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Graph theory is a field of mathematics that studies the properties and applications of graphs, which are structures consisting of vertices (or nodes) connected by edges (or lines). Graphs are used to model relationships in various disciplines such as computer science, biology, social sciences, and, of course, mathematics.

Understanding graph theory begins with familiarizing oneself with basic terms, such as vertices, edges, paths, cycles, and connected components. Two graphs are considered to be isomorphic if they contain the same number of vertices connected in the same way; that is, they are identical in structure, even if the actual objects and connections are different in some way. Nonisomorphic graphs, on the other hand, do not share these structural characteristics, thus representing fundamentally different networks, regardless of the labels of vertices or the visual representation of the edges.

When analyzing nonisomorphic graphs, we typically focus on their properties like the number of vertices and edges, the degree of each vertex, symmetry, and the presence of subgraphs. Each of these properties can significantly influence the behavior and utility of the graph in question, which is why graph theory remains a critical area of study within discrete mathematics.

The degree of a vertex in a graph is defined as the number of edges that are connected to it. For any given vertex, the degree represents its immediate number of connections to other vertices within the graph. This concept is central in graph theory as it helps to determine various properties of graphs.

For example, in the exercise of drawing nonisomorphic graphs with given vertex degrees, paying close attention to each vertex's degree leads to the creation of distinct graphs. A vertex with a higher degree becomes a more central or pivotal point within the graph. In the creation of nonisomorphic graphs, ensuring that the vertices meet the specified degree conditions is crucial, since having even one vertex with a differing degree can lead to a completely different graph structure.

It's also worth noting that some graphs have special properties related to degrees, such as regular graphs, where all vertices have the same degree; or bipartite graphs which can be split into two groups of vertices such that each edge connects a vertex from one group to a vertex in the other, and the degrees of the vertices often follow predictable patterns.

Discrete mathematics is an area of mathematics that is concerned with distinct and separate values; it is the mathematical language of computer science and encompasses a wide range of topics, including logic, set theory, combinatorics, and graph theory. Problems in discrete mathematics are typically countable, often finite, and deal with discrete elements.

Graph theory is considered a part of discrete mathematics, and problems such as determining nonisomorphic graphs with certain conditions are classic examples of exercises found in this field. Discrete mathematics is essential for algorithm formulation, analysis of networks, cryptographic systems, and various other computational applications.

From a teaching perspective, exercises in discrete mathematics, such as creating nonisomorphic graphs, allow students to engage with abstract concepts in a tangible way. They also help in developing problem-solving skills that are crucial for computer science and information technology. By composing graphs based on certain criteria, one learns how discrete structures behave and interact, which is foundational to further study in both discrete mathematics and computer science.