We raise some questions about graph polynomials, highlighting concepts and phenomena that may merit consideration in the development of a general theory. Our questions are mainly of three types: When do graph polynomials have reduction relations (simple linear recursions based on local operations), perhaps in a wider class of combinatorial objects? How many levels of reduction relations does a graph polynomial need in order to express it in terms of trivial base cases? For a graph polynomial, how are properties such as equivalence and factorisation reflected in the structure of a graph? We illustrate our discussion with a variety of graph polynomials and other invariants. This leads us to reflect on the historical origins of graph polynomials. We also introduce some new polynomials based on partial colourings of graphs and establish some of their basic properties.

翻译：我们提出了一些关于图多项式的问题，旨在强调在构建一般理论时可能需要考虑的概念与现象。我们的问题主要分为三类：图多项式何时具有归约关系（基于局部操作的简单线性递归），或许在更广泛的组合对象类中亦然？一个图多项式需要多少层归约关系才能用平凡基例表示？对于图多项式，等价性、因式分解等性质如何反映在图的结构中？我们通过各类图多项式及其他不变量来阐述讨论。这促使我们反思图多项式的历史起源。此外，我们基于图的部分着色引入了一些新多项式，并建立了它们的基本性质。