The weak tensor product was introduced by Snevily as a way to construct new graphs that admit $α$-labelings from a pair of known $α$-graphs. In this article, we show that this product and the application to $α$-labelings can be generalized by considering as a second factor of the product, a family $Γ$ of bipartite $(p,q)$-graphs, $p$ and $q$ fixed. The only additional restriction that we should consider is that for every $F\in Γ$, there exists and $α$-labeling $f_F$ with $f_F(V(F))=L\cup H$, where $L,H \subset [0,q]$ are the stable sets induced by the characteristic of $f_F$ and they do not depend on $F$. We also obtain analogous applications to near $α$-labelings and bigraceful labelings.
Quivers with subadditive labelings: classification and integrability
第二作者引入了箭图的严格次可加、次可加和弱次可加标号，推广了Vinberg对无向图的定义。在我们以前的工作中，我们已经表明，箭图与严格次加标签正是表现Zamolodchikov周期性的箭图。在本文中，我们分类所有的箭图与次加标签。我们猜想他们表现出某种形式的可积性，即，作为$T$ -系统动力学进行，在每个顶点的值满足线性递归。相反，我们表明，在这个意义上的每一个可积的必然是我们的分类中的$19$项目之一。对于$\hat A \otimes A$类型的箭图，我们用圆柱多米诺平铺的分拆函数表示递归的系数，称为\emph{Goncharov-Kenyon Hamiltonians}。我们还考虑热带$T$ -系统的类型$\hat A \otimes A$和解释如何仿射切片表现出孤子的行为，即孤子的分辨率和速度守恒。在整个过程中，我们推测如何在纸上的结果预计推广从$\hat A \otimes A$到所有其他颤抖在我们的分类。
图$G$的$L(1,1)$ -标记是从$\{0,1 \cdots, k \}$到$G$的顶点的标签分配，使得相邻或具有共同邻居的两个顶点接收不同的标签。$G$的$λ_1^1-$编号$λ_1^1(G)$是$G$允许$L(1,1)$标记的最小值$k$。对于所有正的$m, n \geq 3$，我们建立了圈$C_m \times C_n$的直积的$λ_1^1-$数，其中两个$m,n$都是偶数或其中一个是偶数而另一个是奇数.
