# Generators for common graphs

Graphs.jl defines generators for various families of deterministic and random graphs.

## Full docs

Graphs.SimpleGraphs.euclidean_graphMethod
euclidean_graph(N, d; rng=nothing, seed=nothing, L=1., p=2., cutoff=-1., bc=:open)

Generate N uniformly distributed points in the box $[0,L]^{d}$ and return a Euclidean graph, a map containing the distance on each edge and a matrix with the points' positions.

Examples

julia> using Graphs

julia> g, dists = euclidean_graph(5, 2, cutoff=0.3);

julia> g
{5, 4} undirected simple Int64 graph

julia> dists
Dict{Graphs.SimpleGraphs.SimpleEdge{Int64},Float64} with 4 entries:
Edge 1 => 5 => 0.205756
Edge 2 => 5 => 0.271359
Edge 2 => 4 => 0.247703
Edge 4 => 5 => 0.168372
source
Graphs.SimpleGraphs.euclidean_graphMethod
euclidean_graph(points)

Given the d×N matrix points build an Euclidean graph of N vertices and return a graph and Dict containing the distance on each edge.

Optional Arguments

• L=1: used to bound the d dimensional box from which points are selected.
• p=2
• bc=:open

Implementation Notes

Defining the d-dimensional vectors x[i] = points[:,i], an edge between vertices i and j is inserted if norm(x[i]-x[j], p) < cutoff. In case of negative cutoff instead every edge is inserted. For p=2 we have the standard Euclidean distance. Set bc=:periodic to impose periodic boundary conditions in the box $[0,L]^d$.

Examples

julia> using Graphs

julia> pts = rand(3, 10); # 10 vertices in R^3

julia> g, dists = euclidean_graph(pts, p=1, bc=:periodic) # Taxicab-distance (L^1);

julia> g
{10, 45} undirected simple Int64 graph
source
Graphs.SimpleGraphs.SimpleDiGraphMethod
SimpleDiGraph{T}(nv, ne; rng=nothing, seed=nothing)

Construct a random SimpleDiGraph{T} with nv vertices and ne edges. The graph is sampled uniformly from all such graphs. If seed >= 0, a random generator is seeded with this value. If not specified, the element type T is the type of nv.

erdos_renyi

Examples

julia> using Graphs

julia> SimpleDiGraph(5, 7)
{5, 7} directed simple Int64 graph
source
Graphs.SimpleGraphs.SimpleGraphMethod
SimpleGraph{T}(nv, ne, edgestream::Channel)

Construct a SimpleGraph{T} with nv vertices and ne edges from edgestream. Can result in less than ne edges if the channel edgestream is closed prematurely. Duplicate edges are only counted once. The element type is the type of nv.

source
Graphs.SimpleGraphs.SimpleGraphMethod
SimpleGraph{T}(nv, ne, smb::StochasticBlockModel)

Construct a random SimpleGraph{T} with nv vertices and ne edges. The graph is sampled according to the stochastic block model smb. The element type is the type of nv.

source
Graphs.SimpleGraphs.SimpleGraphMethod
SimpleGraph{T}(nv, ne; rng=nothing, seed=nothing)

Construct a random SimpleGraph{T} with nv vertices and ne edges. The graph is sampled uniformly from all such graphs. If seed >= 0, a random generator is seeded with this value. If not specified, the element type T is the type of nv.

erdos_renyi

Examples

julia> using Graphs

julia> SimpleGraph(5, 7)
{5, 7} undirected simple Int64 graph
source
Graphs.SimpleGraphs.StochasticBlockModelType
StochasticBlockModel{T,P}

A type capturing the parameters of the SBM. Each vertex is assigned to a block and the probability of edge (i,j) depends only on the block labels of vertex i and vertex j.

The assignment is stored in nodemap and the block affinities a k by k matrix is stored in affinities.

affinities[k,l] is the probability of an edge between any vertex in block k and any vertex in block l.

