Expression for the Number of Spanning Trees of Line Graphs of Arbitrary Connected Graphs

For any graph G, let be the number of spanning trees of G, be the line graph of G, and for any nonnegative integer r, be the graph obtained from G by replacing each edge e by a path of length connecting the two ends of e. In this article, we obtain an expression for in terms of spanning trees of G by a combinatorial approach. This result generalizes some known results on the relation between and and gives an explicit expression if G is of order and size in which s vertices are of degree 1 and the others are of degree k. Thus we prove a conjecture on for such a graph G.     

Cu(OTf)2-Catalyzed 1,2,3-triazole-ring-controlled selective phenolic O–H bond methylthiomethylation

Copper-catalyzed selective methylthiomethylation of 1,4-disubstituted 1,2,3-triazole phenols using DMSO as a methylthiomethylation reagent was achieved. Controlled by the triazole ring, the ortho-phenolic hydroxyl in N(1) aryl can be selectively methylthiomethylated, generating functionalized 1,4-disubstituted 1,2,3-triazoles bearing aryl methylthiomethyl ether fragment.     

非线性广义系统平衡点的定性分析

本文研究非线性广义系统平衡点的定性行为．利用系统的正则性，证明了可以通过其线性系统的定性行为研究非线性系统的定性性态，为定性研究非线性系统提供了一种方法．     

On the Permanental Polynomials of Some Graphs

Let G be a simple graph with adjacency matrix A(G) and (G,x) the permanental polynomial of G. Let G × H denotes the Cartesian product of graphs G and H. Inspired by Kleins idea to compute the permanent of some matrices (Mol. Phy. 31 (3) (1976) 811–823), in this paper in terms of some orientation of graphs we study the permanental polynomial of a type of graphs. Here are some of our main results.1.If G is a bipartite graph containing no subgraph which is an even subdivision of K _2,3, then G has an orientation G ^e such that (G,x) = det (xI-A(G ^e )), where A(G ^e ) denotes the skew adjacency matrix of G ^e.2.Let G be a 2-connected outerplanar bipartite graph with n vertices. Then there exists a 2-connected outerplanar bipartite graph with 2n+2 vertices such that (G,x) is a factor of .3.Let T be an arbitrary tree with n vertices. Then , where ₁ , ₂ , ..., _n are the eigenvalues of T.     

Enumeration of perfect matchings of a type of Cartesian products of graphs

Let G be a graph and let Pm(G) denote the number of perfect matchings of G.We denote the path with m vertices by P_m and the Cartesian product of graphs G and H by G×H. In this paper, as the continuance of our paper [W. Yan, F. Zhang, Enumeration of perfect matchings of graphs with reflective symmetry by Pfaffians, Adv. Appl. Math. 32 (2004) 175-188], we enumerate perfect matchings in a type of Cartesian products of graphs by the Pfaffian method, which was discovered by Kasteleyn. Here are some of our results:1. Let T be a tree and let C_n denote the cycle with n vertices. Then Pm(C₄×T)=∏(2+α²), where the product ranges over all eigenvalues α of T. Moreover, we prove that Pm(C₄×T) is always a square or double a square.2. Let T be a tree. Then Pm(P₄×T)=∏(1+3α²+α⁴), where the product ranges over all non-negative eigenvalues α of T.3. Let T be a tree with a perfect matching. Then Pm(P₃×T)=∏(2+α²), where the product ranges over all positive eigenvalues α of T. Moreover, we prove that Pm(C₄×T)=[Pm(P₃×T)]².     

On the number of matchings of graphs formed by a graph operation

Let G be a simple graph. Define R(G) to be the graph obtained from G by adding a new vertex e* corresponding to each edge e = (a,b) of G and by joining each new vertex e* to the end vertices a and b of the edge e corresponding to it. In this paper, we prove that the number of matchings of R(G) is completely determined by the degree sequence of vertices of G.     

非易失铁电存储器的进展和若干问题

铁电薄膜与半导体集成,产生了新一代非易失存储器。它的功耗之小,写入速度之快,可重写次数之多以及抗辐照能力之强是目前任何一种半导体存储器所不及的。文章介绍了非易失铁电存储器的原理、特点、进展和应用,并讨论了这类存储器在进入大规模商业生产时所面临的若干材料、工艺和器件失效等问题。     

1,1-二氟乙烷和1,2-二氟乙烷分子内原子间非键相互作用

用从头算势诱导极小二乘拟合(PD/LSF)和(exp-6-1)函数研究了1,1-二氟乙烷和1,2-二氟乙烷分子内非键原子间的相互作用。计算结果表明, 同分异构体的能量稳定性差, 在于分子内非键原子间的相互作用。该模式提供了一种简单、实用的研究分子内非键相互作用的方法。