关于5-正则Lin-Lu-Yau Ricci平坦图的Lei-Bai猜想
On the Lei--Bai conjecture on $5$-regular Lin--Lu--Yau Ricci-flat graphs
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中文总结 AI 辅助
本文针对5-正则Lin-Lu-Yau Ricci平坦图的Lei-Bai猜想，构造了无限族既不同构于RF也无平凡笛卡尔积分解的连通图，否定了该猜想，表明此类图比此前认为的更丰富。
中文摘要 AI 辅助
我们研究由Lin、Lu和Yau引入的Ricci曲率。若图中每条边的曲率均为零，则称该图为Ricci平坦图。Lei和Bai通过证明每一个此类5-正则对称Ricci平坦图均同构于特定的72顶点图RF，对其进行了分类，并提出猜想：每一个5-正则Ricci平坦图要么同构于RF，要么具有非平凡笛卡尔积分解。本文中，我们通过构造无限族连通5-正则Ricci平坦图，对该猜想予以否定，此类图中无一同构于RF，也无一个具有非平凡笛卡尔积分解。这表明将分类从对称情形推广至一般5-正则Ricci平坦图的猜想不成立，此类图的丰富程度远超出此前猜想。为确立这些结果，我们利用Lin-Lu-Yau曲率的最优分配公式验证所构造图的Ricci平坦性。
英文摘要
We study the Ricci curvature introduced by Lin, Lu, and Yau. A graph is called Ricci-flat if every edge has curvature zero. Lei and Bai classified $5$-regular symmetric Ricci-flat graphs by proving that every such graph is isomorphic to a particular $72$-vertex graph $\RF$, and conjectured that every $5$-regular Ricci-flat graph is either isomorphic to $\RF$ or admits a nontrivial Cartesian product decomposition. In this paper, we disprove this conjecture by constructing an infinite family of connected $5$-regular Ricci-flat graphs, none of which is isomorphic to $\RF$ or admits a nontrivial Cartesian product decomposition. This shows that the conjectured extension of the classification from the symmetric setting to general $5$-regular Ricci-flat graphs fails and that the class of such graphs is substantially richer than previously conjectured. To establish these results, we use an optimal-assignment formulation of Lin--Lu--Yau curvature to verify the Ricci-flatness of the constructed graphs.
