Question

The converse \(D^{\prime}\) of a digraph \(D\) is obtained from \(D\) by reversing the direction of each arc. (i) Give an example of a digraph that is isomorphic to its converse. (ii) What is the connection between the adjacency matrices of \(D\) and \(D ?\)

Short answer

A simple example of a digraph that is isomorphic to its converse would be a digraph with two vertices and an arc going from the first vertex to the second. The adjacency matrices of a digraph and its converse are transposes of each other.

Step-by-step solution

Understand the problem

To tackle this problem, first get a clear understanding of the terms 'converse of a digraph', 'isomorphic', and 'adjacency matrix'. A digraph is simply a directed graph. Its converse is obtained by reversing the direction of each arc. Two graphs are isomorphic if there is a bijection between their vertex sets that preserves adjacency. The adjacency matrix of a graph is a square matrix where the entry in the i-th row and j-th column is equal to the number of edges between vertices i and j.

Find a digraph that is isomorphic to its converse

A simple example of a digraph that is isomorphic to its converse would be a digraph with two vertices and an arc going from the first vertex to the second. If we reverse the direction of the arc to get the converse, the resulting digraph is isomorphic to the original. This is because there is a bijection (in this case, the identity) between the vertex sets that preserves adjacency.

Analyze the connection between the adjacency matrix of a digraph and its converse

The adjacency matrices of a digraph and its converse are transposes of each other. This is because, if there is an edge from vertex i to vertex j in the original digraph, then there is an edge from vertex j to vertex i in the converse. So, the entry that was in the i-th row and j-th column of the original adjacency matrix moves to the j-th row and i-th column of the adjacency matrix of the converse, and vice versa.

Key concepts

Converse of a Digraph

A digraph, or directed graph, consists of vertices connected by directed edges, known as arcs. Creating the converse of a digraph involves reversing the direction of every arc in the original graph. For example, if a digraph has an arc from vertex A to vertex B, its converse will have an arc from vertex B to vertex A. This operation essentially swaps the roles of the start and end points of each arc, resulting in a new digraph that mirrors the connections of the original.
To illustrate, imagine a simple digraph with two vertices, labeled 1 and 2, and a single directed arc from 1 to 2. The converse of this digraph will have a directed arc from 2 to 1. This basic transformation helps us analyze the properties and structures of directed graphs in various applications.

Isomorphic Graphs

Isomorphic graphs are a pair of graphs that, although they might look different due to their structures or vertex arrangements, share the same connectivity. In simpler terms, an isomorphism between two graphs exists if there is a one-to-one correspondence between their vertex sets and this mapping conserves the adjacency of vertices.

• Bijection: The concept of a bijection is essential here. It is a function that pairs each element of one set with exactly one element of another set, and vice versa.
• Preserving adjacency: This means that two vertices connected by an edge in one graph remain connected post-mapping in the other.

An example of isomorphic digraphs would be two digraphs each containing two vertices, where in one graph there is a directed edge from vertex A to B, and in the other, a directed edge from C to D. They are isomorphic if vertices A and C, and B and D are mutually mapped while preserving the directional relationship.

Adjacency Matrix

An adjacency matrix is a compact way of representing a graph, useful for both undirected and directed graphs. It is a square matrix used to represent the connections between vertices with rows and columns corresponding to the graph’s vertices.
For a directed graph, if there is a direct edge from vertex i to vertex j, the matrix entry at row i and column j is usually marked with a 1 (or with a value representing the edge weight if the graph is weighted). Otherwise, it is marked with a 0.
This matrix offers an algebraic representation of the graph, enabling quick assessment of connectivity and providing a foundational structure for further operations like transpose, to analyze the converse of a digraph.

Transpose of a Matrix

The transpose of a matrix is a fundamental matrix operation, wherein rows of the original matrix become columns in the new matrix, and vice versa. For an adjacency matrix of a digraph, transposing involves flipping the matrix over its diagonal.

• Matrix transposition for digraphs: Consider a matrix where the element at the i-th row and j-th column represents an arc from node i to j. In its transpose, this entry moves to the j-th row and i-th column, effectively reversing the direction of the arcs.
• Transpose operation: This operation is crucial in obtaining the adjacency matrix of the converse of a digraph.

The relation between the adjacency matrices of a digraph and its converse is established explicitly through this transposition, illustrating how the network's directional information is reorganized to reveal the graph's inverse structure.