arXiv 2401.085282024-01math.CO

聚合物的pebbling数

Pebbling number of polymers

Fatemeh Aghaei, Saeid Alikhani

备注：: 15 pages, 9 figures

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AI总结：

本文研究了由单体单元经点附着构成的聚合物图的pebbling数，通过定义pebbling配置和步骤，探讨了使任意配置可解的最小pebble数量。

摘要：

设$G=(V,E)$为一个简单图。函数$f:V\rightarrow \mathbb{N}\cup \{0\}$称为$G$顶点上的pebbling配置，量$\vert f\vert=\sum_{u\in V}f(u)$称为$f$的权重，即分配给顶点的pebble总数。从顶点$u$到其邻居$v$的pebbling步骤使$f(u)$减少二，使$f(v)$增加一。若对于每个顶点$v$，都存在一个（可能为空的）pebbling移动序列，使得最终在$v$上有一个pebble，则称该pebbling配置$f$是可解的。pebbling数$π(G)$等于最小的数$k$，使得每个权重$\vert f\vert = k$的pebbling配置$f$都是可解的。设$G$是由两两不相交的连通图$G_1,...,G_k$通过选择$G_1$的一个顶点、$G_2$的一个顶点并等同这两个顶点而构成的连通图，然后归纳地继续此过程。我们称$G$为polymer graph，由单体单元$G_1,...,G_k$经点附着得到。本文研究了一些聚合物的pebbling数。

英文摘要：

Let $G=(V,E)$ be a simple graph. A function $f:V\rightarrow \mathbb{N}\cup \{0\}$ is called a configuration of pebbles on the vertices of $G$ and the quantity $\vert f\vert=\sum_{u\in V}f(u)$ is called the weight of $f$ which is just the total number of pebbles assigned to vertices. A pebbling step from a vertex $u$ to one of its neighbors $v$ reduces $f(u)$ by two and increases $f(v)$ by one. A pebbling configuration $f$ is said to be solvable if for every vertex $ v $, there exists a sequence (possibly empty) of pebbling moves that results in a pebble on $v$. The pebbling number $ π(G) $ equals the minimum number $ k $ such that every pebbling configuration $ f $ with $ \vert f\vert = k $ is solvable. Let $ G $ be a connected graph constructed from pairwise disjoint connected graphs $ G_1,...,G_k $ by selecting a vertex of $ G_1 $, a vertex of $ G_2 $, and identifying these two vertices. Then continue in this manner inductively. We say that $ G $ is a polymer graph, obtained by point-attaching from monomer units $ G_1,...,G_k $. In this paper, we study the pebbling number of some polymers.

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