Cayley digraphs of finite abelian groups and monomial ideals

In the study of double-loop computer networks, the diagrams known as L-shapes arise as a graphical representation of an optimal routing for every graph’s node. The description of these diagrams provides an efficient method for computing the diameter and the average minimum distance of the corresponding graphs. We extend these diagrams to multiloop computer networks. For each Cayley digraph with a finite abelian group as vertex set, we define a monomial ideal and consider its representations via its minimal system of generators or its irredundant irreducible decomposition. From this last piece of information, we can compute the graph’s diameter and average minimum distance. That monomial ideal is the initial ideal of a certain lattice with respect to a graded monomial ordering. This result permits the use of Gr¨obner bases for computing the ideal and finding an optimal routing. Finally, we present a family of Cayley digraphs parametrized by their diameter d, all of them associated to irreducible monomial ideals

The multidimensional Manhattan networks

The $n$-dimensional Manhattan network $M_n$---a special case of $n$-regular digraph---is formally defined and some of its structural properties are studied. In particular, it is shown that $M_n$ is a Cayley digraph, which can be seen as a subgroup of the $n$-dim version of the wallpaper group $pgg$. These results induce a useful new presentation of $M_n$, which can be applied to design a (shortest-path) local routing algorithm and to study some other metric properties. Also it is shown that the $n$-dim Manhattan networks are Hamiltonian and, in the standard case (that is, dimension two), they can be decomposed in two arc-disjoint Hamiltonian cycles. Finally, some results on the connectivity and distance-related parameters of $M_n$, such as the distribution of the node  distances and the diameter are presented