On Friendly Index Sets of Cyclic Silicates
Received:October 16, 2014  Revised:July 08, 2015
Key Word: vertex labeling   friendly labeling   cordiality   friendly index set   cycle   CS$(n, m)$   arithmetic progression
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11371109).
 Author Name Affiliation Zhenbin GAO College of Science, Harbin Engineering University, Heilongjiang 150001, P. R. China Sinmin LEE Deptartment of Computer Science, San Jose State University, San Jose CA95192, USA Guangyi SUN College of Science, Harbin Engineering University, Heilongjiang 150001, P. R. China Geechoon LAU Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA (Segamat Campus), Johor 85000, Malaysia
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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$(n, m)$.