环状硅酸盐图的友好指数集

Let G be a graph with vertex set V(G) and edge set E(G). A labeling f : V(G) →Z2 induces an edge labeling f*: E(G) → Z2 defined by f*(xy) = f(x) + f(y), for each edge xy ∈ E(G). For i ∈ Z2, let vf(i) = |{v ∈ V(G) : f(v) = i}| and ef(i) = |{e ∈ E(G) : f*(e) =i}|. A labeling f of a graph G is said to be friendly if |vf(0)- vf(1)| ≤ 1. The friendly index set of the graph G, denoted FI(G), is defined as {|ef(0)- ef(1)|: the vertex labeling f is friendly}. This is a generalization of graph cordiality. We investigate the friendly index sets of cyclic silicates CS(n, m).     

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…     

具有5n个点的完全5部图的分类及5图部的色性

§ 1　IntroductionAll graphs considered here are finite and simple.For notations and terminology notdefined here,we refer to [1].For a graph G,by V(G) and E(G) we mean the vertex- setand edge- setof G,respectively.By N3(G) we denote the number of triangles in G.L et S bea set of sedges in G.By G- S (or G- s) we denote the graph obtained from G by deletingall edges in S.L et K (n1 ,n2 ,...,nt) be a complete t- partite graph.We denote by K- sn1 ,n2 ,...,ntthe family of graphs which are …     

单圈图的最大特征值的上界的改进

Let G（V, E） be a unicyclic graph, Cm be a cycle of length m and Cm G, and ui ∈ V（Cm）. The G - E（Cm） are m trees, denoted by Ti, i = 1, 2,..., m. For i = 1, 2,..., m, let eui be the excentricity of ui in Ti and ec = max{eui ： i = 1, 2 , m}. Let κ = ec＋1. Forj = 1,2,...,k- 1, let δij = max{dv ： dist（v, ui） = j,v ∈ Ti}, δj = max{δij ： i = 1, 2,..., m}, δ0 = max{dui ： ui ∈ V（Cm）}. Then λ1（G）≤max{max 2≤j≤k-2 （√δj-1-1＋√δj-1）,2＋√δ0-2,√δ0-2＋√δ1-1}. If G ≌ Cn, then the equality holds, where λ1 （G） is the largest eigenvalue of the adjacency matrix of G.     

The fractional metric dimension of permutation graphs

Let G =(V(G), E(G)) be a graph with vertex set V(G) and edge set E(G). For two distinct vertices x and y of a graph G, let RG{x, y} denote the set of vertices z such that the distance from x to z is not equa l to the distance from y to z in G. For a function g defined on V(G) and for U■V(G), let g(U) =∑s∈Ug(s). A real-valued function g : V(G) → [0, 1] is a resolving function of G if g(RG{x, y}) ≥ 1 for any two distinct vertices x, y ∈ V(G). The fractional metric dimension dimf(G)of a graph G is min{g(V(G)) : g is a resolving function of G}. Let G1 and G2 be disjoint copies of a graph G, and let σ : V(G1) → V(G2) be a bijection. Then, a permutation graph Gσ =(V, E) has the vertex set V = V(G1) ∪ V(G2) and the edge set E = E(G1) ∪ E(G2) ∪ {uv | v = σ(u)}. First,we determine dimf(T) for any tree T. We show that 1 dimf(Gσ) ≤1/2(|V(G)| + |S(G)|) for any connected graph G of order at least 3, where S(G) denotes the set of support vertices of G. We also show that, for any ε 0, there exists a permutation graph Gσ such that dimf(Gσ)- 1 ε. We give examples showing that neither is there a function h1 such that dimf(G) h1(dimf(Gσ)) for all pairs(G, σ), nor is there a function h2 such that h2(dimf(G)) dimf(Gσ) for all pairs(G, σ). Furthermore,we investigate dimf(Gσ) when G is a complete k-partite graph or a cycle.     

关于H-cordial图和semi-H-cordial图的若干结论（英文）

I. Cahit calls a graph H-cordial if it is possible to label the edges with the numbers from the set{1,-1} in such a way that, for some k, at each vertex v the sum of the labels on the edges incident with v is either k or-k and the inequalities |v(k)-v(-k)| ≤ 1 and|e(1)-e(-1)| ≤ 1 are also satisfied. A graph G is called to be semi-H-cordial, if there exists a labeling f, such that for each vertex v, |f(v)| ≤ 1, and the inequalities |e_f(1)-e_f(-1)| ≤ 1 and |vf(1)-vf(-1)| ≤ 1 are also satisfied. An odd-degree(even-degree) graph is a graph that all of the vertex is odd(even) vertex. Three conclusions were proved:(1) An H-cordial graph G is either odd-degree graph or even-degree graph;(2) If G is an odd-degree graph, then G is H-cordial if and only if |E(G)| is even;(3) A graph G is semi-H-cordial if and only if |E(G)| is even and G has no Euler component with odd edges.     

