What's the difference between a sparse and a dense graph?

A sparse graph has relatively few edges compared to the maximum possible, while a dense graph has many edges close to the maximum.

In graph theory, a branch of mathematics and computer science, graphs are used to represent relationships or connections between objects. These graphs are made up of vertices (or nodes) and edges (or arcs) that connect these vertices. The distinction between sparse and dense graphs is based on the number of edges in relation to the number of vertices.

A sparse graph is one where the number of edges is close to the minimal number of edges. In other words, there are relatively few edges compared to the number of vertices. For example, if you have a graph with 10 vertices, the minimum number of edges would be 9 (if the graph is a tree), and any number close to this would make the graph sparse. Sparse graphs are often represented as adjacency lists, which is a more space-efficient method when the number of edges is low.

On the other hand, a dense graph is one where the number of edges is close to the maximum possible number of edges. For a simple undirected graph, the maximum number of edges is n(n-1)/2, where n is the number of vertices. For a directed graph, the maximum is n(n-1). So, if you have a graph with 10 vertices, the maximum number of edges would be 45 (for an undirected graph) or 90 (for a directed graph), and any number close to this would make the graph dense. Dense graphs are often represented as adjacency matrices, which is a more time-efficient method when the number of edges is high.

In practical terms, the distinction between sparse and dense graphs can have significant implications for the efficiency of algorithms. Algorithms that work well on sparse graphs may perform poorly on dense graphs, and vice versa. Therefore, understanding the nature of the graph you're working with can help you choose the most appropriate algorithm and data structure.