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Chapter 8: Problem 12

Write the incidence matrix of each graph. The complete graph on five vertices \(K_{5}\)

Short Answer

Expert verified
The incidence matrix for \(K_5\) is a 5x10 matrix where each row has exactly four 1's and each column has exactly two 1's. Details of which entry is 1 and which is 0 depend on how the edges are labelled.

Step by step solution

01

Initialize the Matrix

This step involves denoting all 5 vertices (1-5) and all 10 edges (1-10). Initialize a 5x10 matrix with all entries as 0.
02

Populate the Matrix

Start filling in the incidence matrix. For each edge, identify the two vertices it is connecting. For these two vertices, mark 1 under the column for that edge. Do this for all the 10 edges.
03

Confirm The Incidence Matrix

Once the matrix is populated, take a step back and make sure every row (vertex) has four 1's (since each vertex in a K5 graph has 4 edges connected to it) and every column (edge) has exactly two 1's (since each edge connects exactly two vertices). If your matrix has this property, then it is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Introduction to Complete Graph
In the study of graph theory, a complete graph is a special type of graph in which every pair of distinct vertices is connected by a unique edge. This means that there are no two vertices in the graph that lack a direct connection.
  • A complete graph with \( n \) vertices contains exactly \( \frac{n(n-1)}{2} \) edges.
  • This is because each vertex connects to \( n-1 \) other vertices, but each connection (or edge) is shared between two vertices.
The concept of a complete graph is important because it provides a baseline in graph theory. It represents the maximum number of connections possible for a given number of vertices. Therefore, it can help in understanding more complex, less connected graphs by comparison.
Understanding Vertices and Edges
Vertices and edges are the core elements of any graph. Without these, a graph is just a theoretical concept.
  • Vertices: These are the points or nodes of the graph. In a social network graph, a vertex might represent a person.
  • Edges: These lines connect pairs of vertices. In the same social network, an edge might represent a friendship or interaction between people.
In mathematics, vertices and edges provide the basic structure of a graph.
  • The degree of a vertex is the number of edges connected to it.
  • In a complete graph, each vertex connects to every other vertex, giving it the maximum possible degree.
By understanding vertices and edges, one can comprehend how complex networks are structured, helping us analyze various real-world systems better.
Exploring the K5 Graph
The \( K_5 \) graph is a specific example of a complete graph, where \( n = 5 \).
  • This means it has 5 vertices and each vertex is connected to every other vertex.
  • Thus, the \( K_5 \) graph contains \( \binom{5}{2} = 10 \) edges in total.
In a \( K_5 \) graph:
  • Each vertex has a degree of 4, meaning it connects directly to 4 other vertices.
  • The incidence matrix for \( K_5 \) captures these connections in a grid format.
  • Each row of the matrix represents a vertex, and each column signifies an edge.
This incidence matrix holds significant importance in graph theory as it provides a clear and organized way to describe the structure of the graph. Understanding the \( K_5 \) graph helps in appreciating the broader concept of graph connectivity and its applications in computing and networking.

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Most popular questions from this chapter

Write an algorithm that finds the lengths of the shortest paths between all vertex pairs in a simple, connected, weighted graph having \(n\) vertices in time \(O\left(n^{3}\right)\).

Show that a dag has at least one vertex with no out edges [i.e. there is at least one vertex \(v\) such that there are no edges of the form \((v, w)]\).

Show that in any simple, connected, planar graph, \(e \leq 3 v-6\).

Give an example of a connected graph such that the removal of any edge results in a graph that is not connected. (Assume that removing an edge does not remove any vertices.)

Is there a graph containing a Hamiltonian cycle for which Algorithm 8.3 .10 always fails; that is, for which whatever the sequence of random guesses made by the algorithm, the Hamiltonian cycle is not found? Prove your answer.
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