Often Voronoi diagrams are constructed from their duals, Delaunay triangulations.
The Delaunay triangulation of a point set in $d$ dimensions can be obtained from the convex hull
of a lifting of the points into $d+1$ dimensions. This relationship is explained in many sources,
including [_The Handbook of Discrete and Computational Geometry_][1] (Chapters 22, 23).

Implementing this on your own is quite a project, so you might first investigate what is available.
Qhull performs the computations; see [www.qhull.org][2].
$d$-dimensional Delaunay triangulation computations are part of [CGAL][3], specifically the module on [dD Convex Hulls and Delaunay Triangulations][4].
You could use [polymake][5] to convert from the Delaunay triangulation to its corresponding Voronoi diagram.
Finally, there are papers written on specific versions of your problem, for example:
["An Explicit Solution for Computing the Euclidean $d$-dimensional Voronoi Diagram of Spheres in a Floating-Point Arithmetic,"][6] by Marina Gavrilova, 2003.

(Incidentally, "Euclidean graphs" play a relatively minor role in this topic.)


  [1]: https://www.routledge.com/Handbook-of-Discrete-and-Computational-Geometry-Third-Edition/Toth-ORourke-Goodman/p/book/9781498711395
  [2]: http://www.qhull.org/
  [3]: https://www.cgal.org/
  [4]: https://doc.cgal.org/Manual/3.2/doc_html/cgal_manual/Convex_hull_d/Chapter_main.html
  [5]: https://polymake.org/doku.php
  [6]: https://doi.org/10.1007/3-540-44842-X_84 "In: Kumar, V., Gavrilova, M.L., Tan, C.J.K., L’Ecuyer, P. (eds) Computational Science and Its Applications — ICCSA 2003. ICCSA 2003. Lecture Notes in Computer Science, vol 2669. Springer, Berlin, Heidelberg. zbMATH review at https://zbmath.org/?q=an:1049.68009"