λ'-最优无三角图的最小边度充分条件

设S是连通图G的一个边割。若G-S不包含孤立点,则称S是G的一个限制边割。图G的最小限制边割的边数称为G的限制边连通度,记为λ'(G).如果图G的限制边连通度等于其最小度,则称图G是最优限制边连通的,简称λ'-最优的。设G是一个n阶的连通无三角图,且最小度δ(G)≥2.文章证明了,若最小边度ξ(G)≥(n/2-2 )(1+1/δ(G)-1),则G是λ'-最优的。并由此推出,若连通无三角图G的最小度δ(G)≥n/4+1,则G是λ'-最优的。最后给出例子说明这些结果给出的边界都是紧的。

超级-λ'无三角图的度和充分条件

设S是连通图G的一个边割。若G-S不包含孤立点,则称S是G的一个限制边割。如果图G的每个最小限制边割恰好分离出图G的一条边,则称图G是超级限制边连通的,简称超级-λ'的。设G是一个阶n≥4的连通无三角图。本文证明了若G中任意满足dist(u,v)=2的点对u,v∈V(G)有d(u)+d(v)≥2[n+2/4]+3,则G是超级-λ'的。最后,举例说明该结论是最好的。

一类无三角正则图的性质

图为无三角正则图,它满足不相邻的顶点恰有两个公共相邻顶点。先从代数的角度去研究它的特征值,得到了它的顶点个数只能取一些特殊的整数,然后证明了其点连通度与边连通度相等,而且存在完美匹配,最后猜想:(1)χ(G)=χ'(G)=k;(2)图G是Hamilton图。

Ramsey数和无三角的Cayley图

记Zn={0,1,…,n}为模n的整数加群,Z*n=Zn\{0}.对一个Z*n逆元封闭的子集A,定义Cayley图Gn(A)为:其点集为Zn,而{x,y}是一条边当且仅当|x-y|∈A.计算了这些Cayley图的独立数至n≤258,改进了Ramsey数r(3,q)的的下界,27≤q≤38.

Laser triangulation measurement of the level in a coal silo

Laser triangulation theory was used to develop a novel contact-free method for measuring the coal level in a silo under harsh environmental conditions found in coal mines, such as the presence of dense dust, high humidity, and low illumination. A laser source and a camera were mounted at the top of the silo. The laser spot projected into the silo was imaged by the camera. The pinhole imaging principle allows the level to be found from the lateral shift of the spot image on the sensor. A pre-calibrated look-up table of the coal depth versus spot position was used to obtain the depth. The measurement accuracy depends on the step size used during pre-calibration. The actual application of a device designed according to these principles shows that it is easy to implement. The detection of the coal level in a silo at the low illumination level found in coal mines is demonstrated.