Kronecker乘积图的超连通性(英文)

设G1和G2是两个连通图,则G1和G2的Kronecker积G1×G2定义如下:V(G1×G2)=V(G1)×V(G2),E(G1×G2)={(u1,v1)(u2,v2):u1u2∈E(G1),v1v2∈E(G2)}.我们证明了G×Kn(n≥4)超连通图当且仅当κ(G)n>δ(G)(n 1),其中G是任意的连通图,Kn是n阶完全图.进一步我们证明了对任意阶至少为3的连通图G,如果κ(G)=δ(G),则G×Kn(n≥3)超连通图.这个结果加强了郭利涛等人的结果.     

n-double图的连通性

设Gl=(V1,E1),G2=(V2,E2)是两个连通图,直积(direct product)(也称为Kronecker product,tensor product和cross product) G1(×)G2的点集为V(G1(×)G2)=V(G1)(×)V(G2),边集为E(G1(×)G2)={(u1,v1)(u2,v2)∶ulu2∈E(G1),vlv2∈E(G2)}.简单图G的n-double图Dn[G]=G(×)Tn,其中n个点的全关系图Tn是完全图Kn在每个点加上一个自环得到的图.在本文中,我们研究了Dn[G]的(边)连通性,超(边)连通性.     

完全二部图K_(1,n),K_(2,n)和K_(3,n)的点强可区别全染色

设f是图G的一个正常全染色.对任意x∈V(G),令C(x)表示与点x相关联或相邻的元素的颜色以及点x的颜色所构成的集合.若对任意u,v∈V(G),u≠v,有C(u)≠C(v),则称.f是图G的一个点强可区别全染色,对一个图G进行点强可区别全染色所需的最少的颜色的数目称为G的点强可区别全色数,记为X_(vst)(G).讨论了完全二部图K_(1,n),K_(2,n)和L_(3,n)的点强可区别全色数,利用组合分析法,得到了当n≥3时,X_(vst)(K_(1,n)=n+1,当n≥4时,X_(vst)(K_(2,n)=n+2,当n≥5时,X_(vst)(K_(3,n))=n+2.     

完全图及完全二部图的点可区别Ⅳ-全染色

设G是简单图,图G的一个k-点可区别Ⅵ-全染色(简记为k-VDIVT染色),f是指一个从V(G)∪E(G)到{1,2,…,k}的映射,满足:()uv,uw∈E(G),v≠w,有,f(uv)≠f(uw);()u,V∈V(G),u≠v,有C(u)≠C(v),其中C(u)={f(u)}∪{f(uv)|uv∈E(G)}.数min{k|G有一个k-VDIVT染色}称为图G的点可区别Ⅵ-全色数,记为x_(vt)~(iv)(G).讨论了完全图K_n及完全二部图K_(m,n)的VDIVT色数.     

完全二部图的色性

设 G 是一个图,我们用 V(G)和 E(G)分别表示 G 的顶点集和边集,记 v=|V(G)|,ε=|E(G)|.P(G;λ)是图 G 的色多项式.称图 G 是色唯一的,如果任何图 H,由 P(H;λ)=P(G;λ),推知 H 与 G 同构.c_t(G)表示 G 中长为 k 的圈的个数.用G=(X,Y)表示二部图,K_(m,n)表示两部分的基数分别为 m 和 n 的完全二部图.本文中所有的图都是简单图,没有定义的术语和记号均可在[1]中找到.我们的主要结果是,用     

度为奇数的正则图的上负全控制数

f:v(G)→{一1,0,1}称为图G的负全控制函数,如果对任意点V∈V,均有f[v]≥1,其中 f[v]= ∑,f(u).如果对每个点v∈V,不存在负全控制函数g:V(G)→{-l,0,1),g≠f,满u∈N(v)足g(v)≤f(v),则称f是-个极小负全控制函数.图的上负全控制数F-t(G)=max{w(f)|f,是G的极小负全控制函数},其中w(f)=∑/v∈V(G)f(v).本文研究正则图的上负全控制数,证明了:令G是-个v∈V(G)n阶r-正则图.若r为奇数,则Γt-(G)＜＝r2 1/r2 2r-1n.     

广义图K(n,m)的点强全色数

图G(V,E)的一个正常k-全染色σ称为G(V,E)的一个k-点强全染色,当且仅当v∈V(G),N[v]中的元素着不同颜色,其中N[v]={u vu∈V(G)}∪{v};并且χvTs(G)=m in{k存在G的一个k-点强全染色}称为G的点强全色数.本文确定了完全图Kn的广义图K(n,m)和乘积图Lm×Kn的点强全色数.     

