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 $C_4\Box C_n$.
Categories: math.CO