λ_(3)-最优连通混合Cayley图

对于连通图X=(V,E),如果X-F不连通并且X-F的每个分支至少含k个点,那么边集F⊆E是一个k-限制性边割.图X的k-限制性边连通度λ_(k)(X)为X的最小k-限制性边割的基数.该文给出了混合Cayley图的3-限制性边连通度和λ_(3)-最优性.

Small-World Networks with Unitary Cayley Graphs for Various Energy Generation

Complex networks have been a prominent topic of research for several years,spanning a wide range of fields from mathematics to computer science and also to social and biological sciences.The eigenvalues of the Seidel matrix,Seidel Signless Laplacian matrix,Seidel energy,Seidel Signless Laplacian energy,Maximum and Minimum energy,Degree Sum energy and Distance Degree energy of the Unitary Cayley graphs[UCG]have been calculated.Low-power devices must be able to transfer data across long distances with low delay and reliability.To overcome this drawback a small-world network depending on the unitary Cayley graph is proposed to decrease the delay and increase the reliability and is also used to create and analyze network communication.Small-world networks based on the Cayley graph have a basic construction and are highly adaptable.The simulation result shows that the small-world network based on unitary Cayley graphs has a shorter delay and is more reliable.Furthermore,the maximum delay is lowered by 40%.

Cayley Picture Fuzzy Graphs and Interconnected Networks

Theory of the Cayley graphs is directly linked with the group theory.However,if there are uncertainties on the vertices or edges or both then fuzzy graphs have an extraordinary importance.In this perspective,numbers of generalηizations of fuzzy graphs have been explored in the literature.Among the others,picture fuzzy graph(PFG)has its own importance.A picture fuzzy graph(PFG)is a pair G=(C,D)defined on a H^(*)=(A,B),where C=(ηC,θ_(C),■_(C))is a picture fuzzy set on A and D=(ηD,θ_(D),■_(D))is a picture fuzzy set over the set B∈A×A such that for any edge mn∈ B with ηD(m,n)≤min(ηC(m),ηC(n)),θD(m,n)≤min(θC(m),θC(n))and ■_(D)(m,n)≥max(■_(C)(m),■_(C)(n)).In this manuscript,we introduce the notion of the Cayley picture fuzzy graphs on groups which is the generalization of the picture fuzzy graphs.Firstly,we discuss few important characteristics of the Cayley picture fuzzy graphs.We show that Cayley picture fuzzy graphs are vertex transitive and hence regular.Then,we investigate different types of Cayley graphs induced by the Cayley picture fuzzy graphs by using different types of cuts.We extensively discuss the term connectivity of the Cayley picture fuzzy graphs.Vertex connectivity and edge connectivity of the Cayley picture fuzzy graphs are also addressed.We also investigate the linkage between these two.Throughout,we provide the extensions of some characηteristics of both the PFGs and Cayley fuzzy graphs in the setting of Cayley picture fuzzy graphs.Finally,we provide the model of interconnected networks based on the Cayley picture fuzzy graphs.

A STABILITY RESULT FOR TRANSLATINGSPACELIKE GRAPHS IN LORENTZ MANIFOLDS

In this paper,we investigate spacelike graphs defined over a domain Ω⊂M^(n) in the Lorentz manifold M^(n)×ℝ with the metric−ds^(2)+σ,where M^(n) is a complete Riemannian n-manifold with the metricσ,Ωhas piecewise smooth boundary,and ℝ denotes the Euclidean 1-space.We prove an interesting stability result for translating spacelike graphs in M^(n)×ℝ under a conformal transformation.

A Value for Games Defined on Graphs

Given a graph g=( V,A ) , we define a space of subgraphs M with the binary operation of union and the unique decomposition property into blocks. This space allows us to discuss a notion of minimal subgraphs (minimal coalitions) that are of interest for the game. Additionally, a partition of the game is defined in terms of the gain of each block, and subsequently, a solution to the game is defined based on distributing to each player (node and edge) present in each block a payment proportional to their contribution to the coalition.

