Elements of Graph Theory

Abstract

Graphs are mathematical structures used to model and visualize relations between certain objects. One of the first formulations of a graph theory problem was the famous Konigsberg Problem solved by Leonhard Euler. He proposed a model which reduced the problem to a schematic diagram and then concluded that the graph needed to satisfy some general conditions for the problem to be solved affirmatively. His studies highlighted the importance of understanding graphs and their properties, which later lead to the creation of Graph Theory as a distinct mathematical discipline. In this chapter we first introduce graphs through some illustrative examples and then describe the basic elements of graphs, as well some important graph properties. We then present special types of graphs like trees or networks, together with some specific techniques, problems and algorithms.