Implementation Notes

Graphs are generated by taking random $i,j ∈ V$ and flipping a coin with probability affinities[nodemap[i],nodemap[j]].

source
Graphs.SimpleGraphs.barabasi_albert!Method
barabasi_albert!(g::AbstractGraph, n::Integer, k::Integer)

Create a Barabási–Albert model random graph with n vertices. It is grown by adding new vertices to an initial graph g. Each new vertex is attached with k edges to k different vertices already present in the system by preferential attachment.

Optional Arguments

• rng=nothing: set the Random Number Generator.
• seed=nothing: set the RNG seed.

Examples

julia> using Graphs

julia> g = cycle_graph(4)
{4, 4} undirected simple Int64 graph

julia> barabasi_albert!(g, 16, 3);

julia> g
{16, 40} undirected simple Int64 graph
source
Graphs.SimpleGraphs.barabasi_albertMethod
barabasi_albert(n::Integer, n0::Integer, k::Integer)

Create a Barabási–Albert model random graph with n vertices. It is grown by adding new vertices to an initial graph with n0 vertices. Each new vertex is attached with k edges to k different vertices already present in the system by preferential attachment. Initial graphs are undirected and consist of isolated vertices by default.

Optional Arguments

• is_directed=false: if true, return a directed graph.
• complete=false: if true, use a complete graph for the initial graph.
• rng=nothing: set the Random Number Generator.
• seed=nothing: set the RNG seed.

Examples

julia> using Graphs

julia> barabasi_albert(10, 3, 2)
{10, 14} undirected simple Int64 graph

julia> barabasi_albert(100, Int8(10), 3, is_directed=true, seed=123)
{100, 270} directed simple Int8 graph
source
Graphs.SimpleGraphs.barabasi_albertMethod
barabasi_albert(n, k)

Create a Barabási–Albert model random graph with n vertices. It is grown by adding new vertices to an initial graph with k vertices. Each new vertex is attached with k edges to k different vertices already present in the system by preferential attachment. Initial graphs are undirected and consist of isolated vertices by default.

Optional Arguments

• is_directed=false: if true, return a directed graph.
• complete=false: if true, use a complete graph for the initial graph.
• rng=nothing: set the Random Number Generator.
• seed=nothing: set the RNG seed.

Examples

julia> using Graphs

julia> barabasi_albert(50, 3)
{50, 141} undirected simple Int64 graph

julia> barabasi_albert(100, Int8(10), is_directed=true, complete=true, seed=123)
{100, 990} directed simple Int8 graph
source
Graphs.SimpleGraphs.dorogovtsev_mendesMethod
dorogovtsev_mendes(n)

Generate a random n vertex graph by the Dorogovtsev-Mendes method (with n \ge 3).

The Dorogovtsev-Mendes process begins with a triangle graph and inserts n-3 additional vertices. Each time a vertex is added, a random edge is selected and the new vertex is connected to the two endpoints of the chosen edge. This creates graphs with many triangles and a high local clustering coefficient.

It is often useful to track the evolution of the graph as vertices are added, you can access the graph from the tth stage of this algorithm by accessing the first t vertices with g[1:t].

References

• http://graphstream-project.org/doc/Generators/Dorogovtsev-Mendes-generator/
• https://arxiv.org/pdf/cond-mat/0106144.pdf#page=24

Examples

julia> using Graphs

julia> dorogovtsev_mendes(10)
{10, 17} undirected simple Int64 graph

julia> dorogovtsev_mendes(11, seed=123)
{11, 19} undirected simple Int64 graph
source
Graphs.SimpleGraphs.erdos_renyiMethod
erdos_renyi(n, ne)

Create an Erdős–Rényi random graph with n vertices and ne edges.

Optional Arguments

• is_directed=false: if true, return a directed graph.
• rng=nothing: set the Random Number Generator.
• seed=nothing: set the RNG seed.

Examples

julia> using Graphs

julia> erdos_renyi(10, 30)
{10, 30} undirected simple Int64 graph

julia> erdos_renyi(10, 30, is_directed=true, seed=123)
{10, 30} directed simple Int64 graph
source
Graphs.SimpleGraphs.erdos_renyiMethod
erdos_renyi(n, p)

Create an Erdős–Rényi random graph with n vertices. Edges are added between pairs of vertices with probability p.

