直径不超过4的树的割宽

§ 1　IntroductionThe cutwidth problem for graphs,as well as a class of optimal labeling and embed-ding problems,have significant applications in VLSI designs,network communicationsand other areas (see [2 ] ) .We shall follow the graph-theoretic terminology and notation of [1 ] .Let G=(V,E)be a simple graph with vertex set V,| V| =n,and edge set E.A labeling of G is a bijec-tion f:V→ { 1 ,2 ,...,n} ,which can by regarded as an embedding of G into a path Pn.Fora given labeling f of G,th…     

单圈图的最大特征值序

主要讨论了单圈图按其最大特征值进行排序的问题，确定了该序的前六个图。     

最大2-正则诱导子图的长度

设G是2-连通图，c(G)是图G的最长诱导圈的长度，c′(G)是图G的最长诱导2-正则子图的长度。本文我们用图的特征值给出了c(G)和c′(G)的几个上界。     

Application of directed metallation in synthesis. Part 3: Studies in the synthesis of (±)-semivioxanthin and its analogues

The synthesis of several analogues of (±)-semivioxanthin including five thiophene analogues, using directed metalation are reported. The strategy consisted of the synthesis of functionalized naphthalene or benzo[b]thiophene as building blocks followed by annelation of the pyrone.     

Idealization of a decomposition theorem

In 1986, Tong [13] proved that a function f : (X,τ)→(Y,φ) is continuous if and only if it is α-continuous and A-continuous. We extend this decomposition of continuity in terms of ideals. First, we introduce the notions of regular-I-closed sets, A _I-sets and A _I -continuous functions in ideal topological spaces and investigate their properties. Then, we show that a function f : (X,τ,I)→(Y, φ) is continuous if and only if it is α-I-continuous and A _I-continuous. This revised version was published online in June 2006 with corrections to the Cover Date.     

A note on strongly π-regular rings

Let R be an associative ring with unit and let N(R) denote the set of nilpotent elements of R. R is said to be stronglyπ-regular if for each x∈R, there exist a positive integer n and an element y∈R such that x ⁿ=x ^{n +1} y and xy=yx. R is said to be periodic if for each x∈R there are integers m,n≥ 1 such that m≠n and x ^m=x ⁿ. Assume that the idempotents in R are central. It is shown in this paper that R is a strongly π-regular ring if and only if N(R) coincides with the Jacobson radical of R and R/N(R) is regular. Some similar conditions for periodic rings are also obtained. This revised version was published online in June 2006 with corrections to the Cover Date.     

强色指数的一个新的上界

给出了图的强色指数的一个新的上界，并指出几类恰好达到该上界的图，从而改进了Erodoes和Nesetri的强色指数猜想，在某种意义上证明了这个猜想。     

Fully-angular polyhex chains with minimal total π-electron energy

By Hückel molecular orbital (HMO) theory, the calculation of the total energy of all π-electrons in conjugated hydrocarbons can be reduced to that of E(G)=|λ₁|+|λ₂|+?+|λn|, where λi are the eigenvalues of the corresponding graph G. Denote by Ψn the set of all fully-angular polyhex chains with n hexagons. In this paper, we show that Hn has the minimum total π-electron energy among chains in Ψn, where Hn is the helicene chain.     

Diameter graphs of polygons and the proof of a conjecture of Graham

We show that for an n-gon with unit diameter to have maximum area, its diameter graph must contain a cycle, and we derive an isodiametric theorem for such n-gons in terms of the length of the cycle. We then apply this theorem to prove Graham's 1975 conjecture that the diameter graph of a maximal 2m-gon (m?3) must be a cycle of length 2m−1 with one additional edge attached to it.     

On co-semihereditary rings

A ringR is left co-semihereditary (strongly left co-semihereditary) if every finitely cogenerated factor of a finitely cogenerated (arbitrary) injective leftR-module is injective. A left co-semihereditary ring, which is not strongly left co-semihereditary, is given to answer a question of Miller and Tumidge in the negative. If _R U _S defines a Morita duality,R is proved to be left co-semihereditary (left semihereditmy) if and only ifS is right semihereditary (right co-semihereditary). Assuming thatS⩾R is an almost excellent extension,S is shown to be (strongly) right co-semihereditary if and only ifR is (strongly) right co-semihereditary. Project supported by the National Natural Science Foundation of China.