ABSTRACT
Graph Theory is a rich field of mathematics. Although it dates back to Euler's paper in 1736, much of the work has been done in the last century. This chapter elaborates on what algorithm efficiency means and how mathematicians determine what makes one algorithm more efficient than another. The algorithm complexities were chosen due to their prevalence in graph theory problems. The chapter explores a different use of digraphs that have a very specific underlying structure, one in which a direction is added to each of the edges from a complete graph. It focuses on a new application for digraphs, one in which items are sent through a network. The chapter then focuses on two types of rooted trees that have a strong connection to computer science: breadth-first search trees and depth-first search trees. The main idea behind a depth-first tree is to travel along a path as far as possible from the root of a given graph.
