Signless Laplacian spectral radius for a k-extendable graph

Let $k$ and $n$ be two nonnegative integers with $n\equiv0$ (mod 2), and let $G$ be a graph of order $n$ with a 1-factor. Then $G$ is said to be $k$-extendable for $0\leq k\leq\frac{n-2}{2}$ if every matching in $G$ of size $k$ can be extended to a 1-factor. In this paper, we first establish a lower bound on the signless Laplacian spectral radius of $G$ to ensure that $G$ is $k$-extendable. Then we create some extremal graphs to claim that all the bounds derived in this article are sharp.Comment: 11 page

Sufficient conditions for fractional [a,b]-deleted graphs

Let $a$ and $b$ be two positive integers with $a\leq b$, and let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. Let $h:E(G)\rightarrow[0,1]$ be a function. If $a\leq\sum\limits_{e\in E_G(v)}{h(e)}\leq b$ holds for every $v\in V(G)$, then the subgraph of $G$ with vertex set $V(G)$ and edge set $F_h$, denoted by $G[F_h]$, is called a fractional $[a,b]$-factor of $G$ with indicator function $h$, where $E_G(v)$ denotes the set of edges incident with $v$ in $G$ and $F_h=\{e\in E(G):h(e)>0\}$. A graph $G$ is defined as a fractional $[a,b]$-deleted graph if for any $e\in E(G)$, $G-e$ contains a fractional $[a,b]$-factor. The size, spectral radius and signless Laplacian spectral radius of $G$ are denoted by $e(G)$, $\rho(G)$ and $q(G)$, respectively. In this paper, we establish a lower bound on the size, spectral radius and signless Laplacian spectral radius of a graph $G$ to guarantee that $G$ is a fractional $[a,b]$-deleted graph.Comment: 1

Quantity and quality of China's water from demand perspectives

China is confronted with an unprecedented water crisis regarding its quantity and quality. In this study, we quantified the dynamics of China's embodied water use and chemical oxygen demand (COD) discharge from 2010 to 2015. The analysis was conducted with the latest available water use data across sectors in primary, secondary and tertiary industries and input-output models. The results showed that (1) China's water crisis was alleviated under urbanisation. Urban consumption occupied the largest percentages (over 30%) of embodied water use and COD discharge, but embodied water intensities in urban consumption were far lower than those in rural consumption. (2) The 'new normal' phase witnessed the optimisation of China's water use structures. Embodied water use in light-manufacturing and tertiary sectors increased while those in heavy-manufacturing sectors (except chemicals and transport equipment) dropped. (3) Transformation of China's international market brought positive effects on its domestic water use. China's water use (116-80 billion tonnes (Bts))9 and COD discharge (3.95-2.22 million tonnes (Mts)) embodied in export tremendously decreased while its total export values (11-25 trillion CNY) soared. Furthermore, embodied water use and COD discharge in relatively low-end sectors, such as textile, started to transfer from international to domestic markets when a part of China's production activities had been relocated to other developing countries