Question

Draw the following graphs. (1) \(\mathrm{K}_{4}\) (2) \(\mathrm{K}_{44}\) (3) \(\mathrm{K}_{2}\) (4) \(\mathrm{K}_{3}\) (5) \(\mathrm{K}_{8}\)

Short answer

Draw the complete graphs: (1) \(\mathrm{K}_{4}\) has 4 vertices and 6 edges. (2) \(\mathrm{K}_{4,4}\) (assuming this is the correct notation) is a complete bipartite graph with two sets of 4 vertices, and 16 edges between the sets. (3) \(\mathrm{K}_{2}\) has 2 vertices and 1 edge. (4) \(\mathrm{K}_{3}\) has 3 vertices and 3 edges. (5) \(\mathrm{K}_{8}\) has 8 vertices and 28 edges.

Step-by-step solution

Drawing \(\mathrm{K}_{4}\)

To draw the complete graph \(\mathrm{K}_{4}\), we first place four vertices on the diagram. Then, we draw an edge between each pair of vertices. There will be a total of 6 edges in this graph.

Drawing \(\mathrm{K}_{44}\)

The notation \(\mathrm{K}_{44}\) seems incorrect as it suggests a complete graph with 44 vertices connected to each other. We will assume that this is a typographical error and instead draw \(\mathrm{K}_{4,4}\), which is a complete bipartite graph. A complete bipartite graph is a graph with two disjoint sets of vertices, such that every vertex in one set is connected to every vertex in the other set. To draw \(\mathrm{K}_{4,4}\), place four vertices on one side (set A) and four vertices on the other side (set B). Then, connect each vertex in set A to every vertex in set B with 16 edges in total.

Drawing \(\mathrm{K}_{2}\)

To draw the complete graph \(\mathrm{K}_{2}\), place two vertices on the diagram. Then, draw a single edge between the two vertices.

Drawing \(\mathrm{K}_{3}\)

To draw the complete graph \(\mathrm{K}_{3}\), place three vertices on the diagram, preferably in a triangular arrangement. Then, draw an edge between each pair of vertices, resulting in a total of 3 edges.

Drawing \(\mathrm{K}_{8}\)

Lastly, to draw the complete graph \(\mathrm{K}_{8}\), place eight vertices on the diagram. Then, draw an edge between each pair of vertices. There will be a total of 28 edges in this graph. Remember to ensure that each vertex is connected to every other vertex with a unique edge.

Key concepts

Graph Theory

Graph theory is a field of mathematics and computer science that involves the study of graphs, which are mathematical representations of a set of objects where some pairs of the objects are connected by links. These objects correspond to mathematical abstractions called vertices (or nodes), and the links that connect some pairs of vertices are called edges.

A graph is typically represented by a set of dots (for the vertices) and lines connecting the dots (for the edges). The importance of graph theory lies in its ability to model complex systems and structures in a variety of disciplines such as biology, computer networks, social sciences, and logistics. Graph theory helps to solve problems related to connectivity, optimal path finding, scheduling, and network flows.

Bipartite Graph

A bipartite graph is a special kind of graph in graph theory where the vertices can be divided into two disjoint and independent sets so that every edge connects a vertex from one set to a vertex from the other set. In other words, there are no edges that connect vertices within the same set.

The complete bipartite graph, often denoted as \(\mathrm{K}_{m,n}\), has all possible edges between the two sets of vertices, with \(m\) vertices in one set and \(n\) vertices in the other. For example, a \(\mathrm{K}_{4,4}\) would have each vertex in one set of 4 connected to all vertices in the other set of 4, for a total of \(4 \times 4 = 16\) edges. They are widely used in modeling relationships in databases, matching problems, and network flow.

Vertex

In graph theory, a vertex (also known as a node) is a fundamental part of a graph. Each vertex serves as a connection point where edges meet. Vertices are used to represent entities, while edges represent the relationships or connections between these entities.

The degree of a vertex is the number of edges that connect to it. In a complete graph, every vertex is connected to every other vertex by a single edge. Therefore, in a complete graph with \(n\) vertices, often represented as \(\mathrm{K}_n\), every vertex has a degree of \(n-1\), indicating a highly interconnected structure. Understanding the concept of vertices is crucial in graph theory as it helps decipher the topology and properties of the graph one is working with.