<i>L</i>(2,1)-Labeling of the Brick Product Graphs

A k-L(2,1)-labeling for a graph G is a function such that whenever and whenever u and v are at distance two apart. The λ-number for G, denoted by λ(G), is the minimum k over all k-L(2,1)-labelings of G. In this paper, we show that for or 11, which confirms Conjecture 6.1 stated in [X. Li, V. Mak-Hau, S. Zhou, The L(2,1)-labelling problem for cubic Cayley graphs on dihedral groups, J. Comb. Optim. (2013) 25: 716-736] in the case when or 11. Moreover, we show that? if 1) either (mod 6), m is odd, r = 3, or 2) (mod 3), m is even (mod 2), r = 0.

Some results on the Induced Matching Partition Number of Product Graphs

The induced matching partition number of graph G is the minimum integer k such that there exists a k-partition(V1,V2,…,Vk) of V(G)such that,for each i(1≤i≤k),G[Vi] is 1-regular.In this paper,we study the induced matching partition number of product graphs.We provide a lower bound and an upper bound for the induced matching partition number of product graphs,and exact results are given for some special product graphs.

The Path-Positive Property on the Products of Graphs

本文所讨论的积图是图的笛卡尔积，图的张量积，图的逻辑积和图的强直积四种积图．证明了：①如果Ｇ１和Ｇ２都是连通图，则积图中笛卡尔积，逻辑积和强直积都是道路正图．②图的张量积是道路正图的是图Ｇ１和Ｇ２是一个连通图，Ｇ１［或Ｇ２］有一个奇圈，且ｍａｘ｛λ１μ１，λｎμｍ｝≥２。

On restricted edge-connectivity of replacement product graphs

This paper considers the edge-connectivity and the restricted edge-connectivity of replacement product graphs, gives some bounds on edge-connectivity and restricted edge-connectivity of replacement product graphs and determines the exact values for some special graphs. In particular, the authors further confirm that under certain conditions, the replacement product of two Cayley graphs is also a Cayley graph, and give a necessary and sufficient condition for such Cayley graphs to have maximum restricted edge-connectivity. Based on these results, we construct a Cayley graph with degree d whose restricted edge-connectivity is equal to d + s for given odd integer d and integer s with d 5 and 1 s d- 3, which answers a problem proposed ten years ago.

<i>L</i>(2,1)-Labeling Number of the Product and the Join Graph on Two Fans

L(2,1)-labeling number of the product and the join graph on two fans are discussed in this paper, we proved that L(2,1)-labeling number of the product graph on two fans is?λ(G) ≤ Δ+3 , L(2,1)-labeling number of the join graph on two fans is?λ(G) ≤ 2Δ+3.

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。

Geodetic Number and Geo-Chromatic Number of 2-Cartesian Product of Some Graphs

A set S ⊆ V (G) is called a geodetic set if every vertex of G lies on a shortest u-v path for some u, v ∈ S, the minimum cardinality among all geodetic sets is called geodetic number and is denoted by . A set C ⊆ V (G) is called a chromatic set if C contains all vertices of different colors in G, the minimum cardinality among all chromatic sets is called the chromatic number and is denoted by . A geo-chromatic set Sc ⊆ V (G) is both a geodetic set and a chromatic set. The geo-chromatic number of G is the minimum cardinality among all geo-chromatic sets of G. In this paper, we determine the geodetic number and the geo-chromatic number of 2-cartesian product of some standard graphs like complete graphs, cycles and paths.

Entanglement and Closest Product States of Graph States with 9 to 11 Qubits

The numbers of local complimentary inequivalent graph states for 9, 10 and 11 qubit systems are 440, 3132, 40457, respectively. We calculate the entanglement, the lower and upper bounds of the entanglement and obtain the closest product states for all these graph states. New patterns of closest product states are analyzed.

Product Cordial Graph in the Context of Some Graph Operations on Gear Graph

A graph is said to be a product cordial graph if there exists a function with each edge assign the label , such that the number of vertices with label 0 and the number of vertices with label 1 differ atmost by 1, and the number of edges with label 0 and the number of edges with label 1 differ by atmost 1. We discuss the product cordial labeling of the graphs obtained by duplication of some graph elements of gear graph. Also, we derive some product cordial graphs obtained by vertex switching operation on gear graph.

Edge Product Cordial Labeling of Some Cycle Related Graphs

For a graph having no isolated vertex, a function is called an edge product cordial labeling of graph G, if the induced vertex labeling function defined by the product of labels of incident edges to each vertex is such that the number of edges with label 0 and the number of edges with label 1 differ by at most 1 and the number of vertices with label 0 and the number of vertices with label 1 also differ by at most 1. In this paper, we discuss edge product cordial labeling for some cycle related graphs.

Bounds on Fractional Domination of Some Products of Graphs

Let γ f(G) and γ~t f(G) be the fractional domination number and fractional total domination number of a graph G respectively. Hare and Stewart gave some exact fractional domination number of P n×P m (grid graph) with small n and m . But for large n and m , it is difficult to decide the exact fractional domination number. Motivated by this, nearly sharp upper and lower bounds are given to the fractional domination number of grid graphs. Furthermore, upper and lower bounds on the fractional total domination number of strong direct product of graphs are given.

