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NeighborhoodGraph
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NeighborhoodGraph

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gives the graph neighborhood of a vertex v in the graph g.

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NeighborhoodGraph[g,{a1,a2,…}]

gives the graph neighborhood of the ai that can be vertices, edges, or subgraphs of g.

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gives the graph neighborhood of the vertices and edges that match the pattern patt.

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gives the neighborhood up to distance d.

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NeighborhoodGraph[{vw,…},…]

uses rules vw to specify the graph g.

Details and Options

  • The neighborhood graph for a vertex v is given by vertices adjacent to v and the edges connecting them.
  • The neighborhood graph for an edge e is the neighborhood graph for the vertices of e.
  • The neighborhood graph for a subgraph h is the neighborhood graph for the vertices in h.
  • The neighborhood graph at distance d is the neighborhood graph for the vertices of the neighborhood graph at distance d-1.
  • The default value for d is 1.
  • NeighborhoodGraph works with undirected graphs, directed graphs, multigraphs, and mixed graphs.

Examples

open allclose all

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

Give the neighborhood from vertex 1 in a graph:

In[1]:=1
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https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-mdlp
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-ed09xj
Out[2]=2

From a set of vertices:

In[3]:=3
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-qh2pz
Out[3]=3

Give the neighborhood up to distance k from the vertices:

In[1]:=1
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-eq8ke9
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-dl7ibs
Out[2]=2

Scope  (8)Survey of the scope of standard use cases

NeighborhoodGraph works with undirected graphs:

In[1]:=1
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-bsehov
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-hiej2e
Out[2]=2

Directed graphs:

In[1]:=1
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-j4qwi
In[2]:=2
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-il1mbb
Out[2]=2

Multigraphs:

In[1]:=1
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-2b294c
In[2]:=2
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-g1ka6a
Out[2]=2

Mixed graphs:

In[1]:=1
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-8flvbi
In[2]:=2
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-2lz8ut
Out[2]=2

NeighborhoodGraph works with vertices:

In[1]:=1
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-byzpdy
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-ew1hwk
Out[2]=2

Edges:

In[3]:=3
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-dn0vdv
Out[3]=3

Use rules to specify the graph:

In[1]:=1
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-j8qr45
In[2]:=2
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-bndh30
Out[2]=2

Use patterns to specify a set of vertices:

In[1]:=1
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-fa2kiq
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-8osrc
Out[2]=2

NeighborhoodGraph works with large graphs:

In[1]:=1
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-r17r6n
In[2]:=2
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-cx2zrt
Out[2]=2

Applications  (2)Sample problems that can be solved with this function

Highlight the neighborhood from the vertices in CompleteGraph:

In[1]:=1
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-bucaor
In[2]:=2
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-i3qvb6
Out[2]=2

CycleGraph:

In[3]:=3
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-cqk4jg
In[4]:=4
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https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-ccqhf4
Out[4]=4

StarGraph:

In[5]:=5
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https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-eudkwe
In[6]:=6
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-ndr68l
Out[6]=6

GridGraph:

In[7]:=7
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-lhh69
In[8]:=8
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https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-olkz
Out[8]=8

CompleteKaryTree:

In[9]:=9
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https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-d9echf
In[10]:=10
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https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-vy1cs
Out[10]=10

PathGraph:

In[11]:=11
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https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-fpmixb
In[12]:=12
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https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-bhjfer
Out[12]=12

RandomGraph:

In[13]:=13
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https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-cpvf7c
In[14]:=14
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-faufnk
Out[14]=14

Manipulate the neighborhood of vertices:

In[1]:=1
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-j0bcra
Out[1]=1

Properties & Relations  (2)Properties of the function, and connections to other functions

Use Subgraph to find the neighborhood graph of a set of vertices:

In[1]:=1
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-bajro8
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-mzp5mu
In[3]:=3
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-g7lyuz

Highlight the subgraph:

In[4]:=4
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-cjb35u
Out[4]=4

This is equivalent to:

In[5]:=5
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-ph4ozf
Out[5]=5

The neighborhood of a vertex in a complete graph is the graph itself:

In[1]:=1
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-fzqr0x
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-blrvnt
Out[2]=2

Neat Examples  (2)Surprising or curious use cases

Pick out random neighborhoods from a grid:

In[1]:=1
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https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-csaox8
In[2]:=2
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-j9bhnf
Out[2]=2

Subtract random neighborhoods from a grid:

In[1]:=1
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-cof010
In[2]:=2
✖
https://wolfram.com/xid/0bhgp6jliqylnwjj5sbe-spfkz
Out[2]=2
Wolfram Research (2010), NeighborhoodGraph, Wolfram Language function, https://reference.wolfram.com/language/ref/NeighborhoodGraph.html (updated 2015).
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Wolfram Research (2010), NeighborhoodGraph, Wolfram Language function, https://reference.wolfram.com/language/ref/NeighborhoodGraph.html (updated 2015).

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_neighborhoodgraph, author="Wolfram Research", title="{NeighborhoodGraph}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/NeighborhoodGraph.html}", note=[Accessed: 25-March-2025 ]}

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@misc{reference.wolfram_2025_neighborhoodgraph, author="Wolfram Research", title="{NeighborhoodGraph}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/NeighborhoodGraph.html}", note=[Accessed: 25-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_neighborhoodgraph, organization={Wolfram Research}, title={NeighborhoodGraph}, year={2015}, url={https://reference.wolfram.com/language/ref/NeighborhoodGraph.html}, note=[Accessed: 25-March-2025 ]}

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@online{reference.wolfram_2025_neighborhoodgraph, organization={Wolfram Research}, title={NeighborhoodGraph}, year={2015}, url={https://reference.wolfram.com/language/ref/NeighborhoodGraph.html}, note=[Accessed: 25-March-2025 ]}