联接图中的最长链

一、引言文中未加说明的述语均同于[1]。给定图 G,以 c(G)记其联通分支数,定义h(G)=min{|s|-c(G\S):S(?)V(G),c(G\S)>1},f(G)=min{d(u) d(v):u、v∈v(G),u=v,uv(?)E}。1978年 H.A.Jung 在[2]中证明了,当 f(G)≥n(G)-4,n(G)≥11,h(G)≥0时,G 含哈密顿圈。本文研究了上述参数与图中最长链所含点数 l(G)之间的关系,得到下述结果:     

两类图的L(2.1)-标号数

图G的一个L(2.1)-标号是从顶点集V(G)到非负整数的一个函数f,使得若d(u,v)=1时,有|f(u)-f(v)|≥2;若d(u,v)=2时,有|f(u)-f(v)|≥1.图G的L(2.1)-标号数λ(G)是G的所有L(2.1)-标号下的跨度max{f(v):v∈V(G)}的最小数.图Fn+1*为扇图的路上每个顶点增加一个悬挂边得到的图.图Hn为轮图的圈上每个顶点增加一个悬挂边得到的图.本文确定了图Fn+1*与Hn的L(2.1)-标号数.     

关于图Pκn的均匀染色

对简单图G=〈V,E〉及自然数k,令V(Gk)=V(G),E(Gk)=E(G)U{uv|d(u,v)=k},其中d(u,v)表示G中u,v的距离,称图Gk为G的k方图.本文讨论了路的k方图Pkn的均匀点染色、均匀边染色和均匀邻强边染色,利用图的色数的基本性质和构造染色函数的方法,得到相应的色数Xev(Pkn),Xec(Pkn),Xeas(Pkn).并证明猜想"若图G有m-EASC,则一定有m+1-EASC"对Pkn是正确的.     

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.     

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.     

Hosoya指标在一个闭区间的一类树的刻画(英文)

Let n and d be two positive integers.By B_n,d we denote the graph obtained by identifying an endvertex of path P_d with the center of star S_n-d+1,where n ≥ d + 1.By C_n,d we denote the graph obtained by identifying an endvertex of P_d-1 with the center of Stare S_n-d,and the other endvertex of P_d-1 with the center of S₃ where n ≥ d + 3.By E_n,d,k we denote the graph obtained by identifying the vertex v_k of P(v₁ - v₂ - ··· - v_d+1) with the center of S_n-d.In this paper,we completely characterize all trees T which have diameter at least d（d ≥ 3） and satisfy the following conditions:（i） Z(B_n,d) ≤ Z（T） ≤ Z(E_n,d,3) for n = d + 3;（ii） Z(B_n,d) ≤ Z（T） ≤ Z(C_n,d) for n ≥ d + 4.     

一种场站设置问题中的几个结论

本文研究如下一种场站设置问题:设S是欧空间E~m中由有限个点A_1,A_2,…,A_n组成的集合.d(A_i,A_j)表示点A_i和A_j之间的距离.令σ(S)=Σ_(1≤i相似文献   

色临界图的最大度与色数的一个关系式

研究了一类简单图G的色数x(G)与最大度△(G)的关系,对满足x(G)>(S~2+S)/2的X(G)+S阶色临界图G,证明了x(G)=△(G)+1-S,或等价地,△(G)+1-[((8△(G)+17~(1/2)-3/2]≤X(G)≤△(G)+1,这一结果部分改进了Brooks经典不等式X(G)≤△(G)+1,并完全刻画n+3(n≥4)个顶点的n-临界图的结构。     

三维Boussinesq方程关于水平速度的一个爆破准则

本文主要讨论三维Boussinesq方程当扩散系数κ=0时光滑解的爆破准则.利用Littlewood-Paley分解和能量方法证明了如果方程关于水平速度场ū=（u₁,u₂,0）的水平导数满足▽_hū=（₁ū,₁ū,0）∈L¹（0,T;B0∞,∞（R³））.则解（u,θ）可以连续到T₁>T.     

任意n个完备二分图的并图的k－优美性和算术性

证明了对于正整数k,n,si,ti（si,ti≥2,i=1,2,…,n）,图n/U/i=1,Ksi,ti是k-优美图;对于正整数k,d（d≥2）,k≠0（roodd）及n,si,ti（si,ti≥2,i=1,2,…,n）,图n/U/i=1,Ksi,ti是（k,d）-算术图,前一结论推广了文[6]的相应结果。     

Signed Roman (Total) Domination Numbers of Complete Bipartite Graphs and Wheels

A signed(res. signed total) Roman dominating function, SRDF(res.STRDF) for short, of a graph G =(V, E) is a function f : V → {-1, 1, 2} satisfying the conditions that(i)∑v∈N[v]f(v) ≥ 1(res.∑v∈N(v)f(v) ≥ 1) for any v ∈ V, where N [v] is the closed neighborhood and N(v) is the neighborhood of v, and(ii) every vertex v for which f(v) =-1 is adjacent to a vertex u for which f(u) = 2. The weight of a SRDF(res. STRDF) is the sum of its function values over all vertices.The signed(res. signed total) Roman domination number of G is the minimum weight among all signed(res. signed total) Roman dominating functions of G. In this paper,we compute the exact values of the signed(res. signed total) Roman domination numbers of complete bipartite graphs and wheels.