Optional Arguments

• is_directed=false: if true, return a directed graph.
• rng=nothing: set the Random Number Generator.
• seed=nothing: set the RNG seed.

Examples

julia> using Graphs

julia> erdos_renyi(10, 0.5)
{10, 20} undirected simple Int64 graph
julia> using Graphs

julia> erdos_renyi(10, 0.5, is_directed=true, seed=123)
{10, 49} directed simple Int64 graph
source
Graphs.SimpleGraphs.expected_degree_graphMethod
expected_degree_graph(ω)

Given a vector of expected degrees ω indexed by vertex, create a random undirected graph in which vertices i and j are connected with probability ω[i]*ω[j]/sum(ω).

Optional Arguments

• rng=nothing: set the Random Number Generator.
• seed=nothing: set the RNG seed.

Implementation Notes

The algorithm should work well for maximum(ω) << sum(ω). As maximum(ω) approaches sum(ω), some deviations from the expected values are likely.

References

Examples

# 1)
julia> using Graphs

julia> g = expected_degree_graph([3, 1//2, 1//2, 1//2, 1//2])
{5, 3} undirected simple Int64 graph

julia> print(degree(g))
[3, 0, 1, 1, 1]

# 2)
julia> g = expected_degree_graph([0.5, 0.5, 0.5], seed=123)
{3, 1} undirected simple Int64 graph

julia> print(degree(g))
[1, 0, 1]
source
Graphs.SimpleGraphs.make_edgestreamMethod
make_edgestream(sbm; rng=nothing, seed=nothing)

Take an infinite sample from the Stochastic Block Model sbm. Pass to Graph(nvg, neg, edgestream) to get a Graph object based on sbm.

source
Graphs.SimpleGraphs.nearbipartiteaffinityMethod
nearbipartiteaffinity(sizes, between, intra)

Construct the affinity matrix for a near bipartite SBM. between is the affinity between the two parts of each bipartite community. intra is the probability of an edge within the parts of the partitions.

This is a specific type of SBM with \frac{k}{2} blocks each with two halves. Each half is connected as a random bipartite graph with probabilityintraThe blocks are connected with probabilitybetween.

source
Graphs.SimpleGraphs.newman_watts_strogatzMethod
newman_watts_strogatz(n, k, β)

Return a Newman-Watts-Strogatz small world random graph with n vertices, each with expected degree k(1 + β) (or (k - 1)(1 + β) if k is odd). Edges are randomized per the model based on probability β.

The algorithm proceeds as follows. First, a perfect 1-lattice is constructed, where each vertex has exacly div(k, 2) neighbors on each side (i.e., k or k - 1 in total). Then the following steps are repeated for a hop length i of 1 through div(k, 2).

1. Consider each vertex s in turn, along with the edge to its ith nearest neighbor t, in a clockwise sense.

2. Generate a uniformly random number r. If r < β, s is connected to some vertex d, chosen uniformly at random from the entire graph, excluding s and its neighbors. (Note that t is a valid candidate.)

For β = 0, the graph will remain a 1-lattice, and for β = 1, all edges will be rewired randomly.

Note: In the original paper by Newman and Watts, self-loops and double edges were allowed, which is not the case here. However, for large enough networks and small enough p and k, this should not deviate much from the original model.

Optional Arguments

• is_directed=false: if true, return a directed graph.
• rng=nothing: set the Random Number Generator.
• seed=nothing: set the RNG seed.

Examples

julia> using Graphs

julia> newman_watts_strogatz(10, 4, 0.3)
{10, 26} undirected simple Int64 graph


jldoctest

julia> newmanwattsstrogatz(Int8(10), 4, 0.8, is_directed=true, seed=123) {10, 36} directed simple Int8 graph 

References

source
Graphs.SimpleGraphs.randbnMethod
randbn(n, p; rng=nothing, seed=nothing)

Return a binomially-distributed random number with parameters n and p and optional seed.

