Cycle bases of reduced powers of graphs

TL;DR

This paper introduces the reduced k-th power of a graph as a configuration space for indistinguishable tokens, and provides a method to construct minimum cycle bases for these graphs, with implications for automata and Markov chains.

Contribution

It defines the reduced power of a graph and offers a novel construction for minimum cycle bases, enhancing understanding of their structural properties and applications.

Findings

Constructed minimum cycle bases for reduced graph powers

Provided conditions to prevent violations of microscopic reversibility

Linked cycle bases to automata transition graphs and Markov chains

Abstract

We define what appears to be a new construction. Given a graph $G$ and a positive integer $k$, the reduced $k$th power of $G$, denoted $G^{(k)}$, is the configuration space in which $k$ indistinguishable tokens are placed on the vertices of $G$, so that any vertex can hold up to $k$ tokens. Two configurations are adjacent if one can be transformed to the other by moving a single token along an edge to an adjacent vertex. The reduced power $G^{(k)}$ is the transition graph of the master Markov chain for $k$ identical and indistinguishable stochastic automata with transition graph $G$. We present propositions related to the structural properties of reduced graph powers and, most significantly, provide a construction of minimum cycle bases of $G^{(k)}$. Minimum cycle bases of reduced graph powers are especially relevant to applications that do not allow state-dependent coupling of automata to introduce nonequilibrium steady states. This paper also demonstrates how our minimum cycle basis construction provides conditions that ensure against violations of microscopic reversibility. In addition, the minimum cycle basis construction is an interesting combinatorial problem in itself.