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WeightedAdjacencyMatrix
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WeightedAdjacencyMatrix

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gives the adjacency matrix of edge weights of the graph g.

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uses rules vw to specify the graph g.

Details and Options

  • WeightedAdjacencyMatrix returns a SparseArray object, which can be converted to an ordinary matrix using Normal.
  • An entry wij of the weighted adjacency matrix is the weight of a directed edge from vertex νi to vertex νj. If there is no edge the weight is taken to be 0.
  • An edge without explicit EdgeWeight specified is taken to have weight 1.
  • An undirected edge is interpreted as two directed edges with opposite directions and the same weight.
  • The vertices vi are assumed to be in the order given by VertexList[g].
  • The weighted adjacency matrix for a graph will have dimensions ×, where is the number of vertices.

Examples

open allclose all

Basic Examples  (2)Summary of the most common use cases

The weighted adjacency matrix of an undirected graph:

In[1]:=1
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https://wolfram.com/xid/0bcn9kefk7cgxj5kdixz2-dkm3wv
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bcn9kefk7cgxj5kdixz2-f5ppu9

The weighted adjacency matrix of a directed graph:

In[1]:=1
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https://wolfram.com/xid/0bcn9kefk7cgxj5kdixz2-4qshb
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bcn9kefk7cgxj5kdixz2-d62e0h

Scope  (5)Survey of the scope of standard use cases

The weighted adjacency matrix of an undirected graph is symmetric:

In[1]:=1
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https://wolfram.com/xid/0bcn9kefk7cgxj5kdixz2-brfunn
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bcn9kefk7cgxj5kdixz2-fo7txu

The weighted adjacency matrix of a directed graph can be unsymmetric:

In[1]:=1
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https://wolfram.com/xid/0bcn9kefk7cgxj5kdixz2-becm22
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bcn9kefk7cgxj5kdixz2-pcpon

Use rules to specify the graph:

In[1]:=1
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https://wolfram.com/xid/0bcn9kefk7cgxj5kdixz2-bndh30

The weighted adjacency matrix of the graph with self-loops has diagonal entries:

In[1]:=1
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https://wolfram.com/xid/0bcn9kefk7cgxj5kdixz2-cuz87q
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bcn9kefk7cgxj5kdixz2-jtfkjp

WeightedAdjacencyMatrix works with large graphs:

In[1]:=1
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https://wolfram.com/xid/0bcn9kefk7cgxj5kdixz2-jyefgb
In[2]:=2
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https://wolfram.com/xid/0bcn9kefk7cgxj5kdixz2-fi4a2b
Out[2]=2

Use MatrixPlot to visualize the matrix:

In[3]:=3
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https://wolfram.com/xid/0bcn9kefk7cgxj5kdixz2-ne4gm8
Out[3]=3

Properties & Relations  (4)Properties of the function, and connections to other functions

Rows and columns of the weighted adjacency matrix follow the order given by VertexList:

In[1]:=1
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https://wolfram.com/xid/0bcn9kefk7cgxj5kdixz2-joylju
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bcn9kefk7cgxj5kdixz2-cwu88m
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0bcn9kefk7cgxj5kdixz2-i573me

Use WeightedAdjacencyGraph to construct a graph from a weighted adjacency matrix:

In[1]:=1
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https://wolfram.com/xid/0bcn9kefk7cgxj5kdixz2-clncmi
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bcn9kefk7cgxj5kdixz2-p3vm2
Out[2]=2

The number of rows or columns is equal to the number of vertices:

In[1]:=1
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https://wolfram.com/xid/0bcn9kefk7cgxj5kdixz2-dxrnuc
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bcn9kefk7cgxj5kdixz2-gmegrm
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0bcn9kefk7cgxj5kdixz2-b6qsb8
Out[3]=3

The main diagonals for a loop-free graph are all zeros:

In[1]:=1
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https://wolfram.com/xid/0bcn9kefk7cgxj5kdixz2-jez1rj
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bcn9kefk7cgxj5kdixz2-ddcmod
Out[2]=2
Wolfram Research (2010), WeightedAdjacencyMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/WeightedAdjacencyMatrix.html (updated 2015).
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Wolfram Research (2010), WeightedAdjacencyMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/WeightedAdjacencyMatrix.html (updated 2015).

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_weightedadjacencymatrix, author="Wolfram Research", title="{WeightedAdjacencyMatrix}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/WeightedAdjacencyMatrix.html}", note=[Accessed: 25-March-2025 ]}

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@misc{reference.wolfram_2025_weightedadjacencymatrix, author="Wolfram Research", title="{WeightedAdjacencyMatrix}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/WeightedAdjacencyMatrix.html}", note=[Accessed: 25-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_weightedadjacencymatrix, organization={Wolfram Research}, title={WeightedAdjacencyMatrix}, year={2015}, url={https://reference.wolfram.com/language/ref/WeightedAdjacencyMatrix.html}, note=[Accessed: 25-March-2025 ]}

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@online{reference.wolfram_2025_weightedadjacencymatrix, organization={Wolfram Research}, title={WeightedAdjacencyMatrix}, year={2015}, url={https://reference.wolfram.com/language/ref/WeightedAdjacencyMatrix.html}, note=[Accessed: 25-March-2025 ]}