Graph characterising polynomials Original Research Article

The discovery of a complete set of invariants for graphs, or equivalently, a graph characterising polynomial has been a long outstanding problem in graph theory. In 1980 Balasubrahmanian and Parthasarathy introduced the bivariate permanent polynomial and the frame polynomial and conjectured that these are graph characterising polynomials. In this paper we show that these conjectures imply the Graph Reconstruction Conjectures and suggest a possible approach to settle the bivariate permanent polynomial conjecture using Mnukhinʹs (Acta Applicandae Mathematicae 29 (1992) 83–117) characterisation of a complete set of graph invariants. We also address the problem of determining the number of single vertex components of a graph from its permanent polynomial.