$\ga$-Max Labelings of Graphs

Let $G$ be a graph of order $n$ and size $m$. A $\gamma$-labeling  of $G$ is a one-to-one function $f: V(G) \rightarrow \{0, 1, 2, \ldots, m\}$ that induces an edge-labeling$f': E(G) \rightarrow \{1, 2, \ldots, m\}$ on $G$ defined by$$ f'(e)=|f(u)-f(v)|\, ,\quad \mbox{for each edge $e=uv$  in}\, E(G)\, . $$ The value of $f$ is defined as $${\rm val}(f)=\sum_{e\in E(G)}f'(e)\, . $$ The  maximum value of a \gamma$-labeling of  $G$ is defined as $${\val}_{\max}(G) =  \max \{\val(f): \mbox{$f$ is a $$-labeling of $G$}\};$$ while  the minimum value of a $\gamma$-labeling of  $G$  is $${\val}_{\min}(G) = \min \{\val(f): \mbox{$f$ is a $\gamma$-labeling of $G$}\}.$$ In this paper, we give an alternative short proof by mathematical induction to achieve the formulae for ${\val}_{\max}(K_{r, s})$ and ${\val}_{\max}(K_n)$.