Full Friendly Index Set-$II$

Abstract

Let $G=(V,E)$ be a graph. A vertex labeling $f:V\to {\mathbb{Z}}_2$ induces an edge labeling $f^{\ast }:E\to {\mathbb{Z}}_2$ defined by $f^{\ast }(xy)=f(x)+f(y)$ for each $xy\in E$. For each $i\in {\mathbb{Z}}_2$, define $v_f(i)=|f^{-1}(i)|$ and $e_f(i)=|{f^{\ast }}^{-1}(i)|$. We call $f$ friendly if $|v_f(1)–v_f(0)|\le 1$. The full friendly index set of $G$ is the set of all possible values of $e_f(1)–e_f(0)$, where $f$ is a friendly labeling. In this paper, we study the full friendly index set of the wheel $W_n$, the tensor product of paths $P_2$ and $P_n$, i.e., $P_2\otimes P_n$, and the double star $D(m,n)$.