Question

How many edges has each of the following graphs: (i) \(K_{10}\); (ii) \(K_{5,7}\); (iii) \(Q_{4}\); (v) the Petersen graph?

Short answer

The graphs have the following numbers of edges: (i) \(K_{10}\): 45 edges; (ii) \(K_{5,7}\): 35 edges; (iii) \(Q_{4}\): 32 edges; (v) the Petersen graph: 15 edges.

Step-by-step solution

Calculate the number of edges in \(K_{10}\)

Plugging \(n=10\) into the formula \(n(n-1)/2\), we get \(10 \times (10 - 1) / 2 = 45\)

Calculate the number of edges in \(K_{5,7}\)

Using the formula \(m \times n\), where \(m\) and \(n\) are the numbers of vertices in the two sets (5 and 7 in this case), the number of edges is \(5 \times 7 = 35\)

Calculate the number of edges in \(Q_{4}\)

The formula for the number of edges in a hypercube is \(n2^{n-1}\), so for \(Q_{4}\), it's \(4 \times 2^{3} = 32\)

Identify the number of edges in the Petersen graph

The Petersen graph is a specific predefined graph with 10 vertices and 15 edges. So the number of edges is 15

Key concepts

Complete Graph

In graph theory, a complete graph is one where every pair of vertices is connected by an edge. The concept is simple: if you have a group of friends at a party, a complete graph would mean that everyone has had a conversation with everyone else. The formula to calculate the number of edges in a complete graph, denoted by K_n where n represents the number of vertices, is \( \frac{n(n-1)}{2} \). This is because each vertex can connect to every other vertex, minus itself, and each connection is shared between two vertices.

To visualize this, consider K₁₀, a complete graph with 10 vertices. Here, it's like having a dinner party where 10 people all toast with each other. You wouldn't want anyone to be left out, and this is what happens in a complete graph: no one is left out. Hence, the edges in K₁₀ are \( \frac{10(10-1)}{2} = 45 \) edges in total.

Bipartite Graph

Moving on from a scenario where everyone is connected to everyone, we discuss the bipartite graph. This is the equivalent of having two distinct groups at a social gathering, such as smokers and non-smokers, where interactions only happen between individuals from different groups. Mathematically, a bipartite graph K_m,n has two sets of vertices, with m vertices in the first set and n vertices in the second set. In this scenario, every vertex in the first set is connected to every vertex in the second set, and no two vertices within the same set are connected.

The formula for calculating the edges in a bipartite graph is a straight multiplication of the number of vertices in each set, \( m \times n \). For example, K_5,7 represents a bipartite graph with the sets of 5 and 7 vertices, giving us \( 5 \times 7 = 35 \) edges.

Hypercube Graph

A hypercube graph, denoted by Q_n, can be a mind-bender because it's like a geometric cube extended into higher dimensions. Each vertex represents a corner of a hypercube, and each edge connects one vertex to another. The fascinating aspect here is how the graph expands exponentially as more dimensions are added. The formula to count the edges in a hypercube graph is \( n2^{n-1} \).

When we step into the fourth dimension with Q₄, a 4-dimensional hypercube, it has \( 4 \times 2^{4-1} = 32 \) edges. This is quite extraordinary to visualize, as each dimension doubles the number of vertices and connects them with more edges.

Petersen Graph

Finally, there's the Petersen graph, a very specific and non-intuitive graph compared to the ones previously discussed. It cannot be drawn without crossings in a plain and has particular properties that make it a topic of interest in graph theory. The Petersen graph has 10 vertices and suitably, for its complexity, 15 edges. Unlike the graphs mentioned before, the Petersen graph's structure doesn’t allow a simple formula for deducing the number of edges based on vertices.

Instead, the Petersen graph is known as a famous example of a regular graph: each one of its 10 vertices connects to exactly 3 other vertices, making a total of 15 edges. It is often used to illustrate concepts in graph theory due to its unique properties and has exactly 15 edges, known from its definition.