triangulation = list of three-element lists

Supply a previously computed Delaunay triangulation of P. The input must be a valid Delaunay triangulation in the format returned by ComputationalGeometry[DelaunayTriangulation]: a list of three-element lists of integers, representing triangles in a triangulation of P.

vertices = list of integers, strings or symbols

Specifies the vertices to be used in the generated graph.

weighted = truefalse

If weighted=true, the result is a weighted graph whose edge weights correspond to the Euclidean distance between points. The default is false.

The GabrielGraph(P, opts) command returns the Gabriel graph for the point set P.

The parameter P must be a Matrix or list of lists representing a set of points.

Let $P$ be a set of points in $n$ dimensions, let $p$ and $q$ be arbitrary points from $P$, and let $\mathrm{dist}\left(p\,q\right)$ be the Euclidean distance between $p$ and $q$.

The Gabriel graph the graph whose vertices correspond to points in $P$ and whose edges consist of those pairs $p$ and $q$ from $P$ for which the closed ball centered halfway between $p$ and $q$ with diameter equal to $\mathrm{dist}\left(p-q\right)$ contains no other points from $P$.

Formally, define the ball $B\left(p\,q\right)$ to be those points $r\in P$ such that $\mathrm{dist}\left(r\,\frac{p}{2}+\frac{q}{2}\right)\le \frac{\mathrm{dist}\left(p\,q\right)}{2}$. The Gabriel graph has an edge between $p$ and $q$ if and only if $B\left(p\,q\right)=\varnothing$.

The Gabriel graph has the following relationships with other graphs:

The Euclidean minimum spanning tree on P is a subgraph of the Gabriel graph on P.

The nearest neighbor graph on P is a subgraph of the Gabriel graph on P.

The relative neighborhood graph on P is a subgraph of the Gabriel graph on P.

The Urquhart graph on P is a subgraph of the Gabriel graph on P.

The Gabriel graph on P is a subgraph of the Delaunay graph on P.

$\mathrm{with}\left(\mathrm{GraphTheory}\right)\:$

$\mathrm{with}\left(\mathrm{GeometricGraphs}\right)\:$

$\mathrm{points}≔\mathrm{LinearAlgebra}:-\mathrm{RandomMatrix}\left(60\,2\,\mathrm{generator}=0..100.\,\mathrm{datatype}=\mathrm{float}\left[8\right]\right)$

$\mathrm{points}≔\begin{array}{c}\left[\begin{array}{cc}9.85017697341803& 82.9750304386195\\ 86.0670183749663& 83.3188659363996\\ 64.3746795546741& 73.8671607639673\\ 57.3670557294666& 2.34399775883031\\ 23.6234264844933& 52.6873367387328\\ 47.0027547350003& 22.2459488367552\\ 74.9213491558963& 62.0471820220718\\ 92.1513434709073& 96.3107262637080\\ 48.2319624355944& 63.7563267144141\\ 90.9441877431805& 33.8527464913022\\ ⋮& ⋮\end{array}\right]\\ \hfill \text{60 × 2 Matrix}\end{array}$

$\mathrm{GG}≔\mathrm{GabrielGraph}\left(\mathrm{points}\right)$

$\mathrm{GG}≔\mathrm{Graph\; 1:\; an\; undirected\; graph\; with\; 60\; vertices\; and\; 105\; edges}$

$\mathrm{DrawGraph}\left(\mathrm{GG}\right)$

Gabriel, Kuno Ruben; Sokal, Robert Reuven (1969), "A new statistical approach to geographic variation analysis", Systematic Biology, Society of Systematic Biologists, 18(3): 259–278. doi:10.2307/2412323.

The GraphTheory[GeometricGraphs][GabrielGraph] command was introduced in Maple 2020.

For more information on Maple 2020 changes, see Updates in Maple 2020.