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Product-cordial index and friendly index of regular graphs. (English) Zbl 1293.05342
Summary: Let $$G=(V,E)$$ be a connected simple graph. A labeling $$f:V\to Z_2$$ induces two edge labelings $$f^+, f^*: E \to Z_2$$ defined by $$f^+(xy) = f(x)+f(y)$$ and $$f^*(xy) = f(x)f(y)$$ for each $$xy \in E$$. For $$i \in Z_2$$, let $$v_f(i) = |f^{-1}(i)|$$, $$e_{f^+}(i) = |(f^{+})^{-1}(i)|$$ and $$e_{f^*}(i) = |(f^*)^{-1}(i)|$$. A labeling $$f$$ is called friendly if $$|v_f(1)-v_f(0)| \leq 1$$. For a friendly labeling $$f$$ of a graph $$G$$, the friendly index of $$G$$ under $$f$$ is defined by $$i^+_f(G) = e_{f^+}(1)-e_{f^+}(0)$$. The set $$\{i^+_f(G) \mid f\mathrm{ is a friendly labeling of }G\}$$ is called the full friendly index set of $$G$$. Also, the product-cordial index of $$G$$ under $$f$$ is defined by $$i^*_f(G) = e_{f^*}(1)-e_{f^*}(0)$$. The set $$\{i^*_f(G) \mid f\mathrm{ is a friendly labeling of }G\}$$ is called the full product-cordial index set of $$G$$. In this paper, we find a relation between the friendly index and the product-cordial index of a regular graph. As applications, we will determine the full product-cordial index sets of torus graphs which was asked by H. Kwong et al. [Congr. Numerantium 206, 139–150 (2010; Zbl 1218.05169)] and those of cycles.

##### MSC:
 05C78 Graph labelling (graceful graphs, bandwidth, etc.) 05C25 Graphs and abstract algebra (groups, rings, fields, etc.)