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VertexInDegree
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VertexInDegree

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gives the list of vertex in-degrees for all vertices in the graph g.

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gives the vertex in-degree for the vertex v.

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VertexInDegree[{vw,…},…]

uses rules vw to specify the graph g.

Details

  • The vertex in-degree for a vertex v is the number of incoming directed edges to v.
  • For an undirected graph g, an edge is taken to be both an in-edge and an out-edge. »

Examples

open allclose all

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

Find the in-degree for each vertex:

In[1]:=1
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https://wolfram.com/xid/0dc3e0so8mpu-bdzwy9
Out[1]=1

Find the in-degree for a specified vertex:

In[1]:=1
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https://wolfram.com/xid/0dc3e0so8mpu-1fyu9
Out[1]=1

Scope  (6)Survey of the scope of standard use cases

VertexInDegree works with directed graphs:

In[6]:=6
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https://wolfram.com/xid/0dc3e0so8mpu-fvzxpm
Out[6]=6

Undirected graphs:

In[1]:=1
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https://wolfram.com/xid/0dc3e0so8mpu-fla6ma
Out[1]=1

Multigraphs:

In[1]:=1
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https://wolfram.com/xid/0dc3e0so8mpu-uvnf7h
Out[1]=1

Vertex in-degree for a vertex:

In[1]:=1
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https://wolfram.com/xid/0dc3e0so8mpu-hzoot6
Out[1]=1

Use rules to specify the graph:

In[1]:=1
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https://wolfram.com/xid/0dc3e0so8mpu-bndh30
Out[1]=1

VertexInDegree works with large graphs:

In[1]:=1
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https://wolfram.com/xid/0dc3e0so8mpu-pq9ae
In[2]:=2
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https://wolfram.com/xid/0dc3e0so8mpu-dag496

Applications  (4)Sample problems that can be solved with this function

Highlight the vertex in-degree for directed graphs including CycleGraph:

In[1]:=1
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https://wolfram.com/xid/0dc3e0so8mpu-gzipus
In[2]:=2
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https://wolfram.com/xid/0dc3e0so8mpu-baipzx
In[3]:=3
✖
https://wolfram.com/xid/0dc3e0so8mpu-g2btke
In[4]:=4
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https://wolfram.com/xid/0dc3e0so8mpu-h351s2
Out[4]=4

StarGraph:

In[5]:=5
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https://wolfram.com/xid/0dc3e0so8mpu-h7904
In[6]:=6
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https://wolfram.com/xid/0dc3e0so8mpu-fod79c
In[7]:=7
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https://wolfram.com/xid/0dc3e0so8mpu-yoa55
Out[7]=7

GridGraph:

In[8]:=8
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https://wolfram.com/xid/0dc3e0so8mpu-jbz7i6
In[9]:=9
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https://wolfram.com/xid/0dc3e0so8mpu-j70q6
In[10]:=10
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https://wolfram.com/xid/0dc3e0so8mpu-hp22t8
Out[10]=10

CompleteKaryTree:

In[11]:=11
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https://wolfram.com/xid/0dc3e0so8mpu-gca8u7
In[12]:=12
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https://wolfram.com/xid/0dc3e0so8mpu-fxghik
In[13]:=13
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https://wolfram.com/xid/0dc3e0so8mpu-o6y56
Out[13]=13

PathGraph:

In[14]:=14
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https://wolfram.com/xid/0dc3e0so8mpu-duj4e6
In[15]:=15
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https://wolfram.com/xid/0dc3e0so8mpu-659hg
In[16]:=16
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https://wolfram.com/xid/0dc3e0so8mpu-cpnmy0
Out[16]=16

RandomGraph:

In[17]:=17
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https://wolfram.com/xid/0dc3e0so8mpu-fnxa4i
In[18]:=18
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https://wolfram.com/xid/0dc3e0so8mpu-d5r4aw
In[19]:=19
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https://wolfram.com/xid/0dc3e0so8mpu-f3cto8
Out[19]=19

Show the in-degree histogram for BernoulliGraphDistribution[n,p]:

In[1]:=1
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https://wolfram.com/xid/0dc3e0so8mpu-h85tpk
In[2]:=2
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https://wolfram.com/xid/0dc3e0so8mpu-kmwtyv
Out[2]=2

The in-degree distribution follows BinomialDistribution[n-1,p]:

In[3]:=3
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https://wolfram.com/xid/0dc3e0so8mpu-i06cd3
Out[3]=3

The vertex in-degree distribution for PriceGraphDistribution follows a power-law:

In[1]:=1
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https://wolfram.com/xid/0dc3e0so8mpu-jroi74
In[2]:=2
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https://wolfram.com/xid/0dc3e0so8mpu-b0tydj
Out[2]=2

Create a food chain where an edge indicates what animals and insects eat:

In[1]:=1
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https://wolfram.com/xid/0dc3e0so8mpu-fb5pb0
In[2]:=2
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https://wolfram.com/xid/0dc3e0so8mpu-1ni0oa
In[3]:=3
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https://wolfram.com/xid/0dc3e0so8mpu-bctffq

