A necessary and sufficient condition for bounds on the sum of a list of real numbers and its applications

Let $x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n$ be a list of real numbers, let $s:={\sum }_{i=1}^n{x}_i$ and let $h:\mathbb{N}\to \mathbb{R}$ 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 $\{H_1,\mathrm{.}\mathrm{.}\mathrm{.},H_n\}$ and $\{V_1,\mathrm{.}\mathrm{.}\mathrm{.},V_n\}$ be a decomposition and a partition of $G$, respectively. Let $H_{i,j}$ and $V_{i,j},i\le j,$ be the union of $H_i,\mathrm{.}\mathrm{.}\mathrm{.},H_j$ and $V_i,\mathrm{.}\mathrm{.}\mathrm{.},V_j$, 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 $k$, $1\le k\le n$, respectively. Let $f:E\to \mathbb{R}$ and $g:V\to \mathbb{R}$ be mappings, let the \emph{weight} of $f$ or $g$ on $G$ be ${\mathrm{\Sigma }}_{e\in E}f(e)$ or ${\mathrm{\Sigma }}_{v\in V}g(v)$, 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 $\text{cr}(\odot (T^n))$ of a periodic graph $\odot (T^n)$, $\text{cr}(\odot (T^n))=h(n)$. As applications, we obtained $\text{cr}(C(4n;\{1,4\}))$ of the circulant $C(4n;\{1,4\})$, the paired domination number of $C_5\mathrm{\square }{C}_n$ and the upper total domination number of $C_4\mathrm{\square }{C}_n$.