In this paper,the automorphism group of G is determined,where G is a 4 × 4 upper unitriangular matrix group over Z.Let K be the subgroup of AutG consisting of all elements of AutG which act trivially on G/G,G /ζG and ζG,then (i) InnG ■ K ■ AutG;(ii) AutG/K≌=G1×D8×Z2,where G1=(a,b,c|a4=b2=c2=1,ab=a-1,[a,c]= [b,c]=1 ;(iii) K/Inn G≌=Z×Z×Z. 相似文献
Suppose that a connected graph G has n vertices and m edges, and each edge is contained in some triangle of G. Bounds are established here on the minimum number tmin(G) of triangles that cover the edges of G. We prove that ?(n - 1)/2? ? tmin(G) with equality attained only by 3-cactii and by strongly related graphs. We obtain sharp upper bounds: if G is not a 4-clique, then. The triangle cover number tmin(G) is also bounded above by Γ(G) = m - n + c, the cyclomatic number of a graph G of order n with m edges and c connected components. Here we give a combinatorial proof for tmin(G) ? Γ(G) and characterize the family of all extremal graphs satisfying equality. 相似文献
Viewing the GI/G/c queue as a service system alternating between two basic states—that of a loaded (non-empty) GI/G/1 queue and that of a GI/G/∞ queue (dependent, respectively, on whether all servers in the GI/G/c queue are busy or otherwise)—approximations for the components of the mixture distribution of the steady-state probabilities are derived. The M/G/c queue is separately treated. Two imposed prerequisites, that only minimal prior information about the queue will be required and that no numeric method be needed other than a root-finding algorithm, are strictly adhered to. The accuracy attained is generally satisfactory, while remarkable algebraic simplicity is preserved. 相似文献
一个边割被称为圈边割,如果该边割能分离图的两个不同圈.如果一个图有圈边割,称该图为圈边可分离的.一个圈边可分离图G的最小圈边割的阶数被称为圈边连通度,记作cλ(G).定义:ζ(G)=min{w(X)|X导出G的最短圈},其中w(X)为端点分别在X和V(G)-X中的边的数目.如果一个圈边可分离图G使得cλ(G)=ζ(G)成立,称该图是圈边最优的.Tian和Meng在文章[11]以及Yang et al在文章[15]中研究了两种不同的双轨道图的圈边最优性.本文我们将研究具有两个同阶轨道的双轨道图的圈边连通度. 相似文献
The cycle length distribution of a graph G of order n is a sequence (c1 (G),..., cn (G)), where ci(G) is the number of cycles of length i in G. In general, the graphs with cycle length distribution (c1(G),...,cn(G)) are not unique. A graph G is determined by its cycle length distribution if the graph with cycle length distribution (c1 (G),..., cn (G)) is unique. Let Kn,n+r be a complete bipartite graph and A(∈)E(Kn,n+r). In this paper, we obtain: Let s > 1 be an integer. (1) If r = 2s, n > s(s - 1) + 2|A|, then Kn,n+r - A (A(∈)E(Kn,n+r),|A| ≤ 3) is determined by its cycle length distribution; (2) If r = 2s + 1,n > s2 + 2|A|, Kn,n+r - A (A(∈)E(Kn,n+r), |A| ≤ 3) is determined by its cycle length distribution. 相似文献
对简单图G(V,E),若存在自然数κ(1≤κ≤Δ(G))和映射f:E(G)→{1,2,…,κ}使得对任意相邻两点u,v∈V(G),uv∈E(G),当d(u)=d(v)时,有C(u)=C(u),则f为G的κ-邻点可约边染色(简记为κ-AVREC of G),而x′_(aur)(G)=max{κ|κ-AVREC of G}称为G的邻点可约边染色数.其中C(u)={f(uv)|uv∈E(G)}.证明了联图在若干情况下的邻点可约边染色定理,得到了S_n+S_n,F_n+F_n,W_n+W_n,S_n+F_n,S_n+W_n和F_n+W_n的邻点可约边色数. 相似文献
A total k-coloring c of a graph G is a proper total coloring c of G using colors of the set[k] = {1, 2,..., k}. Let f(u) denote the sum of the color on a vertex u and colors on all the edges incident to u. A k-neighbor sum distinguishing total coloring of G is a total k-coloring of G such that for each edge uv ∈ E(G), f(u) = f(v). By χ nsd(G), we denote the smallest value k in such a coloring of G. Pil′sniak and Wo′zniak conjectured that χ nsd(G) ≤Δ(G) + 3 for any simple graph with maximum degree Δ(G). In this paper, by using the famous Combinatorial Nullstellensatz, we prove that the conjecture holds for any triangle free planar graph with maximum degree at least 7. 相似文献
Let $G$ be a finite group and $\mathfrak{c}(G)$ denote the number of cyclic subgroups of $G$. It is known that the minimal value of $\mathfrak{c}$ on the set of groups of order $n$, where $n$ is a positive integer, will occur at the cyclic group $Z_n$. In this paper, for non-cyclic nilpotent groups $G$ of order $n$, the lower bounds of $\mathfrak{c}(G)$ are established. 相似文献
If G is a graph on n vertices, its Laplacian matrix L(G) = D(G) - A(G) is the difference of the diagonal matrix of vertex degrees and the adjacency matrix. The main purpose of this note is to continue the study of the positive definite, doubly stochastic graph matrix (In + L(G))-1= ω(G) = (wij). If, for example, w(G) = min wij, then w(G)≥0 with equality if and only if G is disconnected and w(G) ≤ l/(n + 1) with equality if and only if G = Kn. If i¦j, then wii ≥2wij, with equality if and only if the ith vertex has degree n - 1. In a sense made precise in the note, max w,, identifies most remote vertices of G. Relations between these new graph invariants and the algebraic connectivity emerge naturally from the fact that the second largest eigenvalue of ω(G) is 1/(1 + a(G)). 相似文献
In this paper, we construct three new sequence spaces $b^{{r,s}}_{0}(G)$, $b^{{r,s}}_{c}(G)$ and $b^{{r,s}}_{\infty}(G)$ and mention some inclusion relations, where $G$ is generalized difference matrix. Moreover, we give Schauder basis of the spaces $b^{{r,s}}_{0}(G)$ and $b^{{r,s}}_{c}(G)$. Afterward, we determine $\alpha-$, $\beta-$ and $\gamma-$duals of those spaces. Finally, we characterize some matrix classes related to the space $b^{{r,s}}_{c}(G)$. 相似文献