608 research outputs found
Signless Laplacian spectral radius for a k-extendable graph
Let and be two nonnegative integers with (mod 2), and let
be a graph of order with a 1-factor. Then is said to be
-extendable for if every matching in of size
can be extended to a 1-factor. In this paper, we first establish a lower
bound on the signless Laplacian spectral radius of to ensure that is
-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 and be two positive integers with , and let be a
graph with vertex set and edge set . Let
be a function. If holds for every
, then the subgraph of with vertex set and edge set
, denoted by , is called a fractional -factor of with
indicator function , where denotes the set of edges incident with
in and . A graph is defined as a
fractional -deleted graph if for any , contains a
fractional -factor. The size, spectral radius and signless Laplacian
spectral radius of are denoted by , and ,
respectively. In this paper, we establish a lower bound on the size, spectral
radius and signless Laplacian spectral radius of a graph to guarantee that
is a fractional -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
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