Adjacency matrix

Definition

An adjacency matrix is a square grid used to represent a finite graph, where each element indicates whether pairs of vertices are adjacent or not in the graph. In labeled graphs, the matrix contains values that reflect the presence or absence of edges between vertices, while in unlabeled graphs, the adjacency matrix can illustrate more general relationships without specific identifiers. This mathematical tool provides a compact way to encode graph information and is essential for various graph algorithms and properties.

5 Must Know Facts For Your Next Test

1. An adjacency matrix for a graph with n vertices is an n x n matrix, where the rows and columns correspond to the vertices.
2. In labeled graphs, the entries of the adjacency matrix are typically 1 (or another value) if there is an edge connecting two vertices and 0 if there isn't.
3. For undirected graphs, the adjacency matrix is symmetric; if vertex i is connected to vertex j, then vertex j is also connected to vertex i, resulting in identical values for (i,j) and (j,i).
4. In directed graphs, the adjacency matrix may not be symmetric because edges have a direction, meaning an entry can indicate a connection from vertex i to vertex j without a reciprocal connection.
5. The concept of an adjacency matrix can extend beyond simple graphs to represent weighted graphs, where entries represent weights instead of mere connectivity.

Review Questions

• How does an adjacency matrix differ when representing labeled versus unlabeled graphs?
  • In labeled graphs, the adjacency matrix explicitly shows connections with distinct identifiers for each vertex, allowing for precise relationships to be represented. In contrast, for unlabeled graphs, the focus is on general connectivity without specific names or labels for the vertices. This means that while both types of matrices convey information about edge presence, labeled matrices provide more detailed structure, while unlabeled matrices emphasize the overall shape and connections in the graph.
• Discuss how you would determine the degree of a vertex using an adjacency matrix and what this tells you about the graph's structure.
  • To determine the degree of a vertex using an adjacency matrix, you would count the number of non-zero entries in the corresponding row or column for that vertex. This count reveals how many edges connect to that vertex. Understanding the degree distribution across vertices gives insights into the graph's structure, indicating whether it has highly connected nodes (hubs) or if it tends to have more isolated vertices, thereby affecting overall properties like connectivity and potential for traversal.
• Evaluate how the properties of symmetry in an adjacency matrix inform us about the type of graph being represented.
  • The symmetry properties of an adjacency matrix reveal whether a graph is undirected or directed. If the matrix is symmetric, it indicates that for every edge from vertex i to vertex j, there is also an edge from j to i, characteristic of undirected graphs. Conversely, if the matrix shows asymmetry with differing entries for (i,j) and (j,i), it signifies a directed graph where relationships are not reciprocal. This evaluation not only helps classify graphs but also impacts algorithmic approaches for traversal and pathfinding within those structures.

Related terms

Graph Theory: A field of mathematics that studies the properties and applications of graphs, which are structures made up of vertices connected by edges.

Incidence Matrix: A matrix representation of a graph that shows the relationship between edges and vertices, indicating which edges are incident to which vertices.

Degree of a Vertex: The number of edges connected to a vertex in a graph, which can be easily derived from the adjacency matrix by counting non-zero entries in the corresponding row or column.