弱可加集指标图：一个创造性的综述
Weak Integer Additive Set-Indexed Graphs: A Creative Review
摘要：
对于有限或无限的非空基集$X$，给定图$G$的{ \em集赋值}或{ \em集标号}是一个内射函数$f:V(G) \to \mathcal{P}(X)$，其中$\mathcal{P}(X)$是集合$X$的幂集.图$G$的集指标是内射集值函数$f:V(G) \to \mathcal{P}(X)$，使得由$f^{\ast}(uv) = f(u){\ast} f(v)$对每个$uv{\in} E(G)$定义的函数$f^{\ast}:E(G)\to \mathcal{P}(X)-\{\emptyset\}$也是内射的。其中$\ast$是集合上的二元运算。整数加性集合索引器（IASI）被定义为内射函数$f:V(G)\to \mathcal{P}({\mathbb{N}_0})$，使得由$g_f (uv) = f(u)+ f(v)$定义的导出函数$g_f:E(G) \to \mathcal{P}(\mathbb{N}_0)$也是内射的，其中$\mathbb{N}_0$是所有非负整数的集合，$\mathcal{P}(\mathbb{N}_0)$是它的幂集.弱IASI是IASI $f$，使得$|f^+(uv)|= \text{max}(f(u),f(v))$。本文对弱整数可加集值图的概念和性质进行了批判性和创造性的综述。
英文摘要：
For a non-empty ground set $X$, finite or infinite, the {\em set-valuation} or {\em set-labeling} of a given graph $G$ is an injective function $f:V(G) \to \mathcal{P}(X)$, where $\mathcal{P}(X)$ is the power set of the set $X$. A set-indexer of a graph $G$ is an injective set-valued function $f:V(G) \to \mathcal{P}(X)$ such that the function $f^{\ast}:E(G)\to \mathcal{P}(X)-\{\emptyset\}$ defined by $f^{\ast}(uv) = f(u){\ast} f(v)$ for every $uv{\in} E(G)$ is also injective., where $\ast$ is a binary operation on sets. An integer additive set-indexer (IASI) is defined as an injective function $f:V(G)\to \mathcal{P}({\mathbb{N}_0})$ such that the induced function $g_f:E(G) \to \mathcal{P}(\mathbb{N}_0)$ defined by $g_f (uv) = f(u)+ f(v)$ is also injective, where $\mathbb{N}_0$ is the set of all non-negative integers and $\mathcal{P}(\mathbb{N}_0)$ is its power set. A weak IASI is an IASI $f$ such that $|f^+(uv)|= \text{max}(f(u),f(v))$. In this paper, we critically and creatively review the concepts and properties of weak integer additive set-valued graphs.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
