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In [[mathematics]], an '''incidence matrix''' is a [[matrix (mathematics)|matrix]] that shows the relationship between two classes of objects.  If the first class is ''X'' and the second is ''Y'', the matrix has one row for each element of ''X'' and one column for each element of ''Y''.  The entry in row ''x'' and column ''y'' is 1 if ''x'' and ''y'' are related (called '''incident''' in this context) and 0 if they are not. There are variations; see below.
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


==Graph theory==
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
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Incidence matrices are mostly used in [[graph theory]].
Registered users will be able to choose between the following three rendering modes:


===Undirected and directed graphs===
'''MathML'''
[[Image:Labeled undirected graph.svg|thumb|250px|An undirected graph]]
:<math forcemathmode="mathml">E=mc^2</math>
In [[graph theory]] an [[undirected graph]] ''G'' has two kinds of incidence matrices: unoriented and oriented.
The '''incidence matrix''' (or '''unoriented incidence matrix''') of ''G'' is a ''n'' &times; ''m'' [[matrix (math)|matrix]] <math>(b_{ij})</math>, where ''n'' and ''m'' are the numbers of [[vertex (graph theory)|vertices]] and [[Edge (graph theory)|edge]]s respectively, such that <math>b_{ij} = 1</math> if the vertex <math>v_i</math> and edge <math>x_j</math> are incident and 0 otherwise.


For example the incidence matrix of the undirected graph shown on the right is a matrix consisting of 4 rows (corresponding to the four vertices, 1-4) and 4 columns (corresponding to the four edges, e1-e4):
<!--'''PNG'''  (currently default in production)
:<math>
:<math forcemathmode="png">E=mc^2</math>
\begin{pmatrix}
  1 & 1 & 1 & 0 \\
  1 & 0 & 0 & 0 \\
  0 & 1 & 0 & 1 \\
  0 & 0 & 1 & 1 \\
\end{pmatrix}
</math>
If we look at the incidence matrix, we see that the sum of each column is equal to 2. This is because each edge has a vertex connected to each end.


The '''incidence matrix''' of a [[directed graph]] ''D'' is a ''n'' &times; ''m'' matrix <math>[b_{ij}]</math> where ''n'' and ''m'' are the number of vertices and edges respectively, such that <math>b_{ij} = -1</math> if the edge <math>x_j</math> leaves vertex <math>v_i</math>, <math>1</math> if it enters vertex <math>v_i</math> and 0 otherwise (Note that many authors use the opposite sign convention.).
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


An '''oriented incidence matrix''' of an undirected graph ''G'' is the incidence matrix, in the sense of directed graphs, of any [[Orientation (graph theory)|orientation]] of ''G''.  That is, in the column of edge ''e'', there is one +1 in the row corresponding to one vertex of ''e'' and one −1 in the row corresponding to the other vertex of ''e'', and all other rows have 0. All oriented incidence matrices of ''G'' differ only by negating some set of columns. In many uses, this is an insignificant difference, so one can speak of ''the'' oriented incidence matrix, even though that is technically incorrect.
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


The oriented or unoriented incidence matrix of a graph ''G'' is related to the [[adjacency matrix]] of its [[line graph]] ''L''(''G'') by the following theorem:
==Demos==


:<math>
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:
A(L(G)) = B(G)^{T}B(G) - 2I_m\
</math>


where <math>A(L(G))</math> is the adjacency matrix of the line graph of ''G'', ''B''(''G'') is the incidence matrix, and <math>I_m</math> is the [[identity matrix]] of dimension m.


The discrete [[Kirchhoff matrix|Laplacian]] (or Kirchhoff matrix) is obtained from the oriented incidence matrix ''M''(''G'') by the formula
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** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


:<math>
==Test pages ==
M(G) M(G)^{T}.
</math>


The integral [[cycle space]] of a graph is equal to the [[null space]] of its oriented incidence matrix, viewed as a matrix over the [[integers]] or [[real numbers|real]] or [[complex numbers]].  The binary cycle space is the null space of its oriented or unoriented incidence matrix, viewed as a matrix over the two-element [[field (mathematics)|field]].
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
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*[[Help:Formula]]


===Signed and bidirected graphs===
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
The incidence matrix of a [[signed graph]] is a generalization of the oriented incidence matrix.  It is the incidence matrix of any [[bidirected graph]] that orients the given signed graph.  The column of a positive edge has a +1 in the row corresponding to one endpoint and a −1 in the row corresponding to the other endpoint, just like an edge in an ordinary (unsigned) graph.  The column of a negative edge has either a +1 or a −1 in both rows.  The line graph and Kirchhoff matrix properties generalize to signed graphs.
==Bug reporting==
 
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
===Multigraphs===
 
The definitions of incidence matrix apply to graphs with [[loop (graph theory)|loops]] and [[multiple edges]].  The column of an oriented incidence matrix that corresponds to a loop is all zero, unless the graph is signed and the loop is negative; then the column is all zero except for ±2 in the row of its incident vertex.
 
===Hypergraphs===
Because the edges of ordinary graphs can only have two vertices (one at each end), the column of an incidence matrix for graphs can only have two non-zero entries. By contrast, a [[hypergraph]] can have multiple vertices assigned to one edge; thus, a general matrix of non-negative integers describes a hypergraph.
 
==Incidence structures==
 
The '''incidence matrix''' of an [[incidence structure]] ''C'' is a ''p'' &times; ''q'' matrix <math>[b_{ij}]</math>, where ''p'' and ''q'' are the number of '''points''' and '''lines''' respectively, such that <math>b_{ij} = 1</math> if the point <math>p_i</math> and line <math>L_j</math> are incident and 0 otherwise. In this case the incidence matrix is also a [[biadjacency matrix]] of the [[Levi graph]] of the structure. As there is a [[hypergraph]] for every Levi graph, and ''vice versa'', the incidence matrix of an incidence structure describes a hypergraph.
 
===Finite geometries===
 
An important example is a [[finite geometry]].  For instance, in a finite plane, ''X'' is the set of points and ''Y'' is the set of lines.  In a finite geometry of higher dimension, ''X'' could be the set of points and ''Y'' could be the set of subspaces of dimension one less than the dimension of ''Y''; or ''X'' could be the set of all subspaces of one dimension ''d'' and ''Y'' the set of all subspaces of another dimension ''e''.
 
===Block designs===
 
Another example is a [[block design]].  Here ''X'' is a finite set of "points" and ''Y'' is a class of subsets of ''X'', called "blocks", subject to rules that depend on the type of design.  The incidence matrix is an important tool in the theory of block designs.  For instance, it is used to prove the fundamental theorem of symmetric 2-designs, that the number of blocks equals the number of points.
 
==References==
*{{citation | last=Diestel | first=Reinhard | title=Graph Theory | series=Graduate Texts in Mathematics | volume=173 | edition=3rd | date=2005 | publisher=Springer-Verlag | isbn=3-540-26183-4}}.
* [[Coxeter|Coxeter, H.S.M.]] ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 (Section 9.2 Incidence matrices, pp.&nbsp;166–171)
* Jonathan L Gross, Jay Yellen, ''Graph Theory and its applications'', second edition, 2006 (p 97, Incidence Matrices for undirect graphs; p 98, incidence matrices for digraphs)
*{{mathworld | urlname = IncidenceMatrix  | title = Incidence matrix }}
 
[[Category:Algebraic graph theory]]
[[Category:Combinatorics]]
[[Category:Matrices]]
[[Category:Graph data structures]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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