A Note on SK, SK₁, SK₂Indices of Interval Weighted Graphs

In this study, the SK, SK1 and SK2 indices are defined on weighted graphs. Then, the SK, SK1 and SK2 indices are defined on interval weighted graphs. Their behaviors are investigated under some graph operations by using these definitions.

Improved frequency standard via weighted graph states

We study the spin squeezing property of weighted graph states,which can be used to improve sensitivity in interferometry.We study the time evolution of spin squeezing under local decoherence acting independently on each qubit.Based on the analysis,the spin squeezing of the weighted graph states is somehow robust in the presence of decoherence and the decoherence limit in the improvement of the interferometric sensitivity is still achievable.Furthermore,one can obtain the optimal improvement of sensitivity by tuning the weighted of each edges of the weighted graph state.

w-DENSITY AND w-BALANCED PROPERTY OF WEIGHTED GRAPHS

The notion of w-density for the graphs with positive weights on vertices and nonnegative weights on edges is introduced.A weighted graph is called w-balanced if its w-density is no less than the w-density of any subgraph of it.In this paper,a good characterization of w-balanced weighted graphs is given.Applying this characterization,many large w-balanced weighted graphs are formed by combining smaller ones.In the case where a graph is not w-balanced,a polynomial-time algorithm to find a subgraph of maximum w-density is proposed.It is shown that the w-density theory is closely related to the study of SEW(G,w) games.

A Dirac Type Condition for Heavy Cycles in Weighted Graphs

In this paper we give a Dirac type condition for heavy cycles in a 3-connected weighted graph, reading that if d^w（v）≥ d for all v ∈ V（G）／{x} and w（uz） = w（vz）, when uz, vz ∈ E（G） and uv ∈/ E（G）. Then G contains either an （x, y）-cycle of weight at least 2d or a Hamilton cycle.

Weighted graph convolutional networks based on network node degree and efficiency

In the study of graph convolutional networks,the information aggregation of nodes is important for downstream tasks.However,current graph convolutional networks do not differentiate the importance of different neighboring nodes from the perspective of network topology when ag-gregating messages from neighboring nodes.Therefore,based on network topology,this paper proposes a weighted graph convolutional network based on network node degree and efficiency(W-GCN)model for semi-supervised node classification.To distinguish the importance of nodes,this paper uses the degree and the efficiency of nodes in the network to construct the impor-tance matrix of nodes,rather than the adjacency matrix,which usually is a normalized symmetry Laplacian matrix in graph convolutional network.So that weights of neighbor nodes can be as-signed respectively in the process of graph convolution operation.The proposed method is ex-amined through several real benchmark datasets(Cora,CiteSeer and PubMed)in the experimen-tal part.And compared with the graph convolutional network method.The experimental results show that the W-GCN model proposed in this paper is better than the graph convolutional net-work model in prediction accuracy and achieves better results.

Dynamic airspace configuration method based on a weighted graph model

This paper proposes a new method for dynamic airspace configuration based on a weighted graph model. The method begins with the construction of an undirected graph for the given airspace, where the vertices represent those key points such as airports, waypoints, and the edges represent those air routes. Those vertices are used as the sites of Voronoi diagram, which divides the airspace into units called as cells. Then, aircraft counts of both each cell and of each air-route are computed. Thus, by assigning both the vertices and the edges with those aircraft counts, a weighted graph model comes into being. Accordingly the airspace configuration problem is described as a weighted graph partitioning problem. Then, the problem is solved by a graph partitioning algorithm, which is a mixture of general weighted graph cuts algorithm, an optimal dynamic load balancing algorithm and a heuristic algorithm. After the cuts algorithm partitions the model into sub-graphs, the load balancing algorithm together with the heuristic algorithm transfers aircraft counts to balance workload among sub-graphs. Lastly, airspace configuration is completed by determining the sector boundaries. The simulation result shows that the designed sectors satisfy not only workload balancing condition, but also the constraints such as convexity, connectivity, as well as minimum distance constraint.

Lower Bounds on the(Laplacian) Spectral Radius of Weighted Graphs

The weighted graphs, where the edge weights are positive numbers, are considered. The authors obtain some lower bounds on the spectral radius and the Laplacian spectral radius of weighted graphs, and characterize the graphs for which the bounds are attained. Moreover, some known lower bounds on the spectral radius and the Laplacian spectral radius of unweighted graphs can be deduced from the bounds.

