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Graph theory is a field of mathematics and computer science that involves the study of graphs, which are mathematical structures used to model pairwise relations between objects. In this context, a graph is a collection of points, known as vertices, connected by lines, called edges. Graph theory is used to solve problems in various disciplines, including biology, computer network design, and sociology.

Nonisomorphic simple graphs are an important concept in graph theory. Two graphs are isomorphic if one can be transformed into the other merely by renaming vertices. Nonisomorphic graphs, on the other hand, cannot be made identical through any renaming of vertices. With three vertices, there is a limited number of nonisomorphic simple graphs, as there are fewer ways to connect the vertices. Simple graphs are graphs without loops (edges connected at both ends to the same vertex) and without multiple edges between the same pair of vertices.

A vertex, also known as a node, is one of the fundamental units from which graphs are built. It represents an entity or a point where edges meet. In the case of nonisomorphic simple graphs with three vertices, you are considering all the possible ways that these points can be connected (or not connected) without forming loops or having more than one edge directly connect two vertices.

In the context of the exercise, step 1 involves a graph with a vertex that is not connected to any other, indicating an isolated entity. Step 2 shows two vertices connected by an edge, with the third vertex remaining separate, suggesting a pairwise but not complete relationship. Finally, step 3 displays a fully connected trio of vertices, where each vertex is directly connected to both of the others, representing a complete, mutual interrelation among all entities.

An edge in graph theory is a line connecting two vertices in a graph, signifying some form of relation or interaction between them. Edges are integral in determining the structure of a graph and can represent diverse concepts such as pathways, relationships, or data flow depending on the application of the graph.

The exercise exemplifies different configurations of edges within simple graphs. Step 1 features no edges, highlighting independence among vertices. Step 2 includes exactly one edge, indicating a solitary connection between two of the vertices. The last step, step 3, illustrates a fully connected scenario with three edges wherein every pair of vertices is joined by a single edge, thus forming a complete network of relations among the vertices. These varying configurations of edges cause the graphs to be nonisomorphic, as no renaming of vertices would make the graphs identical in their connections.