References

• "Non-Uniform Random Variate Generation," Luc Devroye, p. 522. Retrieved via http://www.eirene.de/Devroye.pdf.
• http://stackoverflow.com/questions/23561551/a-efficient-binomial-random-number-generator-code-in-java
source
Graphs.SimpleGraphs.random_configuration_modelMethod
random_configuration_model(n, ks)

Create a random undirected graph according to the configuration model containing n vertices, with each node i having degree k[i].

Optional Arguments

• rng=nothing: set the Random Number Generator.
• seed=nothing: set the RNG seed.
• check_graphical=false: if true, ensure that k is a graphical sequence

(see isgraphical).

Performance

Time complexity is approximately $\mathcal{O}(n \bar{k}^2)$.

Implementation Notes

Allocates an array of $n \bar{k}$ Ints.

source
Graphs.SimpleGraphs.random_orientation_dagMethod
random_orientation_dag(g)

Generate a random oriented acyclical digraph. The function takes in a simple graph and a random number generator as an argument. The probability of each directional acyclic graph randomly being generated depends on the architecture of the original directed graph.

DAG's have a finite topological order; this order is randomly generated via "order = randperm()".

Examples

julia> using Graphs

julia> random_orientation_dag(complete_graph(10))
{10, 45} directed simple Int64 graph

julia> random_orientation_dag(star_graph(Int8(10)), seed=123)
{10, 9} directed simple Int8 graph
source
Graphs.SimpleGraphs.random_regular_digraphMethod
random_regular_digraph(n, k)

Create a random directed regular graph with n vertices, each with degree k.

Optional Arguments

• dir=:out: the direction of the edges for the degree parameter.
• rng=nothing: set the Random Number Generator.
• seed=nothing: set the RNG seed.

Implementation Notes

Allocates an $n × n$ sparse matrix of boolean as an adjacency matrix and uses that to generate the directed graph.

source
Graphs.SimpleGraphs.random_regular_graphMethod
random_regular_graph(n, k)

Create a random undirected regular graph with n vertices, each with degree k.

Optional Arguments

• rng=nothing: set the Random Number Generator.
• seed=nothing: set the RNG seed.

Performance

Time complexity is approximately $\mathcal{O}(nk^2)$.

Implementation Notes

Allocates an array of nk Ints, and . For $k > \frac{n}{2}$, generates a graph of degree $n-k-1$ and returns its complement.

source
Graphs.SimpleGraphs.random_tournament_digraphMethod
random_tournament_digraph(n)

Create a random directed tournament graph with n vertices.

Optional Arguments

• rng=nothing: set the Random Number Generator.
• seed=nothing: set the RNG seed.

Examples

julia> using Graphs

julia> random_tournament_digraph(5)
{5, 10} directed simple Int64 graph

julia> random_tournament_digraph(Int8(10), seed=123)
{10, 45} directed simple Int8 graph
source
Graphs.SimpleGraphs.sbmaffinityMethod
sbmaffinity(internalp, externalp, sizes)

Produce the sbm affinity matrix with internal probabilities internalp and external probabilities externalp.

source
Graphs.SimpleGraphs.static_fitness_modelMethod
static_fitness_model(m, fitness)

Generate a random graph with $|fitness|$ vertices and m edges, in which the probability of the existence of $Edge_{ij}$ is proportional to $fitness_i × fitness_j$.

Optional Arguments

• rng=nothing: set the Random Number Generator.
• seed=nothing: set the RNG seed.

Performance

Time complexity is $\mathcal{O}(|V| + |E| log |E|)$.

References

• Goh K-I, Kahng B, Kim D: Universal behaviour of load distribution in scale-free networks. Phys Rev Lett 87(27):278701, 2001.

