# IncidenceMatrix

gives the vertex-edge incidence matrix of the graph g.

IncidenceMatrix[{vw,}]

uses rules vw to specify the graph g.

# Details

• IncidenceMatrix is also known as the vertex-edge incidence matrix.
• IncidenceMatrix returns a SparseArray object, which can be converted to an ordinary matrix using Normal.
• The incidence matrix for a graph with vertices {v1,,vn} and edges {e1,,em} is an matrix with entries aij given by:
•  0 vi is not incident to ej 1 ej=vivk, ej=vkvi, or ej=vkvi -1 ej=vivk 2 ej=vivi -2 ej=vivi
• The vertices vi are assumed to be in the order given by VertexList[g] and the edges ej are assumed to be in the order given by EdgeList[g].

# Examples

open allclose all

## Basic Examples(2)

The incidence matrix of an undirected graph:

The incidence matrix of a directed graph:

## Scope(5)

The incidence matrix of an undirected graph has no negative entries:

The sum of the entries in any column is 2:

The incidence matrix of a directed graph has some negative entries:

If there are no self-loops, the sum of the entries in any column is 0:

The incidence matrix of a graph with self-loops has some entries equal to 2:

Use rules to specify the graph:

IncidenceMatrix works with large graphs:

Use MatrixPlot to visualize the matrix:

## Properties & Relations(9)

Rows and columns correspond to VertexList and EdgeList order:

The row gives all edge indices incident to the vertex:

The first vertex is incident to edges 2, 3, and 4:

The column gives all vertex indices incident to the edge:

The second edge is incident to vertices 1 and 3:

Use VertexIndex and EdgeIndex to find the indices for vertices and edges:

The incidence matrix can tell whether a vertex and edge are incident:

The incidence matrix for a directed graph indicates the source vertex by and the target vertex by :

Compute the oriented incidence matrix for an undirected graph using DirectedGraph:

The oriented incidence matrix using a random orientation:

The dimensions of the incidence matrix are given by VertexCount and EdgeCount:

Use IncidenceGraph to construct a graph from an incidence matrix:

The adjacency matrix of a line graph can be computed by its IncidenceMatrix:

Wolfram Research (2010), IncidenceMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/IncidenceMatrix.html (updated 2015).

#### Text

Wolfram Research (2010), IncidenceMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/IncidenceMatrix.html (updated 2015).

#### CMS

Wolfram Language. 2010. "IncidenceMatrix." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https://reference.wolfram.com/language/ref/IncidenceMatrix.html.

#### APA

Wolfram Language. (2010). IncidenceMatrix. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/IncidenceMatrix.html

#### BibTeX

@misc{reference.wolfram_2024_incidencematrix, author="Wolfram Research", title="{IncidenceMatrix}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/IncidenceMatrix.html}", note=[Accessed: 10-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_incidencematrix, organization={Wolfram Research}, title={IncidenceMatrix}, year={2015}, url={https://reference.wolfram.com/language/ref/IncidenceMatrix.html}, note=[Accessed: 10-September-2024 ]}