The in-degree corresponds to the number of prey and what an animal preys on:

In[4]:=4
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https://wolfram.com/xid/0dc3e0so8mpu-kvcv66
Out[4]=4

The species with in-degree zero are called basal species or producers:

In[5]:=5
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https://wolfram.com/xid/0dc3e0so8mpu-cb6qpk
Out[5]=5
In[6]:=6
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https://wolfram.com/xid/0dc3e0so8mpu-knn9tb
Out[6]=6

Properties & Relations  (10)Properties of the function, and connections to other functions

The in-degree of an undirected graph is the number of edges incident to each vertex:

In[1]:=1
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https://wolfram.com/xid/0dc3e0so8mpu-lotoz
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dc3e0so8mpu-iw9hke
Out[2]=2

Self-loops are counted twice:

In[4]:=4
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https://wolfram.com/xid/0dc3e0so8mpu-fl4x40
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/0dc3e0so8mpu-52dw
Out[5]=5

Undirected graphs correspond to directed graphs with each edge both an in- and out-edge:

In[1]:=1
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https://wolfram.com/xid/0dc3e0so8mpu-do65mj
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dc3e0so8mpu-djkivd
Out[2]=2

For an undirected graph, the vertex in-degree and out-degree are equal to the vertex degree:

In[1]:=1
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https://wolfram.com/xid/0dc3e0so8mpu-nch4i7
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dc3e0so8mpu-boamnk
Out[2]=2

For a directed graph, the sum of the vertex in-degree and out-degree is the vertex degree:

In[1]:=1
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https://wolfram.com/xid/0dc3e0so8mpu-c7g75m

Put the vertex degree, in-degree, and out-degree before, above, and below the vertex, respectively:

In[2]:=2
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https://wolfram.com/xid/0dc3e0so8mpu-bvebwh
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0dc3e0so8mpu-jrbvj2
Out[3]=3

The sum of the in-degrees of all vertices of an undirected graph is twice the number of edges:

In[1]:=1
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https://wolfram.com/xid/0dc3e0so8mpu-nc90z
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dc3e0so8mpu-co2r0o
Out[2]=2

The sum of the in-degrees of all vertices of a directed graph is equal to the number of edges:

In[1]:=1
✖
https://wolfram.com/xid/0dc3e0so8mpu-lekv8x
Out[1]=1

The vertex in-degrees of an undirected graph can be obtained from the adjacency matrix:

In[1]:=1
✖
https://wolfram.com/xid/0dc3e0so8mpu-0hfe3
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dc3e0so8mpu-cji5i
In[3]:=3
✖
https://wolfram.com/xid/0dc3e0so8mpu-k903ls
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0dc3e0so8mpu-ej029e
Out[4]=4

The vertex in-degrees of a directed graph can be obtained from the adjacency matrix:

In[1]:=1
✖
https://wolfram.com/xid/0dc3e0so8mpu-g2jqah
In[2]:=2
✖
https://wolfram.com/xid/0dc3e0so8mpu-d068v1
In[3]:=3
✖
https://wolfram.com/xid/0dc3e0so8mpu-d6r2qp
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0dc3e0so8mpu-ez0fq2
Out[4]=4

The vertex in-degrees for an undirected graph can be obtained from the incidence matrix:

In[1]:=1
✖
https://wolfram.com/xid/0dc3e0so8mpu-dtruch
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dc3e0so8mpu-clt22t
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0dc3e0so8mpu-gygas4
In[4]:=4
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https://wolfram.com/xid/0dc3e0so8mpu-eanta7
Out[4]=4

A connected directed graph is Eulerian iff every vertex has equal in-degree and out-degree:

In[1]:=1
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https://wolfram.com/xid/0dc3e0so8mpu-hxkdvf
In[2]:=2
✖
https://wolfram.com/xid/0dc3e0so8mpu-d36ld
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0dc3e0so8mpu-cxziv7
Out[3]=3
Wolfram Research (2010), VertexInDegree, Wolfram Language function, https://reference.wolfram.com/language/ref/VertexInDegree.html (updated 2015).
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Wolfram Research (2010), VertexInDegree, Wolfram Language function, https://reference.wolfram.com/language/ref/VertexInDegree.html (updated 2015).

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_vertexindegree, author="Wolfram Research", title="{VertexInDegree}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/VertexInDegree.html}", note=[Accessed: 21-June-2025 ]}

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@misc{reference.wolfram_2025_vertexindegree, author="Wolfram Research", title="{VertexInDegree}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/VertexInDegree.html}", note=[Accessed: 21-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_vertexindegree, organization={Wolfram Research}, title={VertexInDegree}, year={2015}, url={https://reference.wolfram.com/language/ref/VertexInDegree.html}, note=[Accessed: 21-June-2025 ]}

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@online{reference.wolfram_2025_vertexindegree, organization={Wolfram Research}, title={VertexInDegree}, year={2015}, url={https://reference.wolfram.com/language/ref/VertexInDegree.html}, note=[Accessed: 21-June-2025 ]}