Graph Theory Solved Problems

An adjacency matrix is a two-dimensional matrix, with the graph’s vertices as rows and columns.A given intersection is true if those vertices are adjacent, or false if they are not (note: if the graph is directed, be sure to define that relationship in rows vs columns).Graph theory, like any topic, has many specific terms for aspects of a graph.

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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.

This is a consequence of the logical fact that every edge must connect to two vertices — one on each end.

As we have an odd number of vertices, and each of those vertices has an odd number of edges, the total degree of the graph is 21 (AKA 7*3).

Graph theory is a branch of mathematics, first introduced in the 18th century, as a way to model a puzzle.

Graphs are excellent at creating simplified, abstract models of problems.

They also typically require more space than other models, especially with sparse graphs (graphs with “few” edges).

An adjacency matrix needs to reference every vertex against every other vertex, giving O(|V(G)|) space needed.

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.


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