Complement Of Graph Research Articles

A unified construction of semiring-homomorphic graph invariants

It has recently been observed by Zuiddam that finite graphs form a preordered commutative semiring under the graph homomorphism preorder together with join and disjunctive product as addition and multiplication, respectively. This led to a new characterization of the Shannon capacity \(\Theta \) via Strassen’s Positivstellensatz: \(\Theta ({\bar{G}}) = \inf _f f(G)\), where \(f: \mathsf {Graph}\rightarrow \mathbb {R}_+\) ranges over all monotone semiring homomorphisms. Constructing and classifying graph invariants \(\mathsf {Graph}\rightarrow \mathbb {R}_+\) which are monotone under graph homomorphisms, additive under join, and multiplicative under disjunctive product is therefore of major interest. We call such invariants semiring-homomorphic. The only known such invariants are all of a fractional nature: the fractional chromatic number, the projective rank, the fractional Haemers bounds, as well as the Lovász number (with the latter two evaluated on the complementary graph). Here, we provide a unified construction of these invariants based on linear-like semiring families of graphs. Along the way, we also investigate the additional algebraic structure on the semiring of graphs corresponding to fractionalization. Linear-like semiring families of graphs are a new notion of combinatorial geometry different from matroids which may be of independent interest.

On graphs with no induced five‐vertex path or paraglider

AbstractGiven two graphs and , a graph is ‐free if it contains no induced subgraph isomorphic to or . For a positive integer , is the chordless path on vertices. A paraglider is the graph that consists of a chorless cycle plus a vertex adjacent to three vertices of the . In this paper, we study the structure of (, paraglider)‐free graphs, and show that every such graph satisfies , where and are the chromatic number and clique number of , respectively. Our bound is attained by the complement of the Clebsch graph on 16 vertices. More strongly, we completely characterize all the (, paraglider)‐free graphs that satisfies . We also construct an infinite family of (, paraglider)‐free graphs such that every graph in the family has . This shows that our upper bound is optimal up to an additive constant and that there is no ‐approximation algorithm for the chromatic number of (, paraglider)‐free graphs for any .

On Graph theoretic index of Complementary Benzenoids And Macromolecules

Background: A topological index of a graph G is a numerical parameter related to G which characterizes its molecular topology. In the field of QSAR and QSPR research, theoretical properties of the chemical compounds and their molecular topological indices such as distance connectivity indices and degree connectivity indices are used to predict the bioactivity of different molecular compounds. Materials and Methods: Such an approach is different from the traditional QSAR methodology, where one employs selected simpler physico-chemical properties to predict biological activities of molecules. In order to obtain the structure-activity relationships in which theoretical and computational methods are necessary to find appropriate representations of the molecular structure of chemical compounds. These representations are realized through the molecular descriptors. Molecular descriptors are numbers containing structural information derived from the structural representation used for molecules under study. Results: A topological index defined on molecular structure G can be considered as a real valued function f :