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VertexDegree
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VertexDegree

✖

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

✖

gives the vertex degree for the vertex v.

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VertexDegree[{vw,…},…]

uses rules vw to specify the graph g.

Details

  • VertexDegree is also known as valence.
  • The vertex degree for a vertex v is the number of edges incident to v.
  • For a directed graph g, an edge is incident to a vertex whether it is an in-edge or an out-edge.

Examples

open allclose all

Basic Examples  (3)Summary of the most common use cases

Find the degree for each vertex:

In[1]:=1
✖
https://wolfram.com/xid/0b8enotae6-bdzwy9
Out[1]=1

Find the degree for a specified vertex:

In[1]:=1
✖
https://wolfram.com/xid/0b8enotae6-erdgia
Out[1]=1

Vertex degrees of the HararyGraph:

In[1]:=1
✖
https://wolfram.com/xid/0b8enotae6-bjy0qx
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0b8enotae6-iec2l6
Out[2]=2

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

VertexDegree works with undirected graphs:

In[3]:=3
✖
https://wolfram.com/xid/0b8enotae6-fvzxpm
Out[3]=3

Directed graphs:

In[1]:=1
✖
https://wolfram.com/xid/0b8enotae6-fla6ma
Out[1]=1

Multigraphs:

In[1]:=1
✖
https://wolfram.com/xid/0b8enotae6-uvnf7h
Out[1]=1

Vertex degree for a vertex:

In[1]:=1
✖
https://wolfram.com/xid/0b8enotae6-hzoot6
Out[1]=1

Use rules to specify the graph:

In[1]:=1
✖
https://wolfram.com/xid/0b8enotae6-bndh30
Out[1]=1

VertexDegree works with large graphs:

In[1]:=1
✖
https://wolfram.com/xid/0b8enotae6-pq9ae
In[2]:=2
✖
https://wolfram.com/xid/0b8enotae6-f6o29t

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

Highlight the vertex by its vertex degree for CycleGraph:

In[8]:=8
✖
https://wolfram.com/xid/0b8enotae6-gzipus
In[9]:=9
✖
https://wolfram.com/xid/0b8enotae6-baipzx
In[10]:=10
✖
https://wolfram.com/xid/0b8enotae6-g2btke
In[11]:=11
✖
https://wolfram.com/xid/0b8enotae6-h351s2
Out[11]=11

StarGraph:

In[12]:=12
✖
https://wolfram.com/xid/0b8enotae6-h7904
In[13]:=13
✖
https://wolfram.com/xid/0b8enotae6-fod79c
In[14]:=14
✖
https://wolfram.com/xid/0b8enotae6-yoa55
Out[14]=14

GridGraph:

In[15]:=15
✖
https://wolfram.com/xid/0b8enotae6-jbz7i6
In[16]:=16
✖
https://wolfram.com/xid/0b8enotae6-j70q6
In[17]:=17
✖
https://wolfram.com/xid/0b8enotae6-hp22t8
Out[17]=17

CompleteKaryTree:

In[18]:=18
✖
https://wolfram.com/xid/0b8enotae6-gca8u7
In[19]:=19
✖
https://wolfram.com/xid/0b8enotae6-fxghik
In[20]:=20
✖
https://wolfram.com/xid/0b8enotae6-o6y56
Out[20]=20

An unbalanced tree:

In[21]:=21
✖
https://wolfram.com/xid/0b8enotae6-duj4e6
In[22]:=22
✖
https://wolfram.com/xid/0b8enotae6-659hg
In[23]:=23
✖
https://wolfram.com/xid/0b8enotae6-cpnmy0
Out[23]=23

PathGraph:

In[24]:=24
✖
https://wolfram.com/xid/0b8enotae6-6ri66
In[25]:=25
✖
https://wolfram.com/xid/0b8enotae6-fh33ql
In[26]:=26
✖
https://wolfram.com/xid/0b8enotae6-iop80
Out[26]=26

RandomGraph:

In[27]:=27
✖
https://wolfram.com/xid/0b8enotae6-fnxa4i
In[28]:=28
✖
https://wolfram.com/xid/0b8enotae6-d5r4aw
In[29]:=29
✖
https://wolfram.com/xid/0b8enotae6-f3cto8
Out[29]=29

Create a social network:

In[1]:=1
✖
https://wolfram.com/xid/0b8enotae6-lrv5x
In[2]:=2
✖
https://wolfram.com/xid/0b8enotae6-cmlbzy

Find the people with more influence:

In[3]:=3
✖
https://wolfram.com/xid/0b8enotae6-eqnxe
In[4]:=4
✖
https://wolfram.com/xid/0b8enotae6-l9u5c
Out[4]=4

The degree distribution for a Bernoulli random graph follows a BinomialDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0b8enotae6-d1vm7b
Out[1]=1

Generate vertex degrees from 1000 instances of random graphs:

In[2]:=2
✖
https://wolfram.com/xid/0b8enotae6-ft9uun
Out[2]=2

Find the probability that a Bernoulli random graph has max degree greater than 50:

In[3]:=3
✖
https://wolfram.com/xid/0b8enotae6-b521ri
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0b8enotae6-bhpcac
Out[4]=4

The vertex degree distribution for BarabasiAlbertGraphDistribution follows a power-law:

