Graph Theory Solved Problems

The body of graph theory allows mathematicians and computer scientists to apply many known principals, algorithms, and theories to their model. It is composed of two kinds of elements, vertices and edges (sometimes called nodes and links in computer science). Let each vertex represent a team, and let each edge represent a game between teams.Let’s look at using graph theory to quickly solve a problem. I want each of those teams to play exactly 3 games — is this possible? There is a principal, known sometimes as “the handshake lemma”, which states that a graph must have an dd degree).

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Typical notations: Before we get too deep into graph theory or problems, let’s look at the basics of programming using the graph data structure.

There are a few ways to represent graphs in our programs — we’ll look at the most common three, and the basic tradeoffs.

In any graph, number of edges must be half the total degree, but 21 is odd, which means this graph can’t exist.

As it can’t exist, we know that such a tournament bracket can’t either.

Adjacency lists are the typical choice for “general purpose” use, though edge lists or adjacency matrices have their own strengths, which may match a specific use case.

Abstracting graph access is vital if your graph is going to span more than a single function call.

This is well suited to performant lookups of an edge, or listing all edges, but is slow with many other query types.

For example, to find all vertices adjacent to a given vertex, every edge must be examined.

As in any programming context, over-exposing internals leads to over-reliance on knowing the internals, across scopes in the codebase… For example, if you need to list all edges, consider maintaining a separate internal list, rather than iterating over all vertices.

which leads to slow development, and a lot of bugs. There are many libraries that you can use, such as gonum in Go, or networkx in Python, to use pre-built abstractions.


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