On the adjacent vertex-distinguishing equitable edge coloring of graphs

Let G(V, E) be a graph. A k-adjacent vertex-distinguishing equatable edge coloring of G, k-AVEEC for short, is a proper edge coloring f if (1) C(u)≠C(v) for uv ∈ E(G), where C(u) = {f(uv)|uv ∈ E}, and (2) for any i, j = 1, 2,… k, we have ||Ei| |Ej|| ≤ 1, where Ei = {e|e ∈ E(G) and f(e) = i}. χáve (G) = min{k| there exists a k-AVEEC of G} is called the adjacent vertex-distinguishing equitable edge chromatic number of G. In this paper, we obtain the χáve (G) of some special graphs and present a conjecture.     

w-DENSITY AND w-BALANCED PROPERTY OF WEIGHTED GRAPHS

§ 1　IntroductionAll graphsconsidered in this paperare finite undirected ones withoutloops ormultipleedges.Our terminology and notation are standard exceptas indicated.A good reference forany undefined terms is[1 ] .Let G be a graph with vertex set V( G) and edge set E( G) .The density of G is definedbyd( G) =ε( G)ν( G) ,whereν( G) andε( G) denote| V( G) | and| E( G) | ,respectively.G is said to be balanced iffor each subgraph H of G we have d( H )≤ d( G) ,where V( H ) is assum…     

A Counter Example on Faudree-Schelp Conjecture

Let simple graph G=(V, E),V=n,E=m. If there exists a path containing i vertices connecting u and v in V, then property P_i(u,v) will be said to told.For 2≤i≤n, let S_i be the set of all unordered pairs of distinct u and v for which property P_i(u.v) holds, and Let S_1 be the set of all unordered pairs of vertices which are not connected by any path. A graph G satisfies property P_i if |S_i|=n(n-1)/2.     

Bounds of the Estrada index of graphs

Let G be a graph of order n and let λ1,λ2,...,λn be its eigenvalues. The Estrada index[2] of G is defined as EE = EE(G) =∑n i=1 eλi. In this paper, new bounds for EE are established, as well as some relations between EE and graph energy E.     

Complete multipartite decompositions of complete graphs and complete <Emphasis Type="Italic">n</Emphasis>-partite graphs

In this paper, a new concept of an optimal complete multipartite decomposition of type 1 (type 2) of a complete n-partite graph Q _n is proposed and another new concept of a normal complete multipartite decomposition of K _n is introduced. It is showed that an optimal complete multipartite decomposition of type 1 of K _n is a normal complete multipartite decomposition. As for any complete multipartite decomposition of K _n , there is a derived complete multipartite decomposition for Q _n . It is also showed that any optimal complete multipartite decomposition of type 1 of Q _n is a derived decomposition of an optimal complete multipartite decomposition of type 1 of K _n . Besides, some structural properties of an optimal complete multipartite decomposition of type 1 of K _n are given. Supported by the National Natural Science Foundation of China (10271110).     

完全$r$部图乘积上的Graham猜想

图G的Pebbling数f(G)是最小的正整数n,使得不论n个Pebble如何放置在G的顶点上,总可以通过一系列的Pebbling移动把1个Pebble移到任意一点上,其中Pebbling移动是从一个顶点处移走两个Pebble而把其中一个移到与其相邻的一个顶点上.Graham猜测对于任意的连通图G和H有f(G×H)≤f(G)f(H).本文证明对于一个完全γ部图和一个具有2-Pebbleing性质的图来说,Graham猜想成立.作为一个推论,当G和H均为完全γ部图时,Graham猜想成立.     

