可加集值图的创新性研究
A Creative Review on Integer Additive Set-Valued Graphs
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$G$的集合索引是一个内射集值函数$f:V(G) \to \mathcal{P}(X)$，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被定义为内射函数$f:V(G)\to \mathcal{P}({\mathbb{N}_0})$，使得由$f^+ (uv) = f(u)+ f(v)$定义的导出函数$f^+:E(G) \to \mathcal{P}(\mathbb{N}_0)$也是内射的，其中$\mathbb{N}_0$是所有非负整数的集合。本文对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 is defined as an injective function $f:V(G)\to \mathcal{P}({\mathbb{N}_0})$ such that the induced function $f^+:E(G) \to \mathcal{P}(\mathbb{N}_0)$ defined by $f^+ (uv) = f(u)+ f(v)$ is also injective, where $\mathbb{N}_0$ is the set of all non-negative integers. In this paper, we critically and creatively review the concepts and properties of 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。
