The Arrowhead Torus : a Cayley Graph on the 6 - valent Grid

The \arrowhead torus" is a broadcast graph that we deene on the 6-valent grid as a Cayley graph. We borrow the term from Mandelbrot who qualiies in that way one of the Sierpinski's famous fractal constructions. The 6-valent grid H = (V; E) is generated by three families of straight lines. We adopt the isotropic orientation S ! N, NE ! SW, NW ! SE and deene the system of generators S = fs 1 ; s 2 ; s 3 g whose elements are the three respective translations. The multiplication on S deenes a group acting on the vertices of V with a basic set of relations. The arrowhead is the graph of a nite group generated by superimposing a cyclic relation for each direction. The arrowhead interconnection network has several important advantages. It has a bounded valence as a grid and the highest allowed valence for a 2D regular grid. As a Cayley graph, it allows recursive constructions and divide-and-conquer schemes for information dissemination, it is also vertex-transitive hence all routers will behave in a similar way. From construction it will appear nally as a good host for embedding subvalent topologies like the usual grid. Le tore en \sagette" : un graphe de Cayley sur la grille 6-valente R esum e : Le tore dit en \sagette" (ou encore \en pointe-de-eche") est un graphe de diiu-sion que l'on d eenit sur la grille 6-valente comme graphe de Cayley. Le terme est emprunt e a Mandelbrot qui qualiie de cette faa con l'une des c el ebres constructions fractales de Sier-pinski. La grille 6-valente H = (V; E) est engendr ee par trois familles de droites. On adopte l'orientation isotrope S ! N, NE ! SW, NW ! SE et on d eenit le syst eme de g en era-Mots-cl e : r eseaux d'interconnexion, graphes de Cayley, grille hexavalente.