Given a graph $G=(V,E)$, a $1-1$ mapping $f:V\rightarrow\{1,2,...,n\}$ is called a proper numbering of G. Then we know, for a proper mapping $f$, the edgesum of the $s_f(G)$ produced by $f$ is defined as \begin{equation} s_f(G)=\sum_{uv\in E}|f(u)-f(v)|. \end{equation} The edgesum $s(G)$ of a graph $G$ is defined by \begin{equation} s(G) = min\{s_f(G): \mathrm{f\ is\ a\ proper\ numbering\ of\ G}\}. \end{equation}
Then consider a function as the sum of square of edges of $G$ with a given $f$,i.e., \begin{equation} \sum_{uv\in E}|f(u)-f(v)|^2. \end{equation} Is there any name of the minimization problem of this function in mathematics? Or is there any results when $G$ is a path or a cycle, or a Cartesian product of paths or cycles?