BLOW-UP CONDITIONS FOR A SEMILINEAR PARABOLIC SYSTEM ON LOCALLY FINITE GRAPHS

In this paper, we investigate a blow-up phenomenon for a semilinear parabolic system on locally finite graphs. Under some appropriate assumptions on the curvature condition CDE’(n,0), the polynomial volume growth of degree m, the initial values, and the exponents in absorption terms, we prove that every non-negative solution of the semilinear parabolic system blows up in a finite time. Our current work extends the results achieved by Lin and Wu (Calc Var Partial Differ Equ, 2017, 56: Art 102) and Wu (Rev R Acad Cien Serie A Mat, 2021, 115: Art 133).

Algorithm for Visualization of Zero Divisor Graphs of the Ring ℤn Using MAPLE Coding

This research investigates the comparative efficacy of generating zero divisor graphs (ZDGs) of the ring of integers ℤn modulo n using MAPLE algorithm. Zero divisor graphs, pivotal in the study of ring theory, depict relationships between elements of a ring that multiply to zero. The paper explores the development and implementation of algorithms in MAPLE for constructing these ZDGs. The comparative study aims to discern the strengths, limitations, and computational efficiency of different MAPLE algorithms for creating zero divisor graphs offering insights for mathematicians, researchers, and computational enthusiasts involved in ring theory and mathematical computations.

Cayley子集S全为2阶元的点稳定子为F20的5度无核2-正则Cayley图

在具有较高对称性的图中,正则Cayley图是一类特殊的对称图。称一个图Γ为2-正则图,如果Γ的全自同构群AutΓ作用在2-弧集上正则。本文给出了点稳定子为F20的5度无核2-正则Cayley图在Cayley子集全为2阶元情况下的全部分类。

On an Invariant of Tournament Digraphs

To date, it is unknown whether it is possible to construct a complete graph invariant in polynomial time, so fast algorithms for checking non-isomorphism are important, including heuristic algorithms, and for successful implementations of such heuristics, both the tasks of some modification of previously described graph invariants and the description of new invariants remain relevant. Many of the described invariants make it possible to distinguish a larger number of graphs in the real time of a computer program. In this paper, we propose an invariant for a special kind of directed graphs, namely, for tournaments. The last ones, from our point of view, are interesting because when fixing the order of vertices, the number of different tournaments is exactly equal to the number of undirected graphs, also with fixing the order of vertices. In the invariant we are considering, all possible tournaments consisting of a subset of vertices of a given digraph with the same set of arcs are iterated over. For such subset tournaments, the places are calculated in the usual way, which are summed up to obtain the final values of the points of the vertices;these points form the proposed invariant. As we expected, calculations of the new invariant showed that it does not coincide with the most natural invariant for tournaments, in which the number of points is calculated for each participant. So far, we have conducted a small number of computational experiments, and the minimum value of the pair correlation between the sequences representing these two invariants that we found is for dimension 15.

Weak External Bisection of Some Graphs

Let G be a graph. A bipartition of G is a bipartition of V (G) with V (G) = V1 ∪ V2 and V1 ∩ V2 = ∅. If a bipartition satisfies ∥V1∣ - ∣V2∥ ≤ 1, we call it a bisection. The research in this paper is mainly based on a conjecture proposed by Bollobás and Scott. The conjecture is that every graph G has a bisection (V1, V2) such that ∀v ∈ V1, at least half minuses one of the neighbors of v are in the V2;∀v ∈ V2, at least half minuses one of the neighbors of v are in the V1. In this paper, we confirm this conjecture for some bipartite graphs, crown graphs and windmill graphs.

On the (Δ + 2)-Total-Colorability of Planar Graphs with 7-Cycles Containing at Most Two Chords

The Total Coloring Conjecture (TCC) proposes that every simple graph G is (Δ + 2)-totally-colorable, where Δ is the maximum degree of G. For planar graph, TCC is open only in case Δ = 6. In this paper, we prove that TCC holds for planar graph with Δ = 6 and every 7-cycle contains at most two chords.

