Peter J. Cameron, Elizabeth J. Billington, Sanming Zhou
Abstract
Every graph can be associated with a family of homogeneous polynomials, one for every degree, having as many variables as the number of vertices.
These polynomials are related to graceful labellings: a graceful polynomial with all even coefficients is a basic tool, in some cases, for proving
that a graph is non-graceful, and for generating a possibly infinite class of non-graceful graphs. Graceful polynomials also seem interesting in their
own right. In this paper we classify graphs whose graceful polynomial has all even coefficients, for small degrees up to 4. We also obtain some
new examples of non-graceful graphs