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

Abstract

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 $Σ_{e\in E}f(e)$ or $Σ_{v\in V}g(v)$, respectively. Suppose that parameters $λ$ and $ξ$ 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 $λ(G_1)=h(n)$ (respectively, $ξ(G_2)=h(n)$), 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 $C_4\Box C_n$.