Base

The abstract type AbstractMetricGraph comprises the following two types representing non-equilateral and equilateral metric graphs.

MeGraPDE.MetricGraphs.MetricGraph — Type

A type representing a Metric Graph (non-equilateral)

  • G

    Simple combinatorial graph

  • ℓ_vec

    Vector containing the edge lengths

  • coords

    Array containing the coordinates of the vertices; specify 'nothing' if no coordinates available

MeGraPDE.MetricGraphs.EquilateralMetricGraph — Type

A type representing a Metric Graph with equilateral edge lengths

  • G

    Simple combinatorial graph

  • ℓ

    Equilateral edge length

  • coords

    Array containing the coordinates of the vertices; specify 'nothing' if no coordinates available

In the field 'coords', coordinates can be specified that allow plotting the graph in a 3d grid.

The function metric_graph assembles a metric graph by specifying a combinatorial graph G and edge lengths. Coordinates can be set optionally, but are false per default.

MeGraPDE.MetricGraphs.metric_graph — Function
metric_graph(G::SimpleGraph, ℓ_vec::Vector; coord=nothing)

Create metric graph from simple graph 'G' with edge lengths 'ℓ_vec' and optionally assign a coordinate specified in 'coord' to the vertices.

metric_graph(G::SimpleGraph, ℓ::Number; coord=nothing)

Equilateral version with one uniform edge length 'ℓ' assigned to each edge.

It is possible to extended every possible graph to a metric graph by assigning edge length.

Consider for example the Barabasi-Albert graph constructed with [Graphs.jl](https://github.com/JuliaGraphs/Graphs.

using Graphs
G = barabasi_albert(100,3)
{100, 291} undirected simple Int64 graph

You can now either assign an equilateral edge length represented by one number or a vector l_vec with edge lengths to create a metric graph

ℓ = 1
Γ = metric_graph(G, ℓ)
{n=100,m=291,ℓ=1} equilateral metric graph
ℓ_vec = rand(1:5,ne(G))
Γ = metric_graph(G, ℓ_vec)
{n=100, m=291} metric graph

Some minor functions are implemented to quickly access properties of Γ. This list is by far not complete and will be expanded by other frequently used functions.

MeGraPDE.MetricGraphs.edge_length — Function
edge_length(Γ::MetricGraph, j::Int)

Return edge length of edge 'j.'

edge_length(Γ::EquilateralMetricGraph)

Equilateral version.

MeGraPDE.MetricGraphs.vol — Function
vol(Γ::MetricGraph)

Return volume $vol_{\Gamma} = \sum_{e \in \mathcal{E}} \ell_e$.

vol(Γ::EquilateralMetricGraph)

Equilateral version, $vol = m \cdot \ell$.