  Let $x_1,...,x_n$ be a list of real numbers, let $s :=\sum_{i=1}^{n}x_i$ and
let $h:\mathbb{N} \rightarrow \mathbb{R}$ be a function. We gave a necessary
and sufficient condition for $s>h(n)$ (respectively, $s<h(n)$). Let $G=(V,E)$
be a graph, let $\{H_1,...,H_n\}$ and $\{V_1,...,V_n\}$ be a decomposition and
a partition of $G$, respectively. Let $H_{i,j}$ and $V_{i,j}, i\leq j,$ be the
union of $H_i,...,H_j$ and $V_i,...,V_j$, respectively, where subscripts are
taken modulo $n$. $G$ is \emph{generalized periodic} or
\emph{partition-transitive} if for each pair of integers $(i,j)$, $H_{i,i+k}$
and $H_{j,j+k}$ or $V_{i,i+k}$ and $V_{j,j+k}$ are isomorphic for all $k$,
$1\leq k\leq n$, respectively. Let $f:E \rightarrow \mathbb{R}$ and $g:V
\rightarrow \mathbb{R}$ be mappings, let the \emph{weight} of $f$ or $g$ on $G$
be $\Sigma_{e\in E}f(e)$ or $\Sigma_{v\in V}g(v)$, respectively. Suppose that
parameters $\lambda$ and $\xi$ of $G$ can be expressed as the minimum or
maximum weight of specified $f$ and $g$, respectively. Then our conditions
imply a necessary and sufficient condition for $\lambda(G_1)=h(n)$
(respectively, $\xi(G_2)=h(n)$), where $G_1$ is generalized periodic and $G_2$
is partition-transitive. For example, the crossing number
$\textrm{cr}(\odot(T^n))$ of a periodic graph $\odot(T^n)$,
$\textrm{cr}(\odot(T^n))=h(n)$. As applications, we obtained
$\textrm{cr}(C(4n;\{1,4\}))$ of the circulant $C(4n;\{1,4\})$, the paired
domination number of $C_5\Box C_n$ and the upper total domination number of
$C_4\Box C_n$.
