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.

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.

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。

Unicyclic graphs with extremal Lanzhou index

Very recently D.Vukicevic et al.[8]introduced a new topological index for a molecular graph G named Lanzhou index as∑_(u∈V(G))d_(u)d^(2)_(u),where d_(u)and d_(u)denote the degree of vertex u in G and in its complement respectively.Lanzhou index Lz(G)can be expressed as(n-1)M_(1)(G)-F(G),where M_(1)(G)and F(G)denote the first Zagreb index and the forgotten index of G respectively,and n is the number of vertices in G.It turns out that Lanzhou index outperforms M_(1)(G)and F(G)in predicting the logarithm of the octanol-water partition coefficient for octane and nonane isomers.It was shown that stars and balanced double stars are the minimal and maximal trees for Lanzhou index respectively.In this paper,we determine the unicyclic graphs and the unicyclic chemical graphs with the minimum and maximum Lanzhou indices separately.

On the ordering of the Kirchhoff indices of the complements of trees and unicyclic graphs

The Kirchhoff index Kf(G) of a graph G is defined to be the sum of the resistance distances between all pairs of vertices of G. In this paper, we develop a novel method for ordering the Kirchhoff indices of the complements of trees and unicyclic graphs. With this method, we determine the first five maximum values of Kf■ and the first four maximum values of Kf(ū),where ■ and ū are the complements of a tree T and unicyclic graph U, respectively.

A Note on the Nullity of Unicyclic Graphs

The nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum. In this paper we show the expression of the nullity and nullity set of unicyclic graphs with n vertices and girth r, and characterize the unicyclic graphs with extremal nullity.

Ordering of Unicyclic Graphs with Perfect Matchings by Minimal Matching Energies

In 2012, Gutman and Wagner proposed the concept of the matching energy of a graph and pointed out that its chemical applications can go back to the 1970s. The matching energy of a graph is defined as the sum of the absolute values of the zeros of its matching polynomial. Let u and v be the non-isolated vertices of the graphs G and H with the same order, respectively. Let wi?be a non-isolated vertex of graph Gi?where i=1, 2, …, k. We use Gu(k)?(respectively, Hv(k)) to denote the graph which is the coalescence of G (respectively, H) and G1, G2,…, Gk?by identifying the vertices u (respectively, v) and w1, w2,…, wk. In this paper, we first present a new technique of directly comparing the matching energies of Gu(k)?and Hv(k), which can tackle some quasi-order incomparable problems. As the applications of the technique, then we can determine the unicyclic graphs with perfect matchings of order 2n with the first to the ninth smallest matching energies for all n≥211.

Pythagorean Neutrosophic Planar Graphs with an Application in Decision-Making

Graph theory has a significant impact and is crucial in the structure of many real-life situations.To simulate uncertainty and ambiguity,many extensions of graph theoretical notions were created.Planar graphs play a vital role in modelling which has the property of non-crossing edges.Although crossing edges benefit,they have some drawbacks,which paved the way for the introduction of planar graphs.The overall purpose of the study is to contribute to the conceptual development of the Pythagorean Neutrosophic graph.The basic methodology of our research is the incorporation of the analogous concepts of planar graphs in the Pythagorean Neutrosophic graphs.The significant finding of our research is the introduction of Pythagorean Neutrosophic Planar graphs,a conceptual blending of Pythagorean Neutro-sophic and Planar graphs.The idea of Pythagorean Neutrosophic multigraphs and dual graphs are also introduced to deal with the ambiguous situations.This paper investigates the Pythagorean Neutrosophic planar values,which form the edges of the Pythagorean neutrosophic graphs.The concept of Pythagorean Neutrosophic dual graphs,isomorphism,co-weak and weak isomorphism have also been explored for Pythagorean Neutrosophic planar graphs.A decision-making algorithm was proposed with a numerical illustra-tion by using the Pythagorean Neutrosophic fuzzy graph.

On the Spectral Properties of Graphs with Rank 4

Let G be a graph and A(G) the adjacency matrix of G. The spectrum of G is the eigenvalues together with their multiplicities of A(G). Chang et al. (2011) characterized the structures of all graphs with rank 4. Monsalve and Rada (2021) gave the bound of spectral radius of all graphs with rank 4. Based on these results as above, we further investigate the spectral properties of graphs with rank 4. And we give the expressions of the spectral radius and energy of all graphs with rank 4. In particular, we show that some graphs with rank 4 are determined by their spectra.

Estrada index of dynamic random graphs

The Estrada index of a graph G on n vertices is defined by EE(G)=∑^(n)_(i=1)^(eλ_(i)),whereλ_(1),λ_(2),···,λ_(n)are the adjacency eigenvalues of G.We define two general types of dynamic graphs evolving according to continuous-time Markov processes with their stationary distributions matching the Erd¨os-R´enyi random graph and the random graph with given expected degrees,respectively.We formulate some new estimates and upper and lower bounds for the Estrada indices of these dynamic graphs.