Examples

julia> g = static_fitness_model(5, [1, 1, 0.5, 0.1])
{4, 5} undirected simple Int64 graph

julia> edges(g) |> collect
5-element Vector{Graphs.SimpleGraphs.SimpleEdge{Int64}}:
Edge 1 => 2
Edge 1 => 3
Edge 1 => 4
Edge 2 => 3
Edge 2 => 4
julia> using Graphs

julia> g = static_fitness_model(5, [1, 1, 0.5, 0.1], seed=123)
{4, 5} undirected simple Int64 graph

julia> edges(g) |> collect
5-element Vector{Graphs.SimpleGraphs.SimpleEdge{Int64}}:
Edge 1 => 2
Edge 1 => 3
Edge 2 => 3
Edge 2 => 4
Edge 3 => 4
source
Graphs.SimpleGraphs.static_fitness_modelMethod
static_fitness_model(m, fitness_out, fitness_in)

Generate a random directed graph with $|fitness\_out + fitness\_in|$ vertices and m edges, in which the probability of the existence of $Edge_{ij}$ is proportional with respect to $i ∝ fitness\_out$ and $j ∝ fitness\_in$.

Optional Arguments

• rng=nothing: set the Random Number Generator.
• seed=nothing: set the RNG seed.

Performance

Time complexity is $\mathcal{O}(|V| + |E| log |E|)$.

References

• Goh K-I, Kahng B, Kim D: Universal behaviour of load distribution in scale-free networks. Phys Rev Lett 87(27):278701, 2001.

Examples

julia> using Graphs

julia> g = static_fitness_model(6, [1, 0.2, 0.2, 0.2], [0.1, 0.1, 0.1, 0.9]; seed=123)
{4, 6} directed simple Int64 graph

julia> edges(g) |> collect
6-element Vector{Graphs.SimpleGraphs.SimpleEdge{Int64}}:
Edge 1 => 2
Edge 1 => 3
Edge 1 => 4
Edge 2 => 3
Edge 2 => 4
Edge 3 => 4
source
Graphs.SimpleGraphs.static_scale_freeMethod
static_scale_free(n, m, α_out, α_in)

Generate a random graph with n vertices, m edges and expected power-law degree distribution with exponent α_out for outbound edges and α_in for inbound edges.

Optional Arguments

• rng=nothing: set the Random Number Generator.
• seed=nothing: set the RNG seed.
• finite_size_correction=true: determines whether to use the finite size correction

proposed by Cho et al.

Performance

Time complexity is $\mathcal{O}(|V| + |E| log |E|)$.

References

• Goh K-I, Kahng B, Kim D: Universal behaviour of load distribution in scale-free networks. Phys Rev Lett 87(27):278701, 2001.
• Chung F and Lu L: Connected components in a random graph with given degree sequences. Annals of Combinatorics 6, 125-145, 2002.
• Cho YS, Kim JS, Park J, Kahng B, Kim D: Percolation transitions in scale-free networks under the Achlioptas process. Phys Rev Lett 103:135702, 2009.
source
Graphs.SimpleGraphs.static_scale_freeMethod
static_scale_free(n, m, α)

Generate a random graph with n vertices, m edges and expected power-law degree distribution with exponent α.

Optional Arguments

• rng=nothing: set the Random Number Generator.
• seed=nothing: set the RNG seed.
• finite_size_correction=true: determines whether to use the finite size correction

proposed by Cho et al.

Performance

Time complexity is $\mathcal{O}(|V| + |E| log |E|)$.

References

• Goh K-I, Kahng B, Kim D: Universal behaviour of load distribution in scale-free networks. Phys Rev Lett 87(27):278701, 2001.
• Chung F and Lu L: Connected components in a random graph with given degree sequences. Annals of Combinatorics 6, 125-145, 2002.
• Cho YS, Kim JS, Park J, Kahng B, Kim D: Percolation transitions in scale-free networks under the Achlioptas process. Phys Rev Lett 103:135702, 2009.
source
Graphs.SimpleGraphs.stochastic_block_modelMethod
stochastic_block_model(c, n)

Return a Graph generated according to the Stochastic Block Model (SBM).

c[a,b] : Mean number of neighbors of a vertex in block a belonging to block b. Only the upper triangular part is considered, since the lower triangular is determined by $c[b,a] = c[a,b] * \frac{n[a]}{n[b]}$. n[a] : Number of vertices in block a

Optional Arguments

• rng=nothing: set the Random Number Generator.
• seed=nothing: set the RNG seed.

