# VertexCount

VertexCount[g]

gives a count of the number of vertices in the graph g.

VertexCount[g,patt]

gives a count of the number of vertices that match the pattern patt.

VertexCount[{vw,},]

uses rules vw to specify the graph g.

# Details

• VertexCount is also known as the order of the graph.
• VertexCount[g] returns the number of vertices in the graph g.
• VertexCount works with undirected graphs, directed graphs, multigraphs, and mixed graphs.

# Examples

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## Basic Examples(2)

Count the number of vertices:

Count the number of vertices that match a pattern:

## Scope(7)

VertexCount works with undirected graphs:

Directed graphs:

Multigraphs:

Mixed graphs:

Use rules to specify the graph:

Use patterns to count a subset of vertices:

VertexCount works with large graphs:

## Generalizations & Extensions(1)

Count the number of vertices on symbolic graph constructors:

## Applications(3)

The minimum number of edges in a connected graph with vertices is :

A path graph with vertices has exactly edges:

Construct a graph with a random vertex coloring:

Count the vertices with the color blue:

Highlight the vertex by the vertex degree:

Count the vertex with degree 3:

## Properties & Relations(5)

Number of vertices of :

VertexCount can be found using VertexList:

The number of vertices can be found from matrix representations:

The number of rows and columns of an adjacency matrix:

The number of rows of an incidence matrix:

The number of rows or columns of a Kirchhoff matrix:

The number of vertices of the line graph of graph g is equal to the number of edges of g:

Line graph of g:

Isomorphic graphs have the same number of vertices:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_vertexcount, author="Wolfram Research", title="{VertexCount}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/VertexCount.html}", note=[Accessed: 20-June-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_vertexcount, organization={Wolfram Research}, title={VertexCount}, year={2015}, url={https://reference.wolfram.com/language/ref/VertexCount.html}, note=[Accessed: 20-June-2024 ]}