两类满足A（H）=3的图

本文利用文[3]的方法给出了两类满足A(H)=3的图,从而肯定了满足A(H)=3的图是不唯一的。本文还给出了满足A(H)=3的最小图。     

由三角形的高线派生出的等边三角形

文[1]中笔者研究三角形性质时,发现了一个由三角形中线“生成”正三角形的问题,在文末笔者指出三条高线中能否有这种生成问题.最近,我们得到了如下结论.图1定理如图1,△ABC中,H是△ABC的垂心,H A、H B、H C的延长线上分别有点Z、L、M.若AZBC=BLAC=CMAB=33,则△ZLM是等边三角形.证明∵AZ=33a,BL=33b,CM=33c.(以锐角三角形为例)∵AH=2R cos A,∴H Z=2R cos A 13a,同理HM=2R cos C 13c.∵∠AH C=180-°B.ZM2=(2R cos A a3)2 (2R cos C c32) 2(2R cos C c)(2R cos A a3)cos B=4R2(cos2A cos2C 2cos A cos B cos C) …     

图的邻接矩阵的行列式与积和式的递推表达式

设A(G)是简单图G的邻接矩阵,H是由G的独立边和不交圈组成的生成子图的集合,e是H中某个图的独立边,C是H中图的圈,且e∈E(C).记G-e是G的删边子图,G\W是从G中删去导出子图W中的顶点及其关联边后得到的图.那么A(G)的行列式为detA(G)=detA(G-e)-detA(G\e)-2(-1)~(|V(C)|)detA(G\C)A(G)的积和式为perA(G)=perA(G-e)+perA(G\e)+2perA(G\C)这里,C取遍H中图的经过边e的圈.     

求线段总长

如图,已知0是边长为1的正六边形ABcDEF:艺 务中心,0H上AB:于二H,Hl、H2、H3、H4E D是0H的五等分点,过H-、Hz、H。、Ha分别作直线平行=于=AB,且依次交FA、CB于A1、A2、A3、A。,Bl、B2、B3、B4,求线段总长A,B, A。Bz A sB。 AtB。智慧窗参考答案 解答(经编辑部改写) 画一个与梯形F邶C全等的梯形,然A 日后将两个梯形拼成一个平行四边形(如图).显然,所求四条线段总长为 3×4÷2=6.求线段总长@张大授$湖南平江七中!414501     

一道平面几何命题的应用及推广

1　命题及其证明命题　如图 1所示 ,若直线 l⊥线段 AB于 H ,则M1 A2 - MA2 =M1 B2 - MB2 (1)反之 ,若式 (1)成立 ,则 M1 M所在的直线 l⊥AB.图 1证明　∵　 l⊥线段AB,∴　由勾股定理得 :AM21 - AH 2 =H M21 ,AM2 - AH 2 =H M2 .两式相减得AM21 - AM2 =H M21 - H M2 . 1同理可得BM21 - BM2 =H M21 - H M2 . 2由 1、2得AM21 - AM2 =BM21 - BM2 .反过来 ,可设∠ AH M1 =θ,则∠ BH M1 =π -θ,∴　 M1 A2 - AM2　 =AH 2 +H M21 - 2 AH . H M1 cosθ- AH 2 - H M2 +2 AH . MH . cosθ　 =H M21 - H M2 - 2 …     

一组猜想的研究

文1提出了一组猜想,本文将首先证明猜想1和猜想2是正确的.因为要用到二次曲线方程的一般形式,为了不引起混淆,字母有所不同.图1猜想1如图1,椭圆内的蝶形A1A2A3A3中,设过蝶心M的一直线分别和射线A2A4,弧A4A1,弧A2A3,射线A1A3交于G,P,Q,H,则     

e≤2(n-2)条件下的3-优图

记S_3(G)为图G所含有的连通的三结点导出子图的个数。图G称为3-优图,若对任何同点同边数的图H有S_3(G)≥S_3(H),3-优图在网络可靠性综合中起着重要作用。本文给出了在e≤2(n-2)条件下3-优图的刻划,其中n为结点数,e为边数,从而解决了文献[1]结论中提出的问题。     

2002年全国高中数学联赛加试试题另解

1.如图 1 ,在△ ABC中 ,∠ A =60°,AB>AC,点 O是外心 ,两条高 BE、CF交于 H点 ,点 M、N分别在线段 BH、H F上 ,且满足BM =CN,求 MH NHOH 的值 .解法 1　连结 OB、OC.∵　∠ BH C =∠ FH E =1 2 0°,又　∠ BOC =2∠ A =1 2 0°,∴　 B、O、H、C四点共圆 .设∠ OBC =α =3 0°,∠ EBC =β,∠ OBC =∠ OCB =3 0°,∠ EBC =∠ H OC =β.∴　 MH NHOH =BH - BM CN - H COH =BHOH- H COH.由正弦定理 ,在△ OH B中 ,BHOH=sin(1 2 0° β)sin(α -β) .在△ OH C中 ,H COH=sinβsin(α -β) .∴　 M…     

圆内接闭折线垂心的又一个新性质

从闭折线 A1A2 A3… An 的 n个顶点中 ,任意除去 2个顶点 Aj、Ak( 1≤ j相似文献   

一类完全三部图的色唯一性

用P(G,λ)表示简单图G的色多项式.设G是一个给定的简单图,若对任意简单图H,当P(H,λ)=P(G,λ)时都有H和G同构(记为H≌G),则称图G是色唯一的.本文证明了以下结果:设n,k,△都为非负整数,其中k≥0,△∈{4,5},若n≥1/3k~2+1/3△~2-1/3k△-1/3k-1/3△+4/3,则完全三部图K(n,n+△,n+k)是色唯一的.同时还给出了一个猜想.     

