Independence equivalence classes of cycles

TL;DR

This paper completely characterizes the independence equivalence classes of odd cycles, extending previous work that only covered even cycles, and provides a comprehensive understanding of graphs sharing the same independence polynomial.

Contribution

The paper solves the open problem of determining the independence equivalence class of odd cycles, completing the classification for all cycle graphs.

Findings

Independence equivalence classes of odd cycles are fully characterized.

The classification extends the known results from even cycles.

Provides a complete description of graphs with identical independence polynomials for odd cycles.

Abstract

The independence equivalence class of a graph $G$ is the set of graphs that have the same independence polynomial as $G$. Beaton, Brown and Cameron (arXiv:1810.05317) found the independence equivalence classes of even cycles, and raised the problem of finding the independence equivalence class of odd cycles. The problem is completely solved in this paper.