A family of Cayley graphs on the hexavalent grid

Arrowhead and Diamond Diameters

“…The topology of our arrowhead family [1] is quite different. The construction follows a recursive scheme and yields various representations of (directed) digraphs or (undirected) graphs: (Sierpiński-like [15], hexagonal) arrowheads or (lozenged, orthogonal) diamonds.…”

“…The paper is concerned with a family of tori (i.e. arrowhead and diamond) which was defined on the triangular (or "hexavalent") grid [1]. Their related interconnection networks have several important advantages.…”

On Patterns and Dynamics of Rule 22 Cellular Automaton

“…It may turn into something like a natural tree by some diffeomorphism. This tree is embeddable into the 2d diffusion graphs embedded into the triangulate lattice [44,45]: its vertex dust forms the Sierpiński gasket patterns. Sierpiński gasket is often known as a Banach fixed point from some contractive affine transformation into three elements.…”

Complete Rotations in Cayley Graphs

“…In particular, Corollary 3.1 can be applied to digraphs. For example, the digraphs defined as arrowheads in [9] have a complete rotation since they can be defined as the Cayley digraphs on the groups G n = (S|R n ) with S = {s 1 , s 2 , s 3 } and R n = {s 1 s 2 s 3 , s 1 s 2 s…”

“…One can think about asking the following question: if Cay(G, S) is rotational, is it possible to find a symmetric presentation that is also minimal with respect to the inclusion? For example, in the case of arrowheads the presentation of G n given in [9] is minimal but not symmetric: (S|R n ), with R n = {s 1 s 2 s 3 , s 1 s 2 s…”