Following earlier work by Coons-Maraj-Misra-Sorea and Misra-Sullivant, we study colored, undirected Gaussian graphical models, and present a necessary and sufficient condition for such a model to have binomial vanishing ideal. These conditions involve Jordan schemes, a variant of association schemes, well-known structures in algebraic combinatorics. Using association schemes without transitive group action, we refute the conjecture by Coons-Maraj-Misra-Sorea that binomiality implies that the color classes must be orbits under the automorphism group of the colored graph.

翻译：继 Coons-Maraj-Misra-Sorea 与 Misra-Sullivant 的早期工作之后，我们研究了着色的无向高斯图模型，并给出了此类模型具有二项消失理想的充分必要条件。这些条件涉及 Jordan 方案——一种关联方案的变体，后者是代数组合学中众所周知的结构。通过使用不具有传递群作用的关联方案，我们反驳了 Coons-Maraj-Misra-Sorea 的猜想，即二项性意味着着色类必须是着色图自同构群作用下的轨道。