Product-cordial index and friendly index of regular graphs

Abstract

Let $G=(V,E)$ be a connected simple graph‎. ‎A labeling $f‎: ‎Vto Z_2$ 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 $|v_f(1)-v_f(0)| le 1$‎. ‎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 ${i^+_f(G);|;f mbox{ is a friendly labeling of}‎ ‎G}$ 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 ‎${i^*_f(G);|;f mbox{ is a friendly labeling of} G}$ 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 Kwong‎, ‎Lee‎ ‎and Ng in 2010; and those of cycles‎.