering and mathematics (STEM), to prove various theorems including the widely known Fermat’s Little Theorem and even in epidemiology to predict the trend of diseases. Additionally, the directed graphs are also quite significant in supply chain and logistics as they help in determining the shortest distance between given points using the famous Dijkstra’s algorithm (computer-based). Other essential applications of graph theory include in fields like medicine, agriculture, social sciences, and biology among others.

References

Shu-Xi, W., 2012. The improved dijkstra’s shortest path algorithm and its application. Procedia Engineering, 29, pp.1186-1190.

Singh, R.P., 2014. Application of graph theory in computer science and engineering. International Journal of Computer Applications, 104(1).