Triple-loop networks with arbitrarily many minimum distance diagrams

TL;DR

This paper demonstrates that triple-loop networks can have an arbitrarily large number of minimum distance diagrams, contrasting with the limited diagrams in double-loop networks, by exploring their relation to monomial ideals.

Contribution

It introduces the existence of multiple minimum distance diagrams in triple-loop networks and links these diagrams to monomial ideals, expanding understanding of network routing structures.

Findings

Triple-loop networks can have arbitrarily many minimum distance diagrams.

Double-loop networks have at most two such diagrams.

The relation between diagrams and monomial ideals is established.

Abstract

Minimum distance diagrams are a way to encode the diameter and routing information of multi-loop networks. For the widely studied case of double-loop networks, it is known that each network has at most two such diagrams and that they have a very definite form "L-shape''. In contrast, in this paper we show that there are triple-loop networks with an arbitrarily big number of associated minimum distance diagrams. For doing this, we build-up on the relations between minimum distance diagrams and monomial ideals.