In data structure, the incidence matrix is a representation of a graph. It is a two-dimensional matrix that describes the relationship between vertices and edges in a graph. Each row of the matrix represents a vertex, and each column represents an edge.
Definition
The incidence matrix is defined as follows:
Syntax:
<ul>
<li> Let V be the set of vertices in the graph
<li> Let E be the set of edges in the graph
<li> Let n be the number of vertices in the graph
<li> Let m be the number of edges in the graph
Pseudocode:
<ul>
<li> Create an empty 2D array with dimensions [n][m]
<li> For each vertex v∈V and edge e∈E, assign 1 to matrix[v][e] if v is incident to e, otherwise assign 0
Example
To better understand how an incidence matrix works, let’s consider a simple example:
G = (V, E)
- V = {A, B, C}
- E = {e1, e2}
We can represent this graph using an incidence matrix as follows:
Incidence Matrix:
Properties and Applications
The incidence matrix has several properties and applications:
1. Compact representation:
The incidence matrix provides a compact representation of a graph compared to other representations like adjacency lists or adjacency matrices. It only requires O(n * m) space, where n is the number of vertices and m is the number of edges.
2. Edge-vertex relationship:
The incidence matrix clearly shows the relationship between edges and vertices in a graph.
Each row represents a vertex, and each column represents an edge. The value at matrix[v][e] indicates whether vertex v is incident to edge e.
3. Graph operations:
The incidence matrix enables efficient implementation of various graph operations such as adding or removing vertices or edges, checking if two vertices are adjacent, finding all neighbors of a vertex, etc.
Conclusion
In summary, the incidence matrix is a useful data structure for representing graphs. It provides a compact representation of the edge-vertex relationship in a graph and enables efficient implementation of various graph operations.
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