GQ(4, 2)-extensions, strongly regular case

A strongly regular locallyGQ(4, 2)-graph is a graph with parameters either (126, 45, 12, 8) or (190, 45, 12, 10). The existence and the uniqueness of the corresponding locallyGQ(4, 2)-graph in the first case are well known. We prove that theGQ(4, 2)-hyperoval on ten vertices either is the Petersen graph, or is the Möbius 5-prism, or consists of two (2, 3)-subgraphs connected by three edges. We obtain homogeneousGQ(4, 2)-solutions with a strongly regular point graph; in particular, this implies the negative answer to the question of F. Buekenhout concerning the existence of a locallyGQ(4, 2)-graph with the parameters (190, 45, 12, 10).     

广义θ-图的邻点可区别的全染色

u,v两点间连多于三条内部不相交的路且至多有一条长度为1的图,称为广义θ-图.本文给出了广义θ-图的邻点可区别的全染色.     

(0, mf-k+1)-图中具有正交(0,f)-因子分解的子图

设G是一个简单图, f是定义在V(G)上的整数值函数,且m是大于等于2的整数. 讨论(0, mf-k+1)-图G的正交因子分解, 并且证明了对任意的1≤k≤m, (0, mf-k+1)-图G中存在着一个子图R, 使得R有一个(0,f)-因子分解正交于图G中的任意一个k-子图H.     

On the Determination Problem for P 4-Transformation of Graphs

We prove that the P ₄-transformation is one-to-one on the set of graphs with minimum degree at least 3, and if graphs G and G ^' have minimum degree at least 3 then any isomorphism from the P ₄-graph P ₄(G) to the P ₄-graph P ₄(G ^') is induced by a vertex-isomorphism from G to G ^' unless G and G ^' both belong to a special family of graphs. Supported by NSFC, PCSIRT and the “973” program.     

临界h-边-连通图的临界度

图G称为k-临界h-边-连通的,若h=λ(G)且对每个k顶点集{u1,…,uk}有λ(G-{u1,…,ui})≤λ(G-{u1,…,ui-1})-1,I≤k.若G是k-临界h-边-连通但不(k 1)-临界h-边-连通,则记之为(h*,k*)λ.本文证明了:存在(h*,k*)λ图的充要条件是(1)1≤k≤[(h 1)/2],h≡0,1,2(mod 4);1≤k≤[(h-1)/2],h≡3(mod 4);或(2)k=h,G=Kk 1.     

Indecomposable r-graphs and some other counterexamples

An r-graph is any graph that can be obtained as a conic combination of its own 1-factors. An r-graph G(V, E) is said to be indecomposable when its edge set E cannot be partitioned as E = E₁ ∪ E₂ so that G_i(V, E_i) is an r_i-graph for i = 1, 2 and, for some r₁, r₂. We give an indecomposable r-graph for every integer r ≥ 4. This answers a question raised in [Seymour, Proc London Math Soc 38 (1979, 423–460], and has interesting consequences for the Schrijver System of the T-cut polyhedron to be given in [Rizzi, 1997, to appear]. A graph in which every two 1-factors intersect is said to be poorly matchable. Every poorly matchable r-graph is indecomposable. We show that for every r ≥ 4 that “being indecomposable” does not imply “being poorly matchable.” Next we give a poorly matchable r-graph for every r ≥ 4. The article provides counterexamples to some conjectures of Seymour. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 1–15, 1999     

On the two-dimensional orthogonal drawing of series-parallel graphs

It has been known that every planar 4-graph has a 2-bend 2-D orthogonal drawing, with the only exception being the octahedron, every planar 3-graph has a 1-bend 2-D orthogonal drawing with the only exception being K₄, and every outerplanar 3-graph with no triangles has a 0-bend 2-D orthogonal drawing. We show in this paper that every series-parallel 4-graph has a 1-bend 2-D orthogonal drawing.     

与星2-正交的(0,f)-因子分解

证明了若G是一个(0,mf-m+1)-图,则对其中任意给定的2m-星K1,2m,G有一个(0,f)-因子分解2-正交于K1,2m.     

优美图的一些性质

对于一个(p,q)-图G,如果存在一个单射.f:V(G)→{0,1,…,q},使得边标号集合{f(uv)| uv∈E(G)}={1,2,…,q},其中边标号为f(uv)=|f(u)-f(v)|,那么称G是优美图,并称.f是G的一个优美标号.通过研究若干优美图,得出一些优美图的性质.