For a dynamic version of the SBM see the StochasticBlockModel type and related functions.

source
Graphs.SimpleGraphs.uniform_treeMethod
uniform_tree(n)

Generates a random labelled tree, drawn uniformly at random over the $n^{n-2}$ such trees. A uniform word of length n-2 over the alphabet 1:n is generated (Prüfer sequence) then decoded. See also the prufer_decode function and this page on Prüfer codes.

Optional Arguments

• rng=nothing: set the Random Number Generator.

Examples

julia> using Graphs

julia> uniform_tree(10)
{10, 9} undirected simple Int64 graph
source
Graphs.SimpleGraphs.watts_strogatzMethod
watts_strogatz(n, k, β)

Return a Watts-Strogatz small world random graph with n vertices, each with expected degree k (or k - 1 if k is odd). Edges are randomized per the model based on probability β.

The algorithm proceeds as follows. First, a perfect 1-lattice is constructed, where each vertex has exacly div(k, 2) neighbors on each side (i.e., k or k - 1 in total). Then the following steps are repeated for a hop length i of 1 through div(k, 2).

1. Consider each vertex s in turn, along with the edge to its ith nearest neighbor t, in a clockwise sense.

2. Generate a uniformly random number r. If r ≥ β, then the edge (s, t) is left unaltered. Otherwise, the edge is deleted and rewired so that s is connected to some vertex d, chosen uniformly at random from the entire graph, excluding s and its neighbors. (Note that t is a valid candidate.)

For β = 0, the graph will remain a 1-lattice, and for β = 1, all edges will be rewired randomly.

Optional Arguments

• remove_edges=true: if false, do not remove the original edges.
• is_directed=false: if true, return a directed graph.
• rng=nothing: set the Random Number Generator.
• seed=nothing: set the RNG seed.

Examples

julia> using Graphs

julia> watts_strogatz(10, 4, 0.3)
{10, 20} undirected simple Int64 graph

julia> watts_strogatz(Int8(10), 4, 0.8, is_directed=true, seed=123)
{10, 20} directed simple Int8 graph

References

source
Graphs.SimpleGraphs.smallgraphMethod
smallgraph(s)
smallgraph(s)

Create a small graph of type s. Admissible values for s are:

sgraph type
:bullA bull graph.
:chvatalA Chvátal graph.
:cubicalA Platonic cubical graph.
:desarguesA Desarguesgraph.
:diamondA diamond graph.
:dodecahedralA Platonic dodecahedral graph.
:fruchtA Frucht graph.
:heawoodA Heawood graph.
:houseA graph mimicing the classic outline of a house.
:housexA house graph, with two edges crossing the bottom square.
:icosahedralA Platonic icosahedral graph.
:karateA social network graph called Zachary's karate club.
:krackhardtkiteA Krackhardt-Kite social network graph.
:moebiuskantorA Möbius-Kantor graph.
:octahedralA Platonic octahedral graph.
:pappusA Pappus graph.
:petersenA Petersen graph.
:sedgewickmazeA simple maze graph used in Sedgewick's Algorithms in C++: Graph Algorithms (3rd ed.)
:tetrahedralA Platonic tetrahedral graph.
:truncatedcubeA skeleton of the truncated cube graph.
:truncatedtetrahedronA skeleton of the truncated tetrahedron graph.
:truncatedtetrahedron_dirA skeleton of the truncated tetrahedron digraph.
:tutteA Tutte graph.
source
Graphs.SimpleGraphs.barbell_graphMethod
barbell_graph(n1, n2)

Create a barbell graph consisting of a clique of size n1 connected by an edge to a clique of size n2.

Implementation Notes

Preserves the eltype of n1 and n2. Will error if the required number of vertices exceeds the eltype. n1 and n2 must be at least 1 so that both cliques are non-empty. The cliques are organized with nodes 1:n1 being the left clique and n1+1:n1+n2 being the right clique. The cliques are connected by and edge (n1, n1+1).

Examples

julia> using Graphs

julia> barbell_graph(3, 4)
{7, 10} undirected simple Int64 graph

julia> barbell_graph(Int8(5), Int8(5))
{10, 21} undirected simple Int8 graph
source
Graphs.SimpleGraphs.binary_treeMethod
binary_tree(k::Integer)

Create a binary tree of depth k.

