Your first metric graph

Before you start, go to the installation section and activate MeGraPDE in the current session with the command

using MeGraPDE

Creating a metric graph

Let us first create a combinatorial star graph G with 5 vertices and 4 edges using Graphs.jl

using Graphs
G = star_graph(5)

{5, 4} undirected simple Int64 graph

In order to extend G to an equilateral metric graph, we define the edge length ℓ

ℓ = pi + pi/2

4.71238898038469

G can now be represented as metric graph Γ by applying the function metric_graph

Γ = metric_graph(G, ℓ)

{n=5,m=4,ℓ=4.71238898038469} equilateral metric graph

For a small example like the star graph, vertex coordinates can be assigned that will later allow to visualize Γ in 3d.

coords = [[0,0],
          [ℓ,0],
          [-ℓ,0],
          [0,ℓ],
          [0,-ℓ]]

5-element Vector{Vector{Float64}}:
 [0.0, 0.0]
 [4.71238898038469, 0.0]
 [-4.71238898038469, 0.0]
 [0.0, 4.71238898038469]
 [0.0, -4.71238898038469]

The function metric_graph takes the optional input vertex_coords to specify the vertex coordinates.

Γ = metric_graph(G, ℓ, vertex_coords = coords)

{n=5,m=4,ℓ=4.71238898038469} equilateral metric graph

We may now plot Γ using plot_graph_3d

plot_graph_3d(Γ)

Note

The previous example graph can be assembled using the constructor metric_star_graph and indicating the desired edge length as metric_star_graph(ℓ = pi + pi/2). Several other example graphs are implemented.

Functions on metric graphs

A function u on a metric graph is represented by a vector of functions u_e, specifying u on each edge e.

u = [ x -> -3*sin(x),
    x -> sin(x),
    x -> sin(x),
    x -> sin(x)
    ]

4-element Vector{Function}:
 #1 (generic function with 1 method)
 #2 (generic function with 1 method)
 #3 (generic function with 1 method)
 #4 (generic function with 1 method)

If vertex coordinates are assigned to Γ, a function can be plotted on Γ with

plot_function_3d(Γ, u)