product-cordial index and friendly index of regular graphs

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 $|v_f(1)-v_f(0)| le 1$. 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 ${i^+_f(g) | f is a friendly labeling of g}$ 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 ${i^*_f(g) | f is a friendly labeling of g}$ 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.