环状硅酸盐图的友好指数集

On Friendly Index Sets of Cyclic Silicates

中文关键词:  顶点标号  友好标号  亲切性  友好指数集  圈  $CS(n, m)$  算术级数

英文关键词:vertex labeling  friendly labeling  cordiality  friendly index set  cycle  CS$(n, m)$  arithmetic progression

中文摘要:

      设图$G$ 的顶点集是$V(G)$,边集是 $E(G)$. 一个顶点标号$f: V(G)\rightarrow Z_{2}$导出一个定义为 $f^{*}(xy) = f(x) + f(y)$ 的边标号（$ xy$ 是边）$ f^{*} : E(G)\rightarrow Z_{2}$.对于$i\in Z_{2}$,令$ v_{f}(i) =|\{v\in V(G) : f(v) = i\}|$ 和 $e_{f}(i) = |\{e\in E(G) : f^{*}(e) = i\}|$. 如果$| v_{f}(0)-v_{f}(1) | \leq 1$,则$G$ 的标号$f$ 被称为友好的. 图$G$的一个友好指标集, 记为$FI(G)$, 定义为 $\{|e_{f}(0) - e_{f}(1)|$: 顶点标号$f$ 是友好的$\}$. 这是图亲切性的广义化.我们研究环状硅酸盐图$CS(n, m)$的友好指数集.

英文摘要:

      Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. A labeling $f : V(G)\rightarrow Z_{2}$ induces an edge labeling $ f^{*} : E(G)\rightarrow Z_{2}$ defined by $f^{*}(xy) = f(x) + f(y)$, for each edge $ xy\in E(G)$. For $i \in Z_{2}$, let $ v_{f}(i) =|\{v \in V(G) : f(v) = i\}|$ and $e_{f}(i) = |\{e\in E(G) : f^{*}(e) = i\}|$. A labeling $f$ of a graph $G$ is said to be friendly if $| v_{f}(0)-v_{f}(1) | \leq 1$. The friendly index set of the graph $G$, denoted ${\rm FI}(G)$, is defined as $\{|e_{f}(0) - e_{f}(1)|$: the vertex labeling $f$ is friendly$\}$. This is a generalization of graph cordiality. We investigate the friendly index sets of cyclic silicates CS$(n, m)$.