关于完全三部图K(m,n,r)的色唯一性

本文研究完全三部图K(m,n,r)的色唯一性问题，通过比较两个色等价图的色划分数的方法，得出两个关于K(m,n,r)为色唯一图的一般形式数值条件，基本上解决了K(m,n,r)为色唯一图的判定问题．     

On the Ascending Subgraph Decomposition Problem

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｐａｐｅｒ［１］,Ａｌａｖｉａｎｄｏｔｈｅｒｓｄｅｆｉｎｅｄｔｈｅｃｏｎｃｅｐｔｏｆａｓｃｅｎｄｉｎｇｓｕｂｇｒａｐｈｄｅｃｏｍｐｏｓｉｔｉｏｎ：Ｄｅｆｉｎｉｔｉｏｎ　ＬｅｔＧｂｅａｇｒａｐｈｏｆｐｏｓｉｔｉｖｅｓｉｚｅｑ,ａｎｄｌｅｔｎｂｅｔｈａｔｐｏｓｉｔｉｖｅｉｎｔｅｇｅｒｆｏｒｗｈｉｃｈｎ＋１２ｑ＜ｎ＋２２．ＴｈｅｎＧｉｓｓａｉｄｔｏｈａｖｅａｎａｓｃｅｎｄｉｎｇｓｕｂｇｒａｐｈｄｅｃｏｍｐｏｓｉｔｉｏｎ（ＡＳＤ）ｉｆＧｃａｎｂｅｄｅｃｏｍｐｏｓｅｄ…     

完全多部图的无符号Laplacian特征多项式(英文)

For a simple graph G,let matrix Q（G）=D（G） + A（G） be it’s signless Laplacian matrix and Q G （λ）=det（λI Q） it’s signless Laplacian characteristic polynomial,where D（G） denotes the diagonal matrix of vertex degrees of G,A（G） denotes its adjacency matrix of G.If all eigenvalues of Q G （λ） are integral,then the graph G is called Q-integral.In this paper,we obtain that the signless Laplacian characteristic polynomials of the complete multi-partite graphs G=K（n₁,n₂,···,n_t）.We prove that the complete t-partite graphs K（n,n,···,n）t are Q-integral and give a necessary and sufficient condition for the complete multipartite graphs K（m,···,m）s（n,···,n）t to be Q-integral.We also obtain that the signless Laplacian characteristic polynomials of the complete multipartite graphs K（m,···,m,）s1（n,···,n,）s2（l,···,l）s3.     

K_(2,3)的多重多部图设计的构造

λKn(t)是一个λ重完全多部图,G为一个不带孤立点的简单图.所谓的图设计G-HDλ(tn)是一个序偶(X,B),其中X是Kn(t)的顶点集,B为λKn(t)的一些子图(亦称为区组)构成的集合,使得任一区组均与图G同构,且λKn(t)的任意2个不同点组成的边恰在B的λ个区组中出现.本文讨论了G=K2,3的完全多部图设计存在性问题,证明了存在G-HDλ(tn)当且仅当λn(n-1)t2≡0(mod12),n≥2,nt≥5且(n,,λt)≠(9,1,1),(12,1,1),(3,1,2),(4,1,2).     

完全多部图同构于二面体群的Cayley齐次分解

设Γ=K_(s[t])是一个完全多部图,其中st是一个偶数,则存在一个二面体群R=D_(2n)(n=st/2),使得R能构造出一个同构于K_(s[t])的Cayley图.讨论了当s、t满足什么条件时,完全多部图Γ有同构于Cay(R,S)的齐次分解.     

一类完全多部图的色可选择性

如果一个图G的选择数等于它的色数,则称该图G是色可选择的．在2002年, Ohba给出如下猜想:每一个顶点个数小于等于2X(G) 1的图G是色可选择的．容易发现Ohba猜想成立的条件是当且仅当它对完全多部图成立,但是目前只是就某些特殊的完全多部图的图类证明了Ohba猜想的正确性．在本文我们证明图K6,3,2*(k-6),1*4(k≥6)是色可选择的,从而对图K6,3,2*(k-6),1*4(k≥6)和它们的所有完全k-部子图证明了Ohba猜想成立．     

对于K2+Tm和完全图的Ramsey函数的渐近上界

本文研究了当n趋于无穷大时,关于K2+Tm和完全图Kn的Ramsey数的渐近上界,以及r(K2+Tm,Kn)和r(K1+Tm,Kn)的渐近关系.利用李雨生等人所给出的一个独立数的下界公式,给出了r(K4,Kn)和r(Kk-c,Kn)的渐近上下界,推广了李雨生等人所给出的r(K1+Tm,Kn)的下界.     

Cycle decompositions III: Complete graphs and fixed length cycles

We show that the necessary conditions for the decomposition of the complete graph of odd order into cycles of a fixed even length and for the decomposition of the complete graph of even order minus a 1‐factor into cycles of a fixed odd length are also sufficient. © 2002 John Wiley & Sons, Inc. J Combin Designs 10: 27–78, 2002