Graph: Add adjacency_matrix and laplacian_matrix (#3862)

* Graphs: vertex degree in a graph

* Graphs: Add adjacency_matrix and laplacian_matrix

* Graphs: Make adjacency_matrix work for directed graphs, add some documentation

* Graphs: Add test for laplacian_matrix and adjacency_matrix

* Graphs: Add new functions to docs

---------

Co-authored-by: Michael Joswig <joswig@math.tu-berlin.de>

docs/src/Combinatorics/graphs.md

@@ -51,13 +51,16 @@ rem_vertices!(g::Graph{T}, a::AbstractVector{Int64}) where {T <: Union{Directed,
 
 ## Auxiliary functions
 ```@docs
+adjacency_matrix(g::Graph)
 all_neighbors(g::Graph{T}, v::Int64) where {T <: Union{Directed, Undirected}}
 automorphism_group_generators(g::Graph{T}) where {T <: Union{Directed, Undirected}}
 complete_graph(n::Int64)
 complete_bipartite_graph(n::Int64, m::Int64)
+degree(g::Graph, v::Int)
 edges(g::Graph{T}) where {T <: Union{Directed, Undirected}}
 has_edge(g::Graph{T}, source::Int64, target::Int64) where {T <: Union{Directed, Undirected}}
 has_vertex(g::Graph{T}, v::Int64) where {T <: Union{Directed, Undirected}}
+laplacian_matrix(g::Graph)
 n_edges(g::Graph{T}) where {T <: Union{Directed, Undirected}}
 n_vertices(g::Graph{T}) where {T <: Union{Directed, Undirected}}
 inneighbors(g::Graph{T}, v::Int64) where {T <: Union{Directed, Undirected}}

src/Combinatorics/Graphs/functions.jl

@@ -507,6 +507,23 @@ function neighbors(g::Graph{T}, v::Int64) where {T <: Union{Directed, Undirected
     return [x+1 for x in result]
 end
 
+@doc raw"""
+    degree(g::Graph{T} [, v::Int64]) where {T <: Union{Directed, Undirected}}
+
+Return the degree of the vertex `v` in the graph `g`.
+If `v` is missing, return the list of degrees of all vertices.
+
+# Examples
+```jldoctest
+julia> g = vertex_edge_graph(icosahedron());
+
+julia> degree(g, 1)
+5
+```
+"""
+degree(g::Graph, v::Int64) = length(neighbors(g, v))
+degree(g::Graph) = [ length(neighbors(g, v)) for v in 1:n_vertices(g) ]
+
 
 @doc raw"""
     inneighbors(g::Graph{T}, v::Int64) where {T <: Union{Directed, Undirected}}
@@ -1182,3 +1199,88 @@ function graph_from_edges(edges::Vector{Vector{Int}},
                           n_vertices::Int=-1)
   return graph_from_edges(Undirected, [Edge(e[1], e[2]) for e in edges], n_vertices)
 end
+
+
+
+@doc raw"""
+    adjacency_matrix(g::Graph{T}) where {T <: Union{Directed, Undirected}}
+
+Return an unsigned (boolean) adjacency matrix representing a graph `g`. If `g`
+is undirected, the adjacency matrix will be symmetric. For `g` being directed,
+the adjacency matrix has a `1` at `(u,v)` if there is an edge `u->v`.
+
+# Examples
+Adjacency matrix for a directed graph:
+```jldoctest
+julia> G = Graph{Directed}(3)
+Directed graph with 3 nodes and no edges
+
+julia> add_edge!(G,1,3)
+true
+
+julia> add_edge!(G,1,2)
+true
+
+julia> adjacency_matrix(G)
+3×3 IncidenceMatrix
+[2, 3]
+[]
+[]
+
+
+julia> matrix(ZZ, adjacency_matrix(G))
+[0   1   1]
+[0   0   0]
+[0   0   0]
+```
+
+Adjacency matrix for an undirected graph:
+```jldoctest
+julia> G = vertex_edge_graph(cube(2))
+Undirected graph with 4 nodes and the following edges:
+(2, 1)(3, 1)(4, 2)(4, 3)
+
+julia> adjacency_matrix(G)
+4×4 IncidenceMatrix
+[2, 3]
+[1, 4]
+[1, 4]
+[2, 3]
+
+
+julia> matrix(ZZ, adjacency_matrix(G))
+[0   1   1   0]
+[1   0   0   1]
+[1   0   0   1]
+[0   1   1   0]
+```
+"""
+adjacency_matrix(g::Graph) = Polymake.call_function(:common, Symbol("IncidenceMatrix::new"), nothing, Polymake.common.adjacency_matrix(pm_object(g)))
+
+
+@doc raw"""
+    laplacian_matrix(g::Graph{T}) where {T <: Union{Directed, Undirected}}
+
+Return the Laplacian matrix of the graph `g`. The Laplacian matrix of a graph
+can be written as the difference of `D`, where `D` is a quadratic matrix with
+the degrees of `g` on the diagonal, and the adjacency matrix of `g`. For an
+undirected graph, the Laplacian matrix is symmetric.
+
+# Examples
+```
+julia> G = vertex_edge_graph(cube(2))
+Undirected graph with 4 nodes and the following edges:
+(2, 1)(3, 1)(4, 2)(4, 3)
+
+julia> laplacian_matrix(G)
+[ 2   -1   -1    0]
+[-1    2    0   -1]
+[-1    0    2   -1]
+[ 0   -1   -1    2]
+```
+"""
+function laplacian_matrix(g::Graph)
+  D = diagonal_matrix(degree(g))
+  A = matrix(ZZ, adjacency_matrix(g))
+  return D-A
+end

src/exports.jl

@@ -194,6 +194,7 @@ export add_gluing!
 export add_vertex!
 export add_vertices!
 export adjacency_tree
+export adjacency_matrix
 export adjacent_chamber
 export adjoint_ideal
 export adjunction_process
@@ -915,6 +916,7 @@ export koszul_complex
 export koszul_homology
 export koszul_matrix
 export labeled_matrix_formatted
+export laplacian_matrix
 export lattice_points
 export lattice_volume
 export leading_coefficient

test/Combinatorics/Graph.jl

@@ -161,4 +161,15 @@
         @test n_edges(G2) == 5
 
     end
+
+    @testset "adjacency_matrix laplacian_matrix" begin
+      G0 = Graph{Directed}(3)
+      add_edge!(G0,1,2)
+      add_edge!(G0,1,3)
+      @test matrix(ZZ, adjacency_matrix(G0)) == matrix(ZZ, [0 1 1; 0 0 0; 0 0 0])
+      @test laplacian_matrix(G0) == matrix(ZZ, [2 -1 -1; 0 0 0; 0 0 0])
+      G1 = vertex_edge_graph(cube(2))
+      @test matrix(ZZ, adjacency_matrix(G1)) == matrix(ZZ, [0 1 1 0; 1 0 0 1; 1 0 0 1; 0 1 1 0])
+      @test laplacian_matrix(G1) == matrix(ZZ, [2 -1 -1 0; -1 2 0 -1; -1 0 2 -1; 0 -1 -1 2])
+    end
 end