On z-Superstable and Critical Configurations of Chip Firing Pairs

TL;DR

This paper extends the duality between superstable and critical configurations from graphs to all (L,M)-chip firing pairs, exploring properties of this generalized duality.

Contribution

It introduces a natural extension of the duality map to all (L,M)-chip firing pairs, broadening the scope beyond previous graph-based results.

Findings

Extended duality map to all (L,M)-chip firing pairs

Analyzed properties of the generalized duality map

Connected previous results to a broader class of chip firing pairs

Abstract

It is well known that there is a duality map between the superstable configurations and the critical configurations of a graph. This was extended to all M-matrices in (Guzm\`an-Klivans 2015). We show a natural way to extend this to all $(L,M)$-chip firing pairs introduced in (Guzm\`an-Klivans 2016). In addition, we study various properties of this map.