Independence Equivalence Classes of Paths and Cycles

TL;DR

This paper investigates the structure of graphs sharing the same independence polynomial, revealing unique classes for odd paths, specific classes for even cycles, and partial results for cycles of prime power length.

Contribution

It extends previous work by characterizing independence equivalence classes for paths and cycles, including new results for even cycles and prime power cycles.

Findings

Odd paths have singleton independence equivalence classes.

Even cycles (except C6) have classes of size two.

For prime p ≥ 5, cycles C_{p^n} have classes of size two.

Abstract

The independence polynomial of a graph is the generating polynomial for the number of independent sets of each size. Two graphs are said to be \textit{independence equivalent} if they have equivalent independence polynomials. We extend previous work by showing that independence equivalence class of every odd path has size 1, while the class can contain arbitrarily many graphs for even paths. We also prove that the independence equivalence class of every even cycle consists of two graphs when $n\ge 2$ except the independence equivalence class of $C_6$ which consists of three graphs. The odd case remains open, although, using irreducibility results from algebra, we were able show that for a prime $p \geq 5$ and $n\ge 1$ the independence equivalence class of $C_{p^n}$ consists of only two graphs.