Singularity of Two Kinds of Quadcyclic Peacock Graphs

Let G be a graph. G is singular if and only if the adjacency matrix of graph G is singular. The adjacency matrix of graph G is singular if and only if there is at least one zero eigenvalue. The study of the singularity of graphs is of great significance for better characterizing the properties of graphs. The following definitions are given. There are 4 paths, the starting points of the four paths are bonded into one point, and the ending point of each path is bonded to a cycle respectively, so this graph is called a kind of quadcyclic peacock graph. And in this kind of quadcyclic peacock graph assuming the number of points on the four cycles is a1, a2, a3, a4, and the number of points on the four paths is s1, s2, s3, s4, respectively. This type of graph is denoted by γ (a1, a2, a3, a4, s1, s2, s3, s4), called γ graph. And let γ (a1, a2, a3, a4, 1, 1, 1, 1) = δ (a1, a2, a3, a4), this type four cycles peacock graph called δ graph. In this paper, we give the necessary and sufficient conditions for the singularity of γ graph and δ graph.

A Remark on the Characterization of Triangulated Graphs

In this study, we consider the problem of triangulated graphs. Precisely we give a necessary and sufficient condition for a graph to be triangulated. This gives an alternative characterization of triangulated graphs. Our method is based on the so-called perfectly nested sequences.

On a Class of Semigroup Graphs

Let G = Γ(S) be a semigroup graph, i.e., a zero-divisor graph of a semigroup S with zero element 0. For any adjacent vertices x, y in G, denote C(x,y) = {z∈V(G) | N (z) = {x,y}}. Assume that in G there exist two adjacent vertices x, y, a vertex s∈C(x,y) and a vertex z such that d (s,z) = 3. This paper studies algebraic properties of S with such graphs G = Γ(S), giving some sub-semigroups and ideals of S. It constructs some classes of such semigroup graphs and classifies all semigroup graphs with the property in two cases.

A Note on the Inverse Connected p-Median Problem on Block Graphs

Recently, the inverse connected p-median problem on block graphs G(V,E,w) under various cost functions, say rectilinear norm, Chebyshev norm, and bottleneck Hamming distance. Their contributions include finding a necessary and sufficient condition for the connected p-median problem on block graphs, developing algorithms and showing that these problems can be solved in O(n log n) time, where n is the number of vertices in the underlying block graph. Using similar technique, we show that some results are incorrect by a counter-example. Then we redefine some notations, reprove Theorem 1 and redescribe Theorem 2, Theorem 3 and Theorem 4.

Hopf Algebra of Labeled Simple Graphs

A lot of combinatorial objects have a natural bialgebra structure. In this paper, we prove that the vector space spanned by labeled simple graphs is a bialgebra with the conjunction product and the unshuffle coproduct. In fact, it is a Hopf algebra since it is graded connected. The main conclusions are that the vector space spanned by labeled simple graphs arising from the unshuffle coproduct is a Hopf algebra and that there is a Hopf homomorphism from permutations to label simple graphs.

ON THE GRAPHS OF PRODUCTS OF CONTINUOUS FUNCTIONS AND FRACTAL DIMENSIONS

In this paper,we consider the graph of the product of continuous functions in terms of Hausdorff and packing dimensions.More precisely,we show that,given a real number 1≤β≤2,any real-valued continuous function in C([0,1])can be decomposed into a product of two real-valued continuous functions,each having a graph of Hausdorff dimensionβ.In addition,a product decomposition result for the packing dimension is obtained.This work answers affirmatively two questions raised by Verma and Priyadarshi[14].

图计算体系结构和系统软件关键技术综述

图计算作为分析事物之间关联关系的重要工具,近年来已成为各国政府及公司争夺的关键技术.学术界和工业界在图计算体系结构和系统软件关键技术方面取得了一定进展.然而,现实场景图计算大多具有动态变化、应用需求复杂多样等特征.这给图计算在基础理论、体系架构和系统软件关键技术方面提出了新的需求,同时也带来了新的挑战.为应对这些挑战,科研人员提出了一系列图计算系统或图计算加速器,通过高性能计算、并行计算等技术来优化图计算过程.综述国内外图计算体系结构和系统软件关键技术的研究发展现状,对国内外研究的最新进展进行归纳、比较和分析,并结合国家发展战略和重大应用需求,选取与我国国计民生密切相关的领域,从典型应用分析总结图计算相关技术的行业进展.最后,就未来的技术挑战和研究方向进行展望.