On some cycles in linearized Wenger graphs

Let ${\mathbb{F}}_q$ be the finite field of $q$ elements for prime power $q$ and let $p$ be the character of ${\mathbb{F}}_q$. For any positive integer $m$, the linearized Wenger graph $L_m\left(q\right)$ is defined as follows: it is a bipartite graph with the vertex partitions being two copies of the $(m+1)$-dimensional vector space ${\mathbb{F}}_q^{m+1}$, and two vertices $p=\left(p\left(1\right),\dots ,p(m+1)\right)$ and $l=\left[l\left(1\right),\dots ,l(m+1)\right]$ being adjacent if $p\left(i\right)+l\left(i\right)=p\left(1\right){\left(l\left(1\right)\right)}^{p^{i?2}}$, for all $i=2,3,\dots ,m+1$. In this paper, we show that for any positive integers $m$ and $k$ with $3\le k\le p^2$, $L_m\left(q\right)$ contains even cycles of length $2k$ which is an open problem put forward by Cao et al. (2015).     

High-performance Na1.25V3O8 nanosheets for aqueous zinc-ion battery by electrochemical induced de-sodium at high voltage

Aqueous rechargeable zinc-ion batteries (ARZIBs) are expected to replace organic electrolyte batteries owing to its low price, safe and environmentally friendly characteristics. Herein, we fabricated vanadium-based Na_1.25V₃O₈ nanosheets as a cathode material for ARZIBs, which present a high performance by electrochemical de-sodium at high voltage to form Na₂V₆O₁₆ phase in the first cycle: high capacity of 390 mAh/g at 0.1 A/g, high rate performance (162 mAh/g at 10 A/g) and superior cycle stability (179 mAh/g with a high capacity retention of 88.2% of the maximum capacity after 2000 cycles). In addition, the cell exhibits a high energy density of 416.9 Wh/kg at 143.6 W/kg, suggesting great potential of the as-prepared Na_1.25V₃O₈ nanosheets for ARZIBs     

On cycle lengths in claw-free graphs with complete closure

We show a construction that gives an infinite family of claw-free graphs of connectivity κ=2,3,4,5 with complete closure and without a cycle of a given fixed length. This construction disproves a conjecture by the first author, A. Saito and R.H. Schelp.     

连通、几乎局部连通、强K1,p-约束图的完全圈可扩

借助于新的连通性——几乎局部连通的定义．证明了连通、几乎局部连通、强K1，p--约束图,完全圈可扩,这一结果涵盖了膝延燕及尤海在拟无爪图上的相应结果。     

Fan-Type Theorem for Long Cycles Containing a Specified Edge

In this paper, we prove that if G is 3-connected noncomplete graph of order n satisfying min{max{d(u),d(v)}:d(u,v)=2}=μ, then for each edge e, G has a cycle containing e of length at least min{n,2μ}, unless G is a spanning subgraph of K_μ + K^c_n−μ or K₃+(lK_μ−2∪K_s), where n=l(μ−2)+s+3,1≤s≤μ−2. Partially supported by NNSFC(No. 60172005); Partially supported by NNSFC(No. 10431020);     

CYCLE SPACES OF GRAPHS ON THE SPHERE AND THE PROJECTIVE PLANE

Cycle base theory of a graph has been well studied in abstract mathematical field such matroid theory as Whitney and Tutte did and found many applications in pratical uses such as electric circuit theory and structure analysis, etc. In this paper graph embedding theory is used to investigate cycle base structures of a 2-(edge)-connected graph on the sphere and the projective plane and it is shown that short cycles do generate the cycle spaces in the case of ““““small face-embeddings““““. As applications the authors find the exact formulae for the minimum lengthes of cycle bases of some types of graphs and present several known results. Infinite examples shows that the conditions in their main results are best possible and there are many 3-connected planar graphs whose minimum cycle bases can not be determined by the planar formulae but may be located by re-embedding them into the projective plane.     

On Edge-Colored Graphs Covered by Properly Colored Cycles

We characterize edge-colored graphs in which every edge belongs to some properly colored cycle. We obtain our result by applying a characterization of 1-extendable graphs. Received: April, 2003     

On 3-colorable planar graphs without prescribed cycles

In this paper we prove that every planar graph without cycles of length 4, 5, 6 and 8 is 3-colorable.     

科技进步贡献率测算方法的研究与应用

本文针对我国的统计现状对科技进步贡献率测算方法进行了研究，提出了一套适合现行统计体系的科技进步贡献率测算方法，并用此方法对安徽省1994年的工业、农业及全社会的科技进步贡献率进行了测算。     

The structure of graphs with given lengths of cycles

The cycle spectrum of a given graph G is the lengths of cycles in G. In this paper, we introduce the following problem: determining the maximum number of edges of an n-vertex graph with given cycle spectrum. In particular, we determine the maximum number of edges of an n-vertex graph without containing cycles of lengths three and at least k. This can be viewed as an extension of a well-known result of Erd?s and Gallai concerning the maximum number of edges of an n-vertex graph without containing cycles of lengths at least k. We also determine the maximum number of edges of an n-vertex graph whose cycle spectrum is a subset of two positive integers.