In[1]:=1
✖
https://wolfram.com/xid/0b8enotae6-jroi74
In[2]:=2
✖
https://wolfram.com/xid/0b8enotae6-b0tydj
Out[2]=2

Properties & Relations  (15)Properties of the function, and connections to other functions

The degree of a vertex of an undirected graph is the number of edges incident to the vertex:

In[1]:=1
✖
https://wolfram.com/xid/0b8enotae6-lotoz
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0b8enotae6-iw9hke
Out[2]=2

Self-loops are counted twice:

In[3]:=3
✖
https://wolfram.com/xid/0b8enotae6-fl4x40
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0b8enotae6-52dw
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0b8enotae6-nch4i7
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0b8enotae6-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
✖
https://wolfram.com/xid/0b8enotae6-c7g75m

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

In[2]:=2
✖
https://wolfram.com/xid/0b8enotae6-bvebwh
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b8enotae6-jrbvj2
Out[3]=3

The sum of the degrees of all vertices of a graph is twice the number of edges:

In[1]:=1
✖
https://wolfram.com/xid/0b8enotae6-nc90z
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0b8enotae6-co2r0o
Out[2]=2

Every graph has an even number of vertices with odd degree:

In[1]:=1
✖
https://wolfram.com/xid/0b8enotae6-iuigjp
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0b8enotae6-biu7nm
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b8enotae6-durc4r
Out[3]=3

Connected simple graphs have minimum vertex degree of at least :

In[1]:=1
✖
https://wolfram.com/xid/0b8enotae6-b1eqe
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0b8enotae6-dsdena
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b8enotae6-ehyr7g
Out[3]=3

A graph with minimum vertex degree at least 2 contains a cycle:

In[1]:=1
✖
https://wolfram.com/xid/0b8enotae6-i86r19
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0b8enotae6-7ebzl
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b8enotae6-4d6x3
Out[3]=3

The vertex degrees of an undirected graph can be obtained from its adjacency matrix:

In[1]:=1
✖
https://wolfram.com/xid/0b8enotae6-bbjeit
In[2]:=2
✖
https://wolfram.com/xid/0b8enotae6-cji5i
In[3]:=3
✖
https://wolfram.com/xid/0b8enotae6-k903ls
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0b8enotae6-ej029e
Out[4]=4

The vertex degrees of a directed graph can be obtained from its adjacency matrix:

In[1]:=1
✖
https://wolfram.com/xid/0b8enotae6-ip1730
In[2]:=2
✖
https://wolfram.com/xid/0b8enotae6-d068v1
In[3]:=3
✖
https://wolfram.com/xid/0b8enotae6-d6r2qp
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0b8enotae6-ch7kgq
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0b8enotae6-bz15n2
In[2]:=2
✖
https://wolfram.com/xid/0b8enotae6-clt22t
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b8enotae6-gygas4
In[4]:=4
✖
https://wolfram.com/xid/0b8enotae6-eanta7
Out[4]=4

The vertex degrees for a directed graph can be obtained from the incidence matrix:

In[1]:=1
✖
https://wolfram.com/xid/0b8enotae6-eaeipp
In[2]:=2
✖
https://wolfram.com/xid/0b8enotae6-ckixb5
In[3]:=3
✖
https://wolfram.com/xid/0b8enotae6-ctf2dr
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0b8enotae6-e548vm
Out[4]=4

Each vertex of a -regular graph has the same vertex degree :

In[1]:=1
✖
https://wolfram.com/xid/0b8enotae6-cdwjr4
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0b8enotae6-bjrxkl
Out[2]=2

All vertices of a simple graph have maximum degree less than the number of vertices:

In[1]:=1
✖
https://wolfram.com/xid/0b8enotae6-zn8ie
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0b8enotae6-byf3az
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b8enotae6-px3kye
Out[3]=3

A simple graph without isolated vertices has at least one pair of vertices with equal degrees:

In[1]:=1
✖
https://wolfram.com/xid/0b8enotae6-bjygid
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0b8enotae6-mc9b
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b8enotae6-dl6hhf
Out[3]=3

A connected undirected graph is Eulerian iff every vertex has an even degree:

In[1]:=1
✖
https://wolfram.com/xid/0b8enotae6-jgb5ug
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0b8enotae6-cqry6f
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b8enotae6-egw5c2
Out[3]=3
Wolfram Research (2010), VertexDegree, Wolfram Language function, https://reference.wolfram.com/language/ref/VertexDegree.html (updated 2015).
✖
Wolfram Research (2010), VertexDegree, Wolfram Language function, https://reference.wolfram.com/language/ref/VertexDegree.html (updated 2015).

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_vertexdegree, author="Wolfram Research", title="{VertexDegree}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/VertexDegree.html}", note=[Accessed: 25-March-2025 ]}

✖
@misc{reference.wolfram_2025_vertexdegree, author="Wolfram Research", title="{VertexDegree}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/VertexDegree.html}", note=[Accessed: 25-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_vertexdegree, organization={Wolfram Research}, title={VertexDegree}, year={2015}, url={https://reference.wolfram.com/language/ref/VertexDegree.html}, note=[Accessed: 25-March-2025 ]}

✖
@online{reference.wolfram_2025_vertexdegree, organization={Wolfram Research}, title={VertexDegree}, year={2015}, url={https://reference.wolfram.com/language/ref/VertexDegree.html}, note=[Accessed: 25-March-2025 ]}