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.     

关于环网G（N;S1,S2）的直径

本文首先给出了环网G(N;s_1,s_2)存在有限直径的充要条件;用初等数论方法给出了其直径的简明计算公式;最后讨论了类似文献[3]的环网直径的对偶性问题.     

A note on acyclic domination number in graphs of diameter two

A subset S of the vertex set of a graph G is called acyclic if the subgraph it induces in G contains no cycles. S is called an acyclic dominating set of G if it is both acyclic and dominating. The minimum cardinality of an acyclic dominating set, denoted by γ_a(G), is called the acyclic domination number of G. Hedetniemi et al. [Acyclic domination, Discrete Math. 222 (2000) 151-165] introduced the concept of acyclic domination and posed the following open problem: if δ(G) is the minimum degree of G, is γ_a(G)?δ(G) for any graph whose diameter is two? In this paper, we provide a negative answer to this question by showing that for any positive k, there is a graph G with diameter two such that γ_a(G)-δ(G)?k.     

Measurement of wire diameter by optical diffraction

A combined interference and diffraction pattern, in the form of equidistant interference fringes, resulting from illuminating a vertical metallic wire by a laser beam is analyzed to measure the diameter of four standard wires. The diameters range from 170 to 450 μm. It is found that the error in the diameter measurements increases for small metallic wires and for small distances between the wire and the screen due to scattering effects. The intensity of the incident laser beam was controlled by a pair of sheet polaroids to minimize the scattered radiation. The used technique is highly sensitive, but requires controlled environmental conditions and absence of vibration effects. The expanded uncertainty for k=2 is calculated and found to decrease from U(D)=±1.45 μm for the wire of nominal diameter 170 μm to ±0.57 μm for the diameter 450 μm.     

一个群论公式及其应用

一个群论公式及其应用①肖文俊（厦门大学数学研究所厦门３６１００５）在以下的讨论中，均假定Ｇ为一有限群，Ｓ为Ｇ的一个生成集，１Ｓ，Ｓ＝Ｓ－１．现设Ｌ为群Ｇ的任一子群，｜Ｇ：Ｌ｜＝ｎ，那么有如下公式定理ＡＧ＝Ｌ（１∪Ｓ）Ｓｎ－２．这一公式的证明相当简短...     

There is no upper bound for the diameter of the commuting graph of a finite group

We construct a family of finite 2-groups which have commuting graph of increasing diameter.     

一类3-紧优双环网无限族

到给定顶点数的双环网的最小直径是图论和计算机工作者广泛研究的问题。虽然大多数双环网可以达到紧优,但仍然有一部分双环网不能达到。本文给出了一类3-紧优的双环网无限族。     

关于猜想γ（A）≤（m－1）~2＋1

设Ａ是一个非负矩阵，若存在正整数ｋ，使得Ａ~ｋ＞０，则称Ａ为本原矩阵，而上述ｋ的最小者称为Ａ的本原指数，记作γ（Ａ）．设ｍ为Ａ的最小多项式的次数，ｇ为Ａ的伴随有向图的围长，当ｇ≤ｍ－１时，猜想γ（Ａ）≤（ｍ－１）~２＋１成立。     

图的直径与Betti亏数

对于任意的正数Ｍ以及正整数ｄ≥４,存在直径为ｄ的ｉ－边连通无环图Ｇ使得ζ（Ｇ）≥Ｍ,其中ζ（Ｇ）是Ｇ的Ｂｅｔｔｉ亏数,ｉ＝１,２,３。