# Graph Theory and Its Applications

Graphs are powerful visual tools because they convey information quickly and clearly. They  are widely utilized in both print and online media. Data can often be easier understood when displayed as a graph rather than a table since the graph might illustrate a pattern or contrast. An important part of the world of graphs is the "Graph Theory".

What is Graph Theory?

Graph theory may be defined as the study of graphs. A graph is a form of mathematical structure that is used to depict a certain function by linking a group of points. Graphs may be used to build a pairwise relationship between entities. A graph G(V, E) is a non-linear data structure formed by two sets (V, E), where V is the non-empty set of vertices (points or nodes) and E is the set of edges E is the set of edges (lines or branches) such that there is a mapping f: E V, that is, from the set E to the set of sorted or unordered pairs of V components.

Technically, a graph may be represented using the pair G. (V, E).

Where,

• V represents the finite set vertices

• E represents the finite set edges.

As a result, the graph consists primarily of a non-empty set of edges E and a set of vertices V.

In graph theory, we often use the graph to represent a collection of things that are related in some way. The elements may be characterized as mathematical notions that can be stated using nodes or vertices, and the relationship between pairs of nodes could be expressed using edges.

History of graph theory

• Graph theory began with Euler, who was assigned to find a suitable path across the seven Köningsberg bridges.

• His attempts and final solution to the classic Königsberg bridge problem resulted in the development of graph theory.

• Leonhard Euler was a well-known Swiss mathematician.

• He drew graphs and solved several mathematical problems using a given collection of points or data.

• He claims that graphical representation allows different types of data to be displayed in various formats, such as line graphs, bar graphs, line plots, circular graphs, frequency distributions, and so on.

The study of points and lines is what graph theory is all about.

It is a type of subfield concerned with the study of graphs.

It is possible to demonstrate the mathematical truth through graphical representation.

The graph theory approach can demonstrate the relationship between nodes and edges.

Types of graphs

There are three types of graphs: undirected graphs, directed graphs, and weighted graphs.

Undirected graphs

• Edges are linked to an unordered pair of vertices.

• Formed of a series of vertices joined together by undirected edges.

• All of the graphs edges are bidirectional.

Directed graphs

• Edges are linked to an organized pair of vertices.

• Formed with the assistance of a set of vertices connected by directed edges.

• Edges have a defined direction that corresponds to the vertices.

Weighted graphs

• Edges have a weight linked with them to indicate significant repercussions like price, length, and volume.

• They may either be directed or undirected.

What is the Process of Graph Theory?

• Graph theory is essentially focused on examining the links between various nodes (vertices) and connections (edges).

• The study of graphs over a structure gives solutions to several design, connectivity, efficiency, matching, and operating challenges.

Graph theory is the study of relationships

• Graph theory, given a set of nodes and connections that may abstract anything from urban layouts to digital data, is a useful tool for quantifying and simplifying the numerous moving pieces of dynamic systems.

• Several arrangement, communication, optimization, matching, and operational issues can be solved by studying graphs using a framework.

Real life applications of graph theory

Graphs may be used to simulate many different types of relationships and interactions in physical, biological, social, and information systems, and they have a wide range of applications.

Graph theory is more than just an abstract mathematical idea. There are different applications of graph theory.

• Google Maps makes use of graph theory  in its building and transportation systems.

• A vertex is the junction of two (or more) roads, while an edge is the road that connects two vertices.

• The technique is then used by their navigation system to compute the shortest route between the two vertices.

Computer Networking

• The relationships between networked computers in a computer network adhere to the rules of graph theory.

• Network security also makes use of graph theory.

• The vertex coloring procedure can be deployed to obtain a suitable four-color coloring for the map.

• For every GSM (Grouped Special Mobile) mobile phone network, the vertex coloring procedure may be used to allocate at most four distinct frequencies.

Social networking sites

• Social media is one of the most popular applications of graph theory.

• Graph theory is used by many networking sites such as Facebook, Twitter, LinkedIn, and Instagram to connect individuals with their friends.

• A large graph connects the users of leading social networking websites.

• In the case of social media, the vertices are the users (people), and the edges are the relationships between them, such as followers and subscribers.

Epidemiology

Epidemiologists utilize graph theory to better understand how illnesses spread.  They can detect which persons are most at danger and how to restrict disease transmission by mapping out who is related to whom.

Product suggestion

• The e-commerce portals frequently predict the requirements of users and recommend certain products.

• They employ the graph data structure to connect what users have purchased in the past or what they have searched for in the past.

Traffic lights

• The operation of traffic lights, such as turning red or green and the precise timing between them is due to graph theory.

• The vertex coloring approach is utilized here to resolve time and space problems by determining the chromatic number for the number of cycles required.

Graph theory is widely used in the study and modeling of various applications in various fields. It makes it easy to represent various types of mathematical objects. Graphs provide us with several strategies and flexibility for describing and addressing real-world problems. Graph theory principles are frequently employed in a variety of applications outside of mathematics. In chemistry, they include the study of molecules and atoms as well as the bond formations. Graph theory is employed in biology to help with conservation work. It"s also utilized in computer applications to create graph algorithms.