Graph Theory Explained: 4 Applications of Graph Theory

Graph theory is a branch of mathematics that covers the graphic depiction of data and relationships between objects. These endpoints (also known as a set of vertices or nodes) connect via a number of edges (sometimes referred to as links or lines). People can then use these graphs to solve both abstract and pragmatic mathematical problems in a host of different disciplines.

Swiss mathematician Leonhard Euler provided a general introduction to graph theory toward the tail end of the eighteenth century. Since his initial pioneering work, others like Kazimierz Kuratowski, Paul Erdös, and Dénes König have made substantial contributions to the field as well. Various subsets of graph theory—like extremal graph theory or combinatorics (combinatorial topology)—developed over time to further apply advances to unique scenarios.

Graph theory has applications far beyond the world of discrete mathematics. For instance, much of computer science operates through the usage of graph algorithms and data structures. Chains of code operate like the vertices and edges of a graph. As a user interacts with software, they engage in a form of graph traversal (moving between nodes to unlock certain features of the program).

Businesspeople, economists, and financial professionals also use graphs to depict data points. Although the graphs for such circumstances might be simpler than those in use by mathematicians in advanced graph theory, they still operate off the same basic principles. By using insights from graph theory, professionals in these fields can achieve optimization for their own business models.

There are numerous different ways to depict graphical data. Here are five of the most common types of graphs you might use:

1. 1. Bipartite graphs: This complete graph depicts the relationship between two independent sets of objects. It has an adjacency matrix of (0,1). Mathematicians generally use two colors to represent the various sets of information on graphs like these.
2. 2. Connected graphs: When you have a set of edges connecting every node, you have a connected graph. You can also disconnect certain vertices to see the impact this has on wider data sets in certain types of math problems, although you’ll no longer have a connected graph if you do so.
3. 3. Directed graphs: Sometimes called digraphs, directed graphs connect ordered pairs of vertices asymmetrically. The directed edges might point from one node to another or reach out in both directions.
4. 4. Planar graphs: If you embed the endpoints and edges of a graph in a plane, you create a planar graph. Planar graphs only intersect at their endpoints. This makes it impossible for their edges to cross each other.
5. 5. Undirected graphs: Edges link a number of vertices symmetrically in these simple graphs. Alongside directed graphs, these are among the most common charts in graph theory as a whole.

You can apply graph theory to a host of different mathematical conundrums. Consider these four applications of the discipline:

1. 1. The bridge problem: This Eulerian problem (named for graph theory originator Leonhard Euler) originated in the eighteenth century. In this thought exercise, Euler aimed to show how you could cross all seven bridges of Königsberg (a Prussian city) only one time each while traversing the entire city. Mathematical historians widely consider this to be the first instance of graph theory put into practice.
2. 2. The four-color theorem: Mathematicians use graph coloring to specify different data sets so adjacent vertices look identifiably different. The four-color theorem states you only need four or fewer colors to do so even in the most complex of graphs. See if you can do so without a tutorial.
3. 3. The Hamiltonian path problem: In graph theory, Hamiltonian paths are paths you can take along a graph’s edges to visit each vertex only once. Every graph you encounter offers you a chance to find the shortest path between all vertices. The traveling salesman problem is just one example of this quandary.
4. 4. The subgraph isomorphism problem: Common in computer science, the subgraph isomorphism problem challenges you to find a fixed graph within the confines of a larger, much more complex graph. Software designers use this same skill to comb through large amounts of code to find blocks they need to adjust.

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