03 representasi - incidence matrix

Summary

This video discusses the representation of graphs using incidence matrices, first focusing on undirected graphs. The incidence matrix is explained as an m x n matrix, where m represents edges and n represents vertices. The values in the matrix show the connection between vertices and edges. The video then explores directed graphs, introducing additional matrix values to account for edge directions, with special cases for loops and directionality. The examples provided illustrate how to calculate the incidence matrix for both types of graphs, helping viewers understand their structure and the underlying concepts of graph representation.

Takeaways

• 😀 An incidence matrix is used to represent a graph, where the rows correspond to vertices and the columns represent edges.
• 😀 In an undirected graph, an incidence matrix contains entries of 2, 1, or 0, where 2 indicates an edge connects to the vertex, 1 indicates a connection at either end of the edge, and 0 indicates no connection.
• 😀 The incidence matrix for an undirected graph is symmetric, meaning every edge is represented by two 1s (one for each vertex it connects).
• 😀 For directed graphs, the incidence matrix has four possible values: 2, 1, -1, and 0. 2 indicates a loop, 1 indicates the source vertex, -1 indicates the target vertex, and 0 indicates no connection.
• 😀 In a directed graph's incidence matrix, it’s important to differentiate between the source and target of each edge, unlike undirected graphs where this distinction isn’t needed.
• 😀 To calculate the degree of each vertex, sum the entries in the corresponding rows of the incidence matrix. For directed graphs, consider both incoming and outgoing edges.
• 😀 The example in the script demonstrates an undirected graph with 4 vertices and 9 edges, showing how to fill in the incidence matrix for such a graph.
• 😀 For directed graphs, the process to calculate the incidence matrix requires distinguishing the direction of edges (from source to target) and assigning appropriate values to the matrix.
• 😀 In directed graphs, when an edge is a loop (i.e., it starts and ends at the same vertex), it is represented by a 2 in the incidence matrix, as it connects the vertex to itself.
• 😀 The script also shows how to work with directed graphs with 6 vertices and 10 edges, illustrating how to fill in the incidence matrix and calculate the degree of vertices.
• 😀 The key takeaway is that the incidence matrix offers a compact and systematic way to represent graphs, whether directed or undirected, and can be used to derive important properties like vertex degrees.

Q & A

What is an incidence matrix in graph theory?

An incidence matrix in graph theory represents the relationship between edges and vertices in a graph. It is a matrix where rows correspond to edges and columns correspond to vertices, with entries indicating the connection between an edge and a vertex.

How does the incidence matrix for an undirected graph differ from that of a directed graph?

In an undirected graph, the matrix entries are 0, 1, or 2, where 2 indicates that the edge connects both vertices. In a directed graph, the entries are 1 (for source vertex), -1 (for target vertex), or 0 (for no connection).

What does the value '2' in the incidence matrix of an undirected graph signify?

The value '2' in the incidence matrix of an undirected graph indicates that the edge connects both vertices in the edge pair.

What does a value of '1' in a directed graph's incidence matrix indicate?

A value of '1' in the incidence matrix of a directed graph indicates that the corresponding vertex is the source vertex of the edge.

How is the incidence matrix for a directed loop edge handled?

In the incidence matrix for a directed loop, the matrix entry will be 1 for the source vertex and -1 for the target vertex. A loop connects a vertex to itself, so the matrix reflects this by showing both the source and target as the same vertex.

What does the matrix entry of '0' represent in both undirected and directed graphs?

A matrix entry of '0' indicates that the corresponding vertex is not part of the edge. It means there is no connection between the edge and the vertex.

How do we calculate the degree of a vertex using the incidence matrix?

The degree of a vertex can be calculated by summing the values of the entries in the corresponding column. For undirected graphs, the entries will be 1 or 2, while in directed graphs, the degree is the sum of absolute values of 1's and -1's for the vertex in the column.

Can you give an example of a graph with four vertices and nine edges, and its incidence matrix?

In the example with four vertices and nine edges, the incidence matrix would have 4 rows (representing vertices) and 9 columns (representing edges). Each entry in the matrix corresponds to a specific edge and whether the edge connects to a vertex.

Why is the incidence matrix helpful in computational graph theory?

The incidence matrix is helpful in computational graph theory because it provides a structured way to analyze the relationships between vertices and edges, which is essential for algorithms related to searching, traversal, and other graph-based operations.

How can the incidence matrix help in identifying sources and targets in directed graphs?

The incidence matrix helps identify sources and targets in directed graphs by using positive values (1) to indicate source vertices, negative values (-1) for target vertices, and zeroes to show no connection, making it easier to analyze the direction of edges.