## Other Concepts in Graphs

In this chapter, we covered ways of representing and traversing a graph and looked at shortest path algorithms. Graphs are also an optimal data structure for some problems we haven't mentioned yet. This section aims to introduce some of them.

### Minimum Spanning Trees

A minimum spanning tree of a graph is a subset of the set of edges *E* of a connected graph that connects all vertices together, without any cycles and with the minimum total edge weight. It is a tree because every two vertices in it are connected by exactly one path.

In order to understand the applicability of minimum spanning trees, consider the problem of a telecommunications company that is moving into a new neighborhood. The company wants to connect all the houses, but also wants to minimize the length of cable that it uses in order to cut costs. One way to solve the problem is by computing the minimum spanning tree of a graph whose vertices are the houses of the neighborhood, and the edges between houses...