The g-Good-Neighbor Connectivity of Some Cartesian Product Graphs

The g-good-neighbor connectivity of G is a generalization of the concept of connectivity, which is just for, and an important parameter in measuring the fault tolerance and reliability of interconnection network. Many well-known networks can be constructed by the Cartesian products of some simple graphs. In this paper, we determine the g-good-neighbor connectivity of some Cartesian product graphs. We give the exact value of g-good-neighbor connectivity of the Cartesian product of two complete graphs and for , mesh for , cylindrical grid and torus for .

Rupture Degree of Some Cartesian Product Graphs

The rupture degree of a noncomplete-connected graph G is defined by , where is the number of components of and is the order of the largest component of. In this paper, we determine the rupture degree of some Cartesian product graphs.

Cartesian product over interval valued intuitionistic fuzzy sets

The intuitionistic fuzzy set(IFS) based on fuzzy theory,which is of high efficiency to solve the fuzzy problem, has been introduced by Atanassov. Subsequently, he pushed the research one step further from the IFS to the interval valued intuitionistic fuzzy set(IVIFS). On the basis of fuzzy set(FS), the IFS is a generalization concept. And the IFS is generalized to the IVIFS.In this paper, the definition of the sixth Cartesian product over IVIFSs is first introduced and its some properties are explored.We prove some equalities based on the operation and the relation over IVIFSs. Finally, we present one geometric interpretation and a numerical example of the sixth Cartesian product over IVIFSs.

Linear arboricity of Cartesian products of graphs

A linear forest is a forest whose components are paths. The linear arboricity la （G） of a graph G is the minimum number of linear forests which partition the edge set E（G） of G. The Cartesian product G□H of two graphs G and H is defined as the graph with vertex set V（G□H） = {（u, v）｜ u ∈V（G）, v∈V（H） } and edge set E（G□H） = { （ u, x） （ v, Y）｜u=v and xy∈E（H）, or uv∈E（G） and x=y}. Let Pm and Cm,, respectively, denote the path and cycle on m vertices and K, denote the complete graph on n vertices. It is proved that （Km□Pm）=[n＋1/2]for m≥2,la（Km□Cm）=[n＋2/2],and la（Km□Km）=[n＋m-1/2]. The methods to decompose these graphs into linear forests are given in the proofs. Furthermore, the linear arboricity conjecture is true for these classes of graphs.

Embedding Cartesian Product of Some Graphs in Books

The book embedding of a graph G consists of placing the vertices of G in a line called spine and assigning edges of the graph to pages so that the edges assigned to the same page do not intersect. The number of pages is the minimum number in which the graph can be embedded. In this paper, we study the book embedding of the Cartesian product Pm × Sn, Pm × Wn, Cn × Sm, Cn × Wm, and get an upper bound of their pagenumber.

Cartesian Product Based Transfer Learning Implementation for Brain Tumor Classification

Knowledge-based transfer learning techniques have shown good performance for brain tumor classification,especially with small datasets.However,to obtain an optimized model for targeted brain tumor classification,it is challenging to select a pre-trained deep learning(DL)model,optimal values of hyperparameters,and optimization algorithm(solver).This paper first presents a brief review of recent literature related to brain tumor classification.Secondly,a robust framework for implementing the transfer learning technique is proposed.In the proposed framework,a Cartesian product matrix is generated to determine the optimal values of the two important hyperparameters:batch size and learning rate.An extensive exercise consisting of 435 simulations for 11 state-of-the-art pre-trained DL models was performed using 16 paired hyperparameters from the Cartesian product matrix to input the model with the three most popular solvers(stochastic gradient descent with momentum(SGDM),adaptive moment estimation(ADAM),and root mean squared propagation(RMSProp)).The 16 pairs were formed using individual hyperparameter values taken from literature,which generally addressed only one hyperparameter for optimization,rather than making a grid for a particular range.The proposed framework was assessed using a multi-class publicly available dataset consisting of glioma,meningioma,and pituitary tumors.Performance assessment shows that ResNet18 outperforms all other models in terms of accuracy,precision,specificity,and recall(sensitivity).The results are also compared with existing state-of-the-art research work that used the same dataset.The comparison was mainly based on performance metric“accuracy”with support of three other parameters“precision,”“recall,”and“specificity.”The comparison shows that the transfer learning technique,implemented through our proposed framework for brain tumor classification,outperformed all existing approaches.To the best of our knowledge,the proposed framework is an efficient framework that helped reduce the computational complexity and the time to attain optimal values of two important hyperparameters and consequently the optimized model with an accuracy of 99.56%.

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.

The Signed Domination Number of Cartesian Product of Two Paths

Let G be a finite connected simple graph with vertex set V(G) and edge set E(G). A function f:V(G) → {1,1} is a signed dominating function if for every vertex v∈V(G), the closed neighborhood of v contains more vertices with function values 1 than with −1. The signed domination number γs(G) of G is the minimum weight of a signed dominating function on G. In this paper, we calculate The signed domination numbers of the Cartesian product of two paths Pm and Pn for m = 3, 4, 5 and arbitrary n.

Domination Number of Square of Cartesian Products of Cycles

A set ?is a dominating set of G if every vertex of ?is adjacent to at least one vertex of S. The cardinality of the smallest dominating set of G is called the domination number of G. The square G2 of a graph G is obtained from G by adding new edges between every two vertices having distance 2 in G. In this paper we study the domination number of square of graphs, find a bound for domination number of square of Cartesian product of cycles, and find the exact value for some of them.