基于Cayley-Dickson构造的二阶实矩阵幂的计算方法

提出了一种基于Cayley-Dickson构造的二阶实矩阵方幂的计算方法 .该方法利用Cayley-Dickson构造,将二阶实矩阵转化为四个实数和span{σ_(0),σ_(1),σ_(2),σ_(3)}中四个实2阶基矩阵σ_(0),σ_(1),σ_(2),σ_(3)的乘积之和,并利用这四个实数构成的4维向量的递推公式来计算二阶实矩阵的幂.通过MATLAB程序的验证,结果表明,该方法具有较高的计算效率,有效地解决了二阶实矩阵方幂的精确计算问题.

Revealing the Hidden Mathematical Beauties of the Cayley-Hamilton Method

The inversion of a non-singular square matrix applying a Computer Algebra System (CAS) is straightforward. The CASs make the numeric computation efficient but mock the mathematical characteristics. The algorithms conducive to the output are sealed and inaccessible. In practice, other than the CPU timing, the applied inversion method is irrelevant. This research-oriented article discusses one such process, the Cayley-Hamilton (C.H.) [1]. Pursuing the process symbolically reveals its unpublished hidden mathematical characteristics even in the original article [1]. This article expands the general vision of the original named method without altering its practical applications. We have used the famous CAS Mathematica [2]. We have briefed the theory behind the method and applied it to different-sized symbolic and numeric matrices. The results are compared to the named CAS’s sealed, packaged library commands. The codes are given, and the algorithms are unsealed.

基于改进GraphSAGE算法的浏览器指纹追踪

当前Web追踪领域主要使用浏览器指纹对用户进行追踪。针对浏览器指纹追踪技术存在指纹随时间动态变化、不易长期追踪等问题,提出一种关注节点和边缘特征的改进图采样聚合算法(An Improved Graph SAmple and AGgregatE with Both Node and Edge Features,NE-GraphSAGE)用于浏览器指纹追踪。首先以浏览器指纹为节点、指纹之间特征相似度为边构建图数据。其次对图神经网络中的GraphSAGE算法进行改进使其不仅能关注节点特征,而且能捕获边缘信息并对边缘分类,从而识别指纹。最后将NE-GraphSAGE算法与Eckersley算法、FPStalker算法和LSTM算法进行对比,验证NE-GraphSAGE算法的识别效果。实验结果表明,NE-GraphSAGE算法在准确率和追踪时长上均有不同程度的提升,最大追踪时长可达80天,相比其他3种算法性能更优,验证了NE-GraphSAGE算法对浏览器指纹长期追踪的能力。

奇图O_(k)及其非Cayley性

奇图O_(k)(k≥2)是一类高度对称的、具有独特结构的简单无向图.例如O_(3)被发现是点传递图中既非Hamilton图也非Cayley图的典型例子.该文给出了一些关于奇图的新结果,特别是在奇图中找到了非Cayley高弧传递图的无限族.

A CLASS OF HAMILTONIAN AND EDGE SYMMETRIC CAYLEY GRAPHS ON SYMMETRIC GROUPS

Let S\-n be the symmetric group, g\++\-i=(123i),g\+-\-i=(1i32) and M\++\-n={g\++\-i∶4≤i≤n}, then M\++\-n is a minimal generating set of S\-n ,where n ≥5.It is proved that Cayley graph Cay( S\-n,M\++\-n∪M\+-\-n) is Hamiltonian and edge symmetric.

