Counterexamples to a conjecture of Balasubramanian and Parthasarathy

TL;DR

This paper provides counterexamples to a longstanding conjecture by Balasubramanian and Parthasarathy that the bivariate permanent polynomial uniquely characterizes graphs, showing that different graphs can share the same polynomial.

Contribution

The paper presents the first known counterexamples disproving the conjecture that the bivariate permanent polynomial is a graph characterizing polynomial.

Findings

Counterexamples exist where non-isomorphic graphs share the same bivariate permanent polynomial

The conjecture by Balasubramanian and Parthasarathy is false

Graph polynomials may not always uniquely identify graph isomorphism

Abstract

In 1980, Balasubramanian and Parthasarathy introduced the bivariate permanent polynomials of graphs and conjectured that this graph polynomial is a graph characterising polynomial, that is, any two graphs with the same bivariate permanent polynomial are isomorphic. In this note, we give counter examples to this conjecture.