arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1602.01561math.CO
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$k$ -几何图

$k$-geometric graphs

Penying Rochanakul

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摘要：

一个有$p$顶点和$q$边的有限简单无向图$G = (V, E)$称为正整数$k$的$k$ -几何平均图，如果存在注入$ψ:V(G)\to \{k,k+1,\dots,k+q\}$，使得当每个边$uv\in E(G)$被赋予标签$\lfloor\sqrt{ψ(u)ψ(v)}\rfloor$或$\lceil\sqrt{ψ(u)ψ(v)}\rceil$时，得到的边标签集是$\{k,k+1,...,k+q-1\}$，$ψ$称为$G$的一个\emph{$k$-geometric mean labeling}。特殊情况$k=1$，一个$1$ -几何平均标号称为几何平均标号，一个$1$ -几何平均图称为几何平均图.本文提出了一类新的几何平均图。然后引入了$k$ -几何平均标号，并证明了某些图类是$k$ -几何平均的.我们还研究了几类允许几何平均标号的有限联图。

英文摘要：

A finite, simple and undirected graph $G = (V, E)$ with $p$ vertices and $q$ edges is said to be a $k$-geometric mean graph for a positive integer $k$ if there is an injection $ψ:V(G)\to \{k,k+1,\dots,k+q\}$ such that, when each edge $uv\in E(G)$ is assigned the label $\lfloor\sqrt{ψ(u)ψ(v)}\rfloor$ or $\lceil\sqrt{ψ(u)ψ(v)}\rceil$, the resulting edge label set is $\{k,k+1,...,k+q-1\}$ and $ψ$ is called a \emph{$k$-geometric mean labeling} of $G$. The special case $k=1$, a $1$-geometric mean labeling is called a geometric mean labeling, and a $1$-geometric mean graph is called a geometric mean graph. In this paper, we present new classes of geometric mean graphs. Then we introduce $k$-geometric mean labeling and prove some classes of graphs are $k$-geometric mean. We also study some classes of finite join of graphs that admit geometric mean labeling.

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