Cyclic Sieving Phenomenon for Independent sets of graphs

TL;DR

This paper explores the cyclic sieving phenomenon in the context of independent sets in graphs, providing new examples, formulas, and recursive techniques for various graph classes.

Contribution

It introduces new cyclic sieving phenomena for independent sets in powers of cycle and path graphs, and develops recursive methods for complex graph classes.

Findings

Cyclic sieving phenomena are exhibited for powers of cycle graphs.

A closed formula for the number of independent sets in these graphs is derived.

Recursive techniques are developed for gear, helm, and book graphs.

Abstract

In this paper, we present examples of the cyclic sieving phenomenon coming from studying independent sets in graphs of a fixed size k. Given a graph G, and a cyclic group C acting on the graph, then C also acts on the collection of independent sets of G of a fixed size k. We exhibit cyclic sieving phenomena for a cyclic group acting on the collection of independent sets of powers of cycle graphs. As a corollary, we also find a closed formula for the number of independent sets of a given size in the power of a cycle graph, and in the power of a path. We also show how the graph construction of whiskering can be used to obtain new cyclic sieving phenomena from old phenomena. We also discuss recursive techniques to exhibit cyclic sieving phenomena for the independent sets of gear graphs, helm graphs, and book graphs.