## Applications of graph theory

A famous Swiss mathematician, Leonhard Euler, started the work in the area of graph theory in 1735. The graph theory has been applied in various fields, including electrical engineering (communications and networks), Chemistry (the study of molecules, construction of bonds, and the study of atoms), biochemistry (DNA fragment assembly, genomics), operations research, among other fields. The powerful combinatorial methods of the graph theory have proven significant results in mathematics. Graph theory is indispensable in mathematics. This essay explores two applications of the graph theory in networking. It discusses the application of the graph theory in communication networks by representing networks as binary tree and butterfly networks.

A network involves a collection of terminals, links, and nodes connected to enable users to access the network. Communication networks have terminals with different addresses, and the connections and messages follow the correct route to recipients. Communication networks are made up of three essential components, terminals, processors, and transmission channels. In communication networks, data is transmitted between computers, processors, and other devices. Various mathematical structures can be used to represent communication networks with different switch sizes and count. The graph theory can represent communication networks as a binary tree or butterfly networks.
A binary tree is made up of nodes and is a data structure with different sources, switches, terminals, and destinations. The binary tree has vertices with controls that direct data through the network. A complete binary tree’s diameter with N inputs and outputs is 2 logN +1. The switches have two incoming edges and two outgoing edges with a 2N-1 number of switches. The binary tree has a root switch that is a complete bottleneck. The binary tree organizes complex communication networks naturally that can be divided into a structure that resembles a tree. It, therefore, provides a simple way of representing networks in a way that easy to understand (Heijmans, 1995).

In a butterfly network, the terminals and switches are arranged in a manner that resembles the letter N. the input is usually at the left end while the output is located at the right. In between the inputs and the outputs are the log(N) + 1 switch levels that are uniquely different in a network. The switch size is a 2×2 with a total switch count of N(log(N)+1). The butterfly network has a unique path between each input and output, and the maximum messages passing through a vertex give the network congestion. Butterfly networks also enable a representation of a complex network structure in a way that is easy to understand and manage. The graph theory represents the communication network into nodes and switches that can be varied in size and patterns to extract useful information about networks.

Networking and communication networks are highly complex structures made up of interconnected components. The underlying connection patterns that exist in networks have different topologies that are represented by the graph theory. The theory studies and defines the structure of the networks and design problems arising in the analysis of the networks. The graph theory has advanced knowledge in networking as it represents communication networks in structures that allow comparison in terms of switch count, switch size, and congestion. The graphs also have a vital role in modeling computer networks. Graphs have vertices, and these represent the terminals and processors, while the edged of the graphs represent the transmission channels, fibers, and wires. The graph theory is used to show how a data packet hops from the input through switches to the output terminal (Meyer, 2010).

The graph theory can be used in communication networks for computer network security. Scientists have shown how the vertex cover algorithm can simulate how stealth worms propagate in computers and design ways of protecting networks against such virus attacks. The graph theory can be used to design a minimum vertex cover in graphs to prevent worm propagation and come up with an ideal network defense system. The graph theory can also be used in GSM mobile networks that comprise of cellular networks with hexagonal cells dividing the geographical region covered by the networks. There is a communication tower in each cell that connects the GSM mobile phones within the cells. The mobile phones connect to the network in search of cells in the vicinity. The vertex algorithm can be used to assign different frequencies for the GSM mobile phone networks within and without the cells (Shirinivas, Vetrivel & Elango, 2010).

In conclusion, mathematics and mathematical formulas are pretty visible in various fields, including electronic engineering, computer science, and networking. Mathematics and graph theory help design algorithms and analyze how they can be applied in networking and engineering applications. The graph theory also improves the effectiveness of other algorithms used. Therefore, graph theory is an essential aspect of networking as it represents networking in different ways, including the binary tree and the butterfly network. This paper explores the use of the graph theory in networking, how the applications are used in networking and how the theory has advanced the knowledge in networking.