Heavy Cycles in 2-connected Triangle-free Weighted Graphs

A weighted graph is one in which every edge e is assigned a nonnegative number, called the weight of e. The sum of the weights of the edges incident with a vertex v is called the weighted degree of v, denoted by dw（v）. The weight of a cycle is defined as the sum of the weights of its edges. Fujisawa proved that if G is a 2-connected triangle-free weighted graph such that the minimum weighted degree of G is at least d, then G contains a cycle of weight at least 2d. In this paper, we proved that if G is a 2-connected triangle-free weighted graph of even size such that dw（u） ＋ dw（v） ≥ 2d holds for any pair of nonadjacent vertices u, v ∈ V（G）, then G contains a cycle of weight at least 2d.

Weighted Graph Form of Structures and Its Application in Robustness Analysis

Structural robustness is the concept to evaluate whether local damages to the structure will cause disproportional consequences. It is one of the most important indexes to keep the structural safety, especially to consider a special loading named as "human active damage". In the present paper, the loaded structure is analyzed by a weighted graph. The joints and members of the structure correspond to the vertexes and edges of the graph, and the ratio of the most dangerous stress state to the material strength of each member is treated as the weight of each edge. Based on the quantitative description of the structural topology, the structure graph is expressed as a hierarchical model which is built by a set of vertex-connected units. The local damage can be expressed as the deterioration of the unit(s), while the final possible failure mode of the structure can be obtained by a specific assignment of its weighted graph. In this way, the relationship between the structural behavior and the combined damages of the subordinate units in each hierarchy can be formed as an envelope diagram. This diagram exactly shows the contribution of each subordinate unit to the robustness of the whole structure. Furthermore, the most vulnerable part, as well as the topologic difference between the subordinates, can be found visually.

On Heavy Paths in 2-connected Weighted Graphs

A weighted graph is a graph in which every edge is assigned a non-negative real number. In a weighted graph, the weight of a path is the sum of the weights of its edges, and the weighed degree of a vertex is the sum of the weights of the edges incident with it. In this paper we give three weighted degree conditions for the existence of heavy or Hamilton paths with one or two given end-vertices in 2-connected weighted graphs.

Parametric Transformation of Timed Weighted Marked Graphs:Applications in Optimal Resource Allocation

Timed weighted marked graphs are a subclass of timed Petri nets that have wide applications in the control and performance analysis of flexible manufacturing systems.Due to the existence of multiplicities(i.e.,weights)on edges,the performance analysis and resource optimization of such graphs represent a challenging problem.In this paper,we develop an approach to transform a timed weighted marked graph whose initial marking is not given,into an equivalent parametric timed marked graph where the edges have unitary weights.In order to explore an optimal resource allocation policy for a system,an analytical method is developed for the resource optimization of timed weighted marked graphs by studying an equivalent net.Finally,we apply the proposed method to a flexible manufacturing system and compare the results with a previous heuristic approach.Simulation analysis shows that the developed approach is superior to the heuristic approach.

Drawing Weighted Directed Graph from It's Adjacency Matrix

This paper proposes an algorithm for building weighted directed graph, defmes the weighted directed relationship matrix of the graph, and describes algorithm implementation using this matrix. Based on this algorithm, an effective way for building and drawing weighted directed graphs is presented, forming a foundation for visual implementation of the algorithm in the graph theory.

The <i>H</i>-Decomposition Problem for Graphs

The concept of H-decompositions of graphs was first introduced by Erd?s, Goodman and Pósa in 1966, who were motivated by the problem of representing graphs by set intersections. Given graphs G and H, an H-decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a graph isomorphic to H. Let Ф(n,H) be the smallest number Ф, such that, any graph of order n admits an H-decomposition with at most Ф parts. The exact computation of Ф(n,H) for an arbitrary H is still an open problem. Recently, a few papers have been published about this problem. In this survey we will bring together all the results about H-decompositions. We will also introduce two new related problems, namely Weighted H-Decompositions of graphs and Monochromatic H-Decom- positions of graphs.

A Note on the Spectral Radius of Weighted Signless Laplacian Matrix

A weighted graph is a graph that has a numeric label associated with each edge, called the weight of edge. In many applications, the edge weights are usually represented by nonnegative integers or square matrices. The weighted signless Laplacian matrix of a weighted graph is defined as the sum of adjacency matrix and degree matrix of same weighted graph. In this paper, a brief overview of the notation and concepts of weighted graphs that will be used throughout this study is given. In Section 2, the weighted signless Laplacian matrix of simple connected weighted graphs is considered, some upper bounds for the spectral radius of the weighted signless Laplacian matrix are obtained and some results on weighted and unweighted graphs are found.