Examples

julia> using Graphs

julia> binary_tree(4)
{15, 14} undirected simple Int64 graph

julia> binary_tree(Int8(5))
{31, 30} undirected simple Int8 graph
source
Graphs.SimpleGraphs.circular_ladder_graphMethod
circular_ladder_graph(n)

Create a circular ladder graph consisting of 2n nodes and 3n edges. This is also known as the prism graph.

Implementation Notes

Preserves the eltype of the partitions vector. Will error if the required number of vertices exceeds the eltype. n must be at least 3 to avoid self-loops and multi-edges.

Examples

julia> using Graphs

{6, 9} undirected simple Int64 graph

{8, 12} undirected simple Int8 graph
source
Graphs.SimpleGraphs.clique_graphMethod
clique_graph(k, n)

Create a graph consisting of n connected k-cliques.

Examples

julia> using Graphs

julia> clique_graph(4, 10)
{40, 70} undirected simple Int64 graph

julia> clique_graph(Int8(10), Int8(4))
{40, 184} undirected simple Int8 graph
source
Graphs.SimpleGraphs.complete_bipartite_graphMethod
complete_bipartite_graph(n1, n2)

Create an undirected complete bipartite graph with n1 + n2 vertices.

Examples

julia> using Graphs

julia> complete_bipartite_graph(3, 4)
{7, 12} undirected simple Int64 graph

julia> complete_bipartite_graph(Int8(3), Int8(4))
{7, 12} undirected simple Int8 graph
source
Graphs.SimpleGraphs.complete_digraphMethod
complete_digraph(n)

Create a directed complete graph with n vertices.

Examples

julia> using Graphs

julia> complete_digraph(5)
{5, 20} directed simple Int64 graph

julia> complete_digraph(Int8(6))
{6, 30} directed simple Int8 graph
source
Graphs.SimpleGraphs.complete_graphMethod
complete_graph(n)

Create an undirected complete graph with n vertices.

Examples

julia> using Graphs

julia> complete_graph(5)
{5, 10} undirected simple Int64 graph

julia> complete_graph(Int8(6))
{6, 15} undirected simple Int8 graph
source
Graphs.SimpleGraphs.complete_multipartite_graphMethod
complete_multipartite_graph(partitions)

Create an undirected complete bipartite graph with sum(partitions) vertices. A partition with 0 vertices is skipped.

Implementation Notes

Preserves the eltype of the partitions vector. Will error if the required number of vertices exceeds the eltype.

Examples

julia> using Graphs

julia> complete_multipartite_graph([1,2,3])
{6, 11} undirected simple Int64 graph

julia> complete_multipartite_graph(Int8[5,5,5])
{15, 75} undirected simple Int8 graph
source
Graphs.SimpleGraphs.cycle_digraphMethod
cycle_digraph(n)

Create a directed cycle graph with n vertices.

Examples

julia> using Graphs

julia> cycle_digraph(3)
{3, 3} directed simple Int64 graph

julia> cycle_digraph(Int8(5))
{5, 5} directed simple Int8 graph
source
Graphs.SimpleGraphs.cycle_graphMethod
cycle_graph(n)

Create an undirected cycle graph with n vertices.

Examples

julia> using Graphs

julia> cycle_graph(3)
{3, 3} undirected simple Int64 graph

julia> cycle_graph(Int8(5))
{5, 5} undirected simple Int8 graph
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Graphs.SimpleGraphs.double_binary_treeMethod
double_binary_tree(k::Integer)

Create a double complete binary tree with k levels.

References

• Used as an example for spectral clustering by Guattery and Miller 1998.

Examples

julia> using Graphs

julia> double_binary_tree(4)
{30, 29} undirected simple Int64 graph

julia> double_binary_tree(Int8(5))
{62, 61} undirected simple Int8 graph
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Graphs.SimpleGraphs.gridMethod
grid(dims; periodic=false)

Create a $|dims|$-dimensional cubic lattice, with length dims[i] in dimension i.