Connectivities of Minimal Cayley Coset Digraphs

we prove that the Connectivities of Minimal Cayley Coset Digraphs are their regular degrees. Connectivity of transitive digraphs and a combinatorial propertyof finite groups Ann., Discrete Math., 8 1980 61--64 Meng Jixiang and Huang Qiongxiang On the connectivity of Cayley digraphs, to appear Sabidussi, G. Vertex transitive graphs Monatsh. Math., 68 1969 426--438 Watkins, M. E. Connectivity of transitive graphs J. Combin. Theory, 8 1970 23--29 Zemor, G. On positive and negative atoms of Cayley digraphs Discrete Applied Math., 23 1989 193--195 Department of Mathematics,Xinjiang University,Urumpi 830046.APPLIED MATHEMATICS 3. Statement of Inexact Method Here we assume F to be continuousely differentiable. Inexact Newton method was first studied in the solution of smooth equations (see ). Now, such a technique has been widely used in optimizations, nonlinear complementarity problems and nonsmooth equations (see, and , etc.) In order to establish the related inexact methods,we introduce a nonlinear operator T(x): R n R n . Its components are defined as follows: (T(x)p) i=[HL(2:1,Z;2,Z] (x k+p k) i, if i∈(x k), H i(x k)+ min {(p k) i,F i(x k) Tp k}, if i∈(x k), F i(x k)+F i(x k) Tp k, i∈(x k).(3.1) Then, it is clear that the subproblem (2.5) turns to T(x k)p k=0.(3.2) In inexact algorithm, we determine p k in the followinginexact way ( see ). ‖T(x k)p k‖ υ k‖H(x k)‖,(3.3) where υ k is a given positive sequence. It is then obviously that (3.2),or equivalently (2.5), is a special case of (3.3) corresponding to υ k=0 . In particular, (3.3) can be used as a termination rule of the iterative process for solving (2.5). The following proposition shows the existence of λ k satisfying (2.4). Proposition 3.1. Let F be continuously differe ntiable. υ k is chosen so that υ k for some constant ∈(0,1). Then p k generated by (3.3) is a descent direction of θ at x k, and for some constant σ∈(0, min (1/2,1- holds θ(x k)-θ(x k+λ kp k) 2σλ kθ(x k)(3.4) for all sufficiently small λ k>0. Proof For simplification, we omit the lower subscripts k and denote (x k) i , H i(x k) , (BH(x k)p k) i , etc.by x i , H i , (BHp) i , etc. respectively. To estimate the directional derivative of θ at x k along p k , we divide it into three parts: D p k θ(x k)=H T(x k)BH(x k)p k=T 1+T 2+T 3,(3.5) where T 1=Σ i∈α k H i(BHp) i , T 2=Σ i∈β k H i(BHp) i , T 3=Σ i∈γ k H i(BHp) i . Consider i∈α k= k∪α -(x k) . In this case, we always have H i(BH(x)p) i=H i 2+H i(x i+p i) . If i∈ k , then H i(BHp) i -H i 2+|H i‖(T(x)p) i|. If i∈α -(x k) , then x i<0 . We have either x i+p i 0 , or x i+p i<0 . When x i+p i 0 , we get H i(BH(x)p) i -H i 2 .In the later case, x i+p i<0 , so H i(BH(x)p) i=-H i 2+|H i‖x i+p i|. Then, by elementary computation, we deduce that T 1 -Σi∈α kH i 2+Σ i∈α k|H i‖(T(x)p) i|.(3.6) Received March 1, 1995. 1991 MR Subject Classification: 05C25

Hamiltonian Cayley Digraphs on Direct Products of Dihedral Groups

We prove that a Cayley digraph on the direct product of dihedral groups D2n × D2m with outdegree two is Hamiltonian if and only if it is connected.

pq阶群上的次正规边传递Cayley图

分析了阶为两个素数乘积的非交换群上的正规边传递Cayley图,并对这类群上的次正规边传递Cayley图进行了分类.

Continuous-Time Classical and Quantum Random Walk on Direct Product of Cayley Graphs

在这篇论文，我们定义图的直接产品并且为获得在顶点上观察粒子在的概率给一个配方连续时间古典并且量随机散步。在配方，在图的直接产品上观察粒子的概率被概率的增加获得在上相应于亚图，在这个方法对在复杂的图上决定散步的概率有用的地方。用这个方法，我们计算概率连续时间古典并且有限直接产品 Cayley 图的许多上的量随机散步(完全的周期，完全的 K n , 宪章和 n 立方体) 。另外，我们询问古典状态静止一致分发作为 t → 被到达∞ 要不是量，状态总是没满足。