拓扑可加集序列图
Topological Integer Additive Set-Sequential Graphs
摘要：
令$\mathbb{N}_0$表示所有非负整数的集合，并且$X$是$\mathbb{N}_0$的任何非空子集。将$X$的幂集表示为$\mathcal{P}(X)$。图$G$的整数可加集标号（IASL）是一个内射集值函数$f:V(G)\to \mathcal{P}(X)$，使得导出函数$f^+:E(G) \to \mathcal{P}(X)$由$f^+ (uv) = f(u)+ f(v)$定义，其中$f(u)+f(v)$是$f(u)$与$f(v)$的和集.如果相关联的集值边函数$f^+$也是单射的，则这样的IASL被称为整数加性集索引器（IASI）。如果$f(V(G))\cup \{\emptyset\}$是基集$X$的拓扑，则称IASL $f$是拓扑IASL（TIASL）。如果$f(V(G))\cup f^+(E(G))= \mathcal{P}(X)-\{\emptyset\}$，则称IASL为整数加性集序标记（IASSL）。给定图$G$的IASL称为$G$的拓扑整数加集序标号，如果它既是$G$的拓扑整数加集序标号，又是$G$的整数加集序标号.在本文中，我们研究了图$G$承认这种类型的IASL所需的条件，并提出了一些重要的特性，承认这种类型的IASL的图。
英文摘要：
Let $\mathbb{N}_0$ denote the set of all non-negative integers and $X$ be any non-empty subset of $\mathbb{N}_0$. Denote the power set of $X$ by $\mathcal{P}(X)$. An integer additive set-labeling (IASL) of a graph $G$ is an injective set-valued function $f:V(G)\to \mathcal{P}(X)$ such that the induced function $f^+:E(G) \to \mathcal{P}(X)$ is defined by $f^+ (uv) = f(u)+ f(v)$, where $f(u)+f(v)$ is the sumset of $f(u)$ and $f(v)$. If the associated set-valued edge function $f^+$ is also injective, then such an IASL is called an integer additive set-indexer (IASI). An IASL $f$ is said to be a topological IASL (TIASL) if $f(V(G))\cup \{\emptyset\}$ is a topology of the ground set $X$. An IASL is said to be an integer additive set-sequential labeling (IASSL) if $f(V(G))\cup f^+(E(G))= \mathcal{P}(X)-\{\emptyset\}$. An IASL of a given graph $G$ is said to be a topological integer additive set-sequential labeling of $G$, if it is a topological integer additive set-labeling as well as an integer additive set-sequential labeling of $G$. In this paper, we study the conditions required for a graph $G$ to admit this type of IASL and propose some important characteristics of the graphs which admit this type of IASLs.
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