On the Signed Domination Number of the Cartesian Product of Two Directed Cycles

Let D be a finite simple directed graph with vertex set V(D) and arc set A(D). A function ?is called a signed dominating function (SDF) if ?for each vertex . The weight ?of f is defined by . The signed domination number of a digraph D is . Let Cm × Cn denotes the cartesian product of directed cycles of length m and n. In this paper, we determine the exact values of gs(Cm × Cn) for m = 8, 9, 10 and arbitrary n. Also, we give the exact value of gs(Cm × Cn) when m, ?(mod 3) and bounds for otherwise.

Full Friendly Index Sets of Cartesian Products of Two Cycles

Let G =（V, E） be a connected simple graph. A labeling f ： V → Z2 induces an edge labeling f＊ ： E → Z2 defined by f＊（xy） = f（x） ＋f（y） for each xy ∈ E. For i ∈ Z2, let vf（i） = ｜f^-1（i）｜ and ef（i） = ｜f＊^-1（i）｜. A labeling f is called friendly if ｜vf（1） - vf（0）｜ ≤ 1. For a friendly labeling f of a graph G, we define the friendly index of G under f by if（G） = e（1） - el（0）. The set [if（G） ｜ f is a friendly labeling of G} is called the full friendly index set of G, denoted by FFI（G）. In this paper, we will determine the full friendly index set of every Cartesian product of two cycles.

The Crossing Number of the Cartesian Products of Wm with Pn

Most results on crossing numbers of graphs focus on some special graphs, such as the Cartesian products of small graphs with path, star and cycle. In this paper, we obtain the crossing number formula of Cartesian products of wheel Wm with path Pn for arbitrary m ≥ 3 and n ≥ 1.

Dimensional Results for Cartesian Products of Homogeneous Moran Sets

M（J, {ms ＊ ns}, {Cs}） be the collection of Cartesian products of two homogenous Moran sets with the same ratios {cs} Where J = [0, 1] × [0, 1]. Then the maximal and minimal values of the Hausdorff dimensions for the elements in M are obtained without any restriction on {msns} or {cs}.

L（2, 1）-Circular Labelings of Cartesian Products of Complete Graphs

For positive integers j and k with j ≥ k, an L（j, k）-labeling of a graph G is an assignment of nonnegative integers to V（G） such that the difference between labels of adjacent vertices is at least j, and the difference between labels of vertices that are distance two apart is at least k. The span of an L（j, k）-labeling of a graph G is the difference between the maximum and minimum integers it uses. The λj, k-number of G is the minimum span taken over all L（j, k）-labelings of G. An m-（j, k）-circular labeling of a graph G is a function f ： V（G） →{0, 1, 2,..., m - 1} such that ｜f（u） - f（v）｜m ≥ j if u and v are adjacent; and ｜f（u） - f（v）｜m 〉 k ifu and v are at distance two, where ｜x｜m = min{｜xl｜, m-｜x｜}. The minimum integer m such that there exists an m-（j, k）-circular labeling of G is called the σj,k-number of G and is denoted by σj,k（G）. This paper determines the σ2,1-number of the Cartesian product of any three complete graphs.

Adjacent Vertex Distinguishing Incidence Coloring of the Cartesian Product of Some Graphs

An adjacent vertex distinguishing incidence coloring of graph G is an incidence coloring of G such that no pair of adjacent vertices meets the same set of colors.We obtain the adjacent vertex distinguishing incidence chromatic number of the Cartesian product of a path and a path,a path and a wheel,a path and a fan,and a path and a star.

The Crossing Numbers of Cartesian Products of Stars with a 5-Vertex Graph

In this paper, we compute the crossing number of a specific graph Hn, and then by contraction, we obtain the conclusion that cr（G13 × Sn） = 4[n/2] [n-1/2]＋[n/2] . The result fills up the blank of the crossing numbers of Cartesian products of stars with all 5-vertex graphs presented by Marian Klesc.

Paired Domination of Cartesian Products of Graphs

Let γpr（G） denote the paired domination number and G □ H denote the Cartesian product of graphs G and H. In this paper we show that for all graphs G and H without isolated vertex, γpr（G）γpr（H）≤ 7γpr （G □H）.

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.

基于多器官特征融合的枣品种识别方法

针对自然场景下的枣品种识别问题,以枣果为研究对象的机器视觉技术已成为枣品种精准识别的主流方法之一。针对枣品种存在类间差异小、类内差异大的问题,提出了一种基于多器官特征融合的枣品种识别方法。首先利用YOLO v3检测算法将采集的自然场景图像中的枣果和叶片器官分割提取,提出了基于笛卡尔乘积构建两器官组合对的枣品种多样本数据集,然后基于EfficientNetV2网络模型,设计了能够充分学习两器官特征相关性的融合策略来提升模型性能,引入了逐步迁移训练方式以提升枣品种识别效率。最后,在构建的包含20个枣品种数据集上进行了大量实验,得到97.04%的识别准确率,明显优于现有研究结果,并且在训练时间和收敛速度上,本方法也有一定提升。结果表明该方法能够有效融合枣品种枣果和叶片器官的特征信息,可为其他品种识别研究提供参考。

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.