Problem 8 - Applied Discrete Structures

Question

Draw complete undirected graphs with \(1,2,3,4,\) and 5 vertices. How many edges does a \(K_{n},\) a complete undirected graph with \(n\) vertices, have?

Step-by-Step Solution

Answer

The number of edges in a complete graph \( K_n \) is \( \frac{n(n-1)}{2} \).

Step 1: Understanding Complete Graphs

A complete undirected graph is a graph in which each pair of distinct vertices is connected by a unique edge. Let's denote such a graph with \( n \) vertices as \( K_n \). Our task is to draw complete graphs for \( n = 1, 2, 3, 4, \) and \( 5 \), and to find the formula for the number of edges in \( K_n \).

Step 2: Drawing Complete Graphs

- **For \( n = 1 \):** A single vertex with zero edges, \( K_1 \), as no other vertex is available to connect.- **For \( n = 2 \):** Two vertices connected by one edge, forming \( K_2 \).- **For \( n = 3 \):** Three vertices where each vertex is connected to the other two, forming a triangle with 3 edges, \( K_3 \).- **For \( n = 4 \):** Four vertices, where each vertex is connected to every other vertex, forming a six-edged graph, \( K_4 \).- **For \( n = 5 \):** Five vertices, with each pair of vertices connected, producing a ten-edged graph, \( K_5 \).

Step 3: Determine the Number of Edges

The number of edges in a complete graph \( K_n \) with \( n \) vertices is given by the combination formula \( \binom{n}{2} \). This formula represents choosing 2 vertices from \( n \) vertices to form an edge. The formula is \( \binom{n}{2} = \frac{n(n-1)}{2} \).

Step 4: Apply the Formula

Now, apply the formula \( \frac{n(n-1)}{2} \) to each complete graph:- **For \( n = 1 \):** \( \frac{1(1-1)}{2} = 0 \) edges.- **For \( n = 2 \):** \( \frac{2(2-1)}{2} = 1 \) edge.- **For \( n = 3 \):** \( \frac{3(3-1)}{2} = 3 \) edges.- **For \( n = 4 \):** \( \frac{4(4-1)}{2} = 6 \) edges.- **For \( n = 5 \):** \( \frac{5(5-1)}{2} = 10 \) edges.

Key Concepts

Complete Graph

In graph theory, a complete graph is an essential concept. A complete graph is a type of graph where every pair of distinct vertices is connected by a unique edge. This means that all the vertices are interconnected without any missing connections.
Such a graph is commonly denoted as \( K_n \), where "\( n \)" represents the number of vertices. For example, \( K_3 \) is a complete graph with three vertices. Each vertex is directly linked to every other vertex. This results in a triangular structure.
Complete graphs are particularly useful in network design and other fields, as they demonstrate maximal connectivity. They ensure that there is a direct path between any two points in the network, which is often desirable for efficiency and reliability.

Vertices

Vertices, sometimes called nodes, are the fundamental units of a graph. They represent the points or junctions where connections (or edges) are made. In graph theory, understanding vertices is crucial as they form the basis of any network.
In a complete graph \( K_n \), the number of vertices is \( n \). For instance, if \( n = 5 \), there are five vertices in the graph. Each of these vertices connects with every other vertex, illustrating the complete nature of the graph.
Vertices can represent various real-world entities depending on the application. Imagine a vertex as a person in a social network, a computer in a network system, or a city in a map. The versatility of vertices makes them a core concept in studying and utilizing graphs. Understanding their relationships (through edges) is essential for analyzing any graph-based structure.

Edges

Edges are the lines that connect pairs of vertices in a graph. In the context of complete graphs, edges are particularly significant because they ensure full connectivity.
Every complete graph \( K_n \) has the maximum possible number of edges for a given \( n \). For instance, when \( n = 3 \), the graph has three edges forming a triangle, as every vertex links to every other vertex.
Edges can represent various real-world links such as communication lines between devices or paths between locations. Their importance in graph structure is undeniable, as they determine the strength and level of connection within the graph.

Combination Formula

The combination formula is a mathematical tool used to determine the number of ways to choose a subset of items from a larger set. In the context of complete graphs, it helps calculate the number of edges.
The formula \( \binom{n}{2} \) is used specifically to find the number of edges in a complete graph \( K_n \). This is because an edge requires selecting two distinct vertices from the \( n \) available vertices.
The mathematical expression is \( \binom{n}{2} = \frac{n(n-1)}{2} \). For example, for a graph with 5 vertices \( (n=5) \), \( \binom{5}{2} = \frac{5 \times 4}{2} = 10 \). This means there are 10 edges in \( K_5 \). This formula plays an essential role in solving problems related to graph connectivity and design.