Graph theory is an important and fascinating branch of mathematics that has applications in computer technology, biology, sociology, and other areas. It is the study of graphs: two-way relationship models between items represented by mathematical structures. Vertices (also known as nodes) and edges (known as arcs) that join pairs of vertices make up a graph.

While graph theory may sound challenging at first, dedicating time to learning it reveals that it is both applicable and interesting. This theory enables the exploration and understanding of a wide range of phenomena, from social networks to biological systems. By studying the properties of vertices and edges, and the ways in which they are interconnected, graph theory will give valuable insight into the fascinating structure and dynamics of networks.

## Basics

The basic components of graph theory are edges, sometimes known as connections, and vertices, sometimes known as nodes. Entities are represented by vertices, and the connections or interactions between these entities are represented by edges. The number of edges that are connected to a vertex is expressed in degrees. “In-degrees” and “out-degrees” refer to the number of incoming and exiting edges, respectively, in directed graphs. In a graph, paths are groups of edges that join a number of distinct vertices. A circuit (or cycle) is a path that begins and finishes at the same vertex.

## Eulerian Paths & Circuits

In an Eulerian path, every edge is visited exactly once. An Eulerian circuit -or Eulerian cycle- is an Eulerian path that starts and ends at the same vertex. Euler’s Theorem says that a connected graph has an Eulerian circuit if- and only if- every vertex has an *even *degree. A graph has an Eulerian path if- and only if- exactly zero *or* two vertices have an *odd *degree (but the rest have even degrees), ensuring it is traversable. A graph is traversable if you can draw a path between all the vertices without going back over a previous path.

The problem of the Königsberg bridges was used by Euler to demonstrate his theorem. The city of Königsberg (formerly in Germany) had seven bridges connecting various parts of the city, and the problem was to determine whether one could walk through the city crossing each bridge only once. Euler represented this problem as a graph and proved that no such path exists because more than two vertices in the graph had an odd degree.

## Hamiltonian Paths & Circuits

A Hamiltonian path is a path in a graph that visits each vertex exactly once. A Hamiltonian circuit (or cycle) is a Hamiltonian path that begins and ends at the same vertex. Unlike Euler’s theorem, there is no easy way to determine a Hamiltonian path and cycle. However, several necessary and sufficient conditions are known.

For example, Dirac’s theorem says that if every vertex in a graph with *n* vertices (where *n* ≥ 3) has a degree of at least* n*/2, then the graph is Hamiltonian. Ore’s theorem states that a network is Hamiltonian if the sum of the degrees of each pair of vertices that are not close to one another is greater than or equal to n. Let us consider a pentagonal-shaped graph. In this graph, the degree of each vertex is 2. It is a Hamiltonian cycle due to the fact that you may start at any vertex, trace each edge just once, then return to the beginning vertex without revisiting any edges.

## Applications

Graph theory has numerous applications in various fields: computer science, biology, social sciences, operations research, and chemistry. In computer science, graph theory is foundational. It is used in algorithms for searching, network flow problems, and the design of circuits and networks. Graphs also model social networks, internet data structures, and more.

In biology, graph theory allows us to understand molecular structures and the functioning of neural networks. An example would be phylogenetic trees, which represent evolutionary relationships. Chemical graph theory means studying the molecular structure of chemical compounds. In this case, vertices represent atoms, and edges represent bonds between atoms.

In the social sciences, social network analysis uses graph theory to study social structures. Vertices represent individuals or organizations, and edges represent relationships or interactions between them. Graphs are used to model and solve optimization problems such as the “traveling salesman” problem, where the goal is to find the shortest Hamiltonian cycle on a salesperson’s travel route.

## At Online Math Center

Graph theory provides powerful tools for modeling and solving complex problems involving relationships and interactions. Eulerian and Hamiltonian paths and circuits are fundamental concepts within graph theory that help in understanding various properties of graphs and their practical applications. As research continues, the relevance and application of graph theory in solving real-world problems continues to grow, solidifying its importance in both theoretical and applied mathematics.

Students can improve their mathematical skills by practicing concepts like these in the classroom and beyond. At OMC, students can improve their core skills through classes and tutoring. With two difficulty levels every grade, OMC strives to progress each student as far as possible in mathematics – and beyond. Contact OMC now to learn more.