Abstract
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 Gröbner 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.
Highlights
Let Γ be a group and S ⊆ Γ a subset
If S is a subset of ZN such that for every element in S its inverse lies in S, CN (S) is an undirected graph called a circulant graph or distributed multiloop computer network
In this article we present monomial ideals as a natural tool for studying the minimum distance diagram (MDD) of arbitrary Cayley digraphs, provided that the vertex group is finite and CAYLEY DIGRAPHS AND MONOMIAL IDEALS
Summary
Let Γ be a group and S ⊆ Γ a subset. The Cayley digraph associated to (Γ, S) is a directed graph whose vertex set is Γ and whose edge set is {(g, h) ∈ Γ2 | g−1h ∈ S}. We present a new and attractive family of circulant digraphs of arbitrary degree parametrized by the diameter d, with average minimum distance d/2, and whose associated monomial ideals are irreducible. We say that an MDD built following Algorithm 3.1 is an L-shape if the associated monomial ideal is an L-shape.
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