Application of rough graph in relationship mining

Based on the definition of class shortest path in weighted rough graph, class shortest path algorithm in weighted rough graph is presented, which extends classical shortest path algorithm. The application in relationship mining shows effectiveness of it.

An Improved Iterative Reconstruction Algorithm for Traveltime Tomography

Traveltime tomography is a technique to reconstruct acoustic, seismic, or electromagnetic wave-speed distributions from first arrival traveltime data. The ray paths that should be used for tomographic techniques strongly depend on the wave-speed distribution. In this paper, a new method is proposed for finding out the ray paths from Fermat's principle, that means the traveltime of the ray path should be a minimum value. The problem of finding out the ray path is actually an optimum problem. Our new method uses the idea to find out the shortest path in a weighted directed graph to solve the problem. The ray paths found out by this method are used in the iterative reconstruction algorithm. Computer simulation result produced by this reconstruction algorithm is better than that by the conventional ones. It also shows that the new algorithm is effective with good convergency and stability.

Perfect State Transfer on Weighted Abelian Cayley Graphs

Recently,there are extensive studies on perfect state transfer(PST for short)on graphs due to their significant applications in quantum information processing and quantum computations.However,there is not any general characterization of graphs that have PST in literature.In this paper,the authors present a depiction on weighted abelian Cayley graphs having PST.They give a unified approach to describe the periodicity and the existence of PST on some specific graphs.

Symmetry properties of tetraammine platinum(II) with C_(2v) and C_(4v) point groups

Let G be a weighted graph with adjacency matrix A=[aij]. An Euclidean graph associated with a molecule is defined by a weighted graph with adjacency matrix D=[dij], where for i≠j, dij is the Euclidean distance between the nuclei i and j. In this matrix dii can be taken as zero if all the nuclei are equivalent. Otherwise, one may introduce different weights for different nuclei. Balasubramanian (1995) computed the Euclidean graphs and their automorphism groups for benzene, eclipsed and staggered forms of ethane and eclipsed and staggered forms of ferrocene. This paper describes a simple method, by means of which it is possible to calculate the automorphism group of weighted graphs. We apply this method to compute the symmetry of tetraammine platinum(II) with C2v and C4v point groups.

Robust interactive image segmentation via graph-based manifold ranking

Interactive image segmentation aims at classifying the image pixels into foreground and background classes given some foreground and background markers. In this paper, we propose a novel framework for interactive image segmentation that builds upon graph-based manifold ranking model, a graph-based semi-supervised learning technique which can learn very smooth functions with respect to the intrinsic structure revealed by the input data. The final segmentation results are improved by overcoming two core problems of graph construction in traditional models: graph structure and graph edge weights. The user provided scribbles are treated as the must-link and must-not-link constraints. Then we model the graph as an approximatively k-regular sparse graph by integrating these constraints and our extended neighboring spatial relationships into graph structure modeling. The content and labels driven locally adaptive kernel parameter is proposed to tackle the insufficiency of previous models which usually employ a unified kernel parameter. After the graph construction,a novel three-stage strategy is proposed to get the final segmentation results. Due to the sparsity and extended neighboring relationships of our constructed graph and usage of superpixels, our model can provide nearly real-time, user scribble insensitive segmentations which are two core demands in interactive image segmentation. Last but not least, our framework is very easy to be extended to multi-label segmentation,and for some less complicated scenarios, it can even get the segmented object through single line interaction. Experimental results and comparisons with other state-of-the-art methods demonstrate that our framework can efficiently and accurately extract foreground objects from background.

Interpreting the vulnerability of power systems in cascading failures using multi-graph convolutional networks

Analyzing the vulnerability of power systems in cascading failures is generally regarded as a challenging problem. Although existing studies can extract some critical rules, they fail to capture the complex subtleties under different operational conditions. In recent years, several deep learning methods have been applied to address this issue. However, most of the existing deep learning methods consider only the grid topology of a power system in terms of topological connections, but do not encompass a power system’s spatial information such as the electrical distance to increase the accuracy in the process of graph convolution. In this paper, we construct a novel power-weighted line graph that uses power system topology and spatial information to optimize the edge weight assignment of the line graph. Then we propose a multi-graph convolutional network(MGCN) based on a graph classification task, which preserves a power system’s spatial correlations and captures the relationships among physical components. Our model can better handle the problem with power systems that have parallel lines, where our method can maintain desirable accuracy in modeling systems with these extra topology features. To increase the interpretability of the model, we present the MGCN using layer-wise relevance propagation and quantify the contributing factors of model classification.