Some Edge Product Cordial Graphs in the Context of Duplication of Some Graph Elements

For a graph, a function is called an edge product cordial labeling of G, if the induced vertex labeling function is defined by the product of the labels of the incident edges as such that the number of edges with label 1 and the number of edges with label 0 differ by at most 1 and the number of vertices with label 1 and the number of vertices with label 0 differ by at most 1. In this paper, we show that the graphs obtained by duplication of a vertex, duplication of a vertex by an edge or duplication of an edge by a vertex in a crown graph are edge product cordial. Moreover, we show that the graph obtained by duplication of each of the vertices of degree three by an edge in a gear graph is edge product cordial. We also show that the graph obtained by duplication of each of the pendent vertices by a new vertex in a helm graph is edge product cordial.

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.

强乘积图的最小强半径

A strong product graph is denoted by G_(1)■G_(2),where G_(1) and G_(2) are called its factor graphs.This paper gives the range of the minimum strong radius of the strong product graph.And using the relationship between the cartesian product graph G_(1)■G_(2) and the strong product graph G_(1)■G_(2),another different upper bound of the minimum strong radius of the strong product graph is given.

SOME RESULTS ON DOMINATION NUMBER OF PRODUCTS OF GRAPHS

Let G=(V,E) be a simple graph. A subset D of V is called a dominating set of G if for every vertex x∈V-D,x is adjacent to at least one vertex of D . Let γ(G) and γ c(G) denote the domination and connected domination number of G , respectively. In 1965,Vizing conjectured that if G×H is the Cartesian product of G and H , thenγ(G×H)≥γ(G)·γ(H).In this paper, it is showed that the conjecture holds if γ(H) ≠ γ c(H) .And for paths P m and P n , a lower bound and an upper bound for γ(P m×P n) are obtained.

基于图文多模态融合推理的产品创新方案设计方法研究

目的针对当前产品创新设计领域中对基于图像-文本多模态知识支撑创新设计方法研究不足的问题,提出了一套基于图文多模态的产品创新方案设计方法。方法首先,对设计师的设计草图与文本要求进行预处理,然后引入产品设计知识图谱来促进设计思维的发散和创新;其次,通过微调的生成式预训练变换器模型和扩散模型生成产品方案及其概念图;最后,利用深度多模态设计评估模型对产品设计方案的可行性和市场潜力进行评估。结果通过产品设计知识图谱,及深度多模态设计评估模型的引入,该设计流程可以生成富有创新性且具备可行性的产品方案。结论基于图文多模态的产品创新方案设计流程结合了最新的深度学习技术,不仅提高了设计的效率,还为设计师提供了更广阔的创新视角和灵感来源。

面向产品全生命周期管理的知识模型构建

为有效地描述产品全生命周期管理各阶段产生的无组织散乱状态的数据以及数据之间的联系,论文提出了一种基于知识图谱的产品全生命周期数据建模方法。该方法有效地解决了系统数据模型重复存储以及模型间关联性缺失的问题。同时,为提高数据建模效率,论文提出了融合图神经网络和基于注意力机制的LSTM的算法模型以实现数据自动建模。为验证算法模型的有效性,基于产品数据集进行了算法性能对比实验。结果表明,论文算法相较于其他算法具有更好的性能指标,可以应用于产品全生命周期管理(Product Lifecycle Management,PLM)系统自动构建知识模型。

路与星图的强乘积图的容错直径

设路P_(m)与星图S_(1,n-1)的强乘积图为G=P_(m)S_(1,n-1).首先,通过归纳假设和构造内点或边不交路的方法,结合星图的中心性,给出图G的点容错直径D_(w)(G)和边容错直径D′t(G).结果表明,对图G中发生的任意点或边故障,都有D_(w)(G)≤d(G)+2,D′t(G)≤d(G)+1.其次,通过顶点数和边数构造的不等关系,给出两个极大连通图的强乘积图的点容错直径的上界,以及两个非平凡连通图的强乘积图的边容错直径的上界.

经验模态分解-图神经网络算法预测农产品价格

为了提高图神经网络算法对农产品价格预测精度,采用经验模态分解法按时间片轮转抽取农产品历史价格信号,以便对历史价格信号进行特征提取;将原始价格信号分解成多个本征模态函数及残余项,并根据本征模态函数构建样本特征;根据得到的样本特征构建价格预测图结构,将图结构输出的特征信号通过图神经网络的过渡函数和预测函数,通过不断减小损失值输出农产品价格预测结果。结果表明,经验模态分解可以对原始农产品价格信号的本征模态函数分量进行有效分解和提取,从而使经验模态分解-图神经网络算法的农产品价格预测平均绝对误差减小71.4%;相比于其他类型的预测算法,经验模态分解-图神经网络算法对4类农产品价格预测的平均绝对误差更小,最大值仅为2.465。