Optional Arguments

• periodic=false: If true, the resulting lattice will have periodic boundary

condition in each dimension.

Examples

julia> using Graphs

julia> grid([2,3])
{6, 7} undirected simple Int64 graph

julia> grid(Int8[2, 2, 2], periodic=true)
{8, 12} undirected simple Int8 graph

julia> grid((2,3))
{6, 7} undirected simple Int64 graph
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Graphs.SimpleGraphs.ladder_graphMethod
ladder_graph(n)

Create a ladder graph consisting of 2n nodes and 3n-2 edges.

Implementation Notes

Preserves the eltype of n. Will error if the required number of vertices exceeds the eltype.

Examples

julia> using Graphs

{6, 7} undirected simple Int64 graph

{8, 10} undirected simple Int8 graph
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Graphs.SimpleGraphs.lollipop_graphMethod
lollipop_graph(n1, n2)

Create a lollipop graph consisting of a clique of size n1 connected by an edge to a path of size n2.

Implementation Notes

Preserves the eltype of n1 and n2. Will error if the required number of vertices exceeds the eltype. n1 and n2 must be at least 1 so that both the clique and the path have at least one vertex. The graph is organized with nodes 1:n1 being the clique and n1+1:n1+n2 being the path. The clique is connected to the path by an edge (n1, n1+1).

Examples

julia> using Graphs

julia> lollipop_graph(2, 5)
{7, 6} undirected simple Int64 graph

julia> lollipop_graph(Int8(3), Int8(4))
{7, 7} undirected simple Int8 graph
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Graphs.SimpleGraphs.path_digraphMethod
path_digraph(n)

Creates a directed path graph with n vertices.

Examples

julia> using Graphs

julia> path_digraph(5)
{5, 4} directed simple Int64 graph

julia> path_digraph(Int8(10))
{10, 9} directed simple Int8 graph
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Graphs.SimpleGraphs.path_graphMethod
path_graph(n)

Create an undirected path graph with n vertices.

Examples

julia> using Graphs

julia> path_graph(5)
{5, 4} undirected simple Int64 graph

julia> path_graph(Int8(10))
{10, 9} undirected simple Int8 graph
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Graphs.SimpleGraphs.roach_graphMethod
roach_graph(k)

Create a Roach graph of size k.

References

• Guattery and Miller 1998

Examples

julia> using Graphs

julia> roach_graph(10)
{40, 48} undirected simple Int64 graph
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Graphs.SimpleGraphs.star_digraphMethod
star_digraph(n)

Create a directed star graph with n vertices.

Examples

julia> using Graphs

julia> star_digraph(3)
{3, 2} directed simple Int64 graph

julia> star_digraph(Int8(10))
{10, 9} directed simple Int8 graph
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Graphs.SimpleGraphs.star_graphMethod
star_graph(n)

Create an undirected star graph with n vertices.

Examples

julia> using Graphs

julia> star_graph(3)
{3, 2} undirected simple Int64 graph

julia> star_graph(Int8(10))
{10, 9} undirected simple Int8 graph
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Graphs.SimpleGraphs.turan_graphMethod
turan_graph(n, r)

Creates a Turán Graph, a complete multipartite graph with n vertices and r partitions.

Examples

julia> using Graphs

julia> turan_graph(6, 2)
{6, 9} undirected simple Int64 graph

julia> turan_graph(Int8(7), 2)
{7, 12} undirected simple Int8 graph
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Graphs.SimpleGraphs.wheel_digraphMethod
wheel_digraph(n)

Create a directed wheel graph with n vertices.

Examples

julia> using Graphs

julia> wheel_digraph(5)
{5, 8} directed simple Int64 graph

julia> wheel_digraph(Int8(6))
{6, 10} directed simple Int8 graph
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Graphs.SimpleGraphs.wheel_graphMethod
wheel_graph(n)

Create an undirected wheel graph with n vertices.

Examples

julia> using Graphs

julia> wheel_graph(5)
{5, 8} undirected simple Int64 graph

julia> wheel_graph(Int8(6))
{6, 10} undirected simple Int8 graph`
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