<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

The products of graphs discussed in this paper are the following four kinds: the Cartesian product of graphs, the tensor product of graphs, the lexicographic product of graphs and the strong direct product of graphs. It is proved that:① If the graphs G 1 and G 2 are the connected graphs, then the Cartesian product, the lexicographic product and the strong direct product in the products of graphs, are the path positive graphs. ② If the tensor product is a path positive graph if and only if the graph G 1 and G 2 are the connected graphs, and the graph G 1 or G 2 has an odd cycle and max{ λ 1μ 1,λ nμ m}≥2 in which λ 1 and λ n [ or μ 1 and μ m] are maximum and minimum characteristic values of graph G 1 [ or G 2 ], respectively.

Circular L(j,k)-labeling numbers of trees and products of graphs

Let j, k and m be three positive integers, a circular m-L（j, k）-labeling of a graph G is a mapping f： V（G）→{0, 1, …, m-1}such that f（u）-f（v）m≥j if u and v are adjacent, and f（u）-f（v）m≥k if u and v are at distance two,where a-bm=min{a-b,m-a-b}. The minimum m such that there exists a circular m-L（j, k）-labeling of G is called the circular L（j, k）-labeling number of G and is denoted by σj, k（G）. For any two positive integers j and k with j≤k,the circular L（j, k）-labeling numbers of trees, the Cartesian product and the direct product of two complete graphs are determined.

On the Transitivity of the Strong Product of Graphs

Since many large graphs are composed from some existing smaller graphs by using graph operations, say, the Cartesian product, the Lexicographic product and the Strong product. Many properties of such large graphs are closely related to those of the corresponding smaller ones. In this short note, we give some properties of the Strong product of vertex-transitive graphs. In particular, we show that the Strong product of Cayley graphs is still a Cayley graph.

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

In this paper we define direct product of graphs and give a recipe for obtaining probability of observing particle on vertices in the continuous-time classical and quantum random walk. In the recipe, the probability of observing particle on direct product of graph is obtained by multiplication of probability on the corresponding to sub-graphs, where this method is useful to determining probability of walk on compficated graphs. Using this method, we calculate the probability of Continuous-time classical and quantum random walks on many of finite direct product Cayley graphs （complete cycle, complete Kn, charter and n-cube）. Also, we inquire that the classical state the stationary uniform distribution is reached as t→∞ but for quantum state is not always satisfied.

<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.

On 3rd-Dimensional Product of Vertex Measurable Graphs

In this paper, first, a 3rd-dimensional vertex measurable graphs G is defined, which is an extension of the concept that was introduced in [3]. G = G1 × G2 × G3 is a graph defined over algebra ζ1 ×ζz × ζ3, which consists of all vertex sets that produce sub graphs of G. G1,G2, and G3 are three simple graphs, provided that （G1,ζ1）,（G2,ζz）, and （G3,ζ3） are three vertex measure spaces. Second, in order to maximize the edge＇s set, we present an alternative version of the definition of two-dimension Cartesian product of vertex measurable graphs that was given in [3], with preserving the same properties of the graphs and sub graphs that were illustrated.

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.

The Crossing Number of Cartesian Products of Stars with 5-vertex Graphs II

The crossing number of cartesian products of paths and cycles with 5-vertex graphs mostly are known, but only few cartesian products of 5-vertex graphs with star K 1,n are known. In this paper, we will extent those results, and determine the crossing numbers of cartesian products of two 5-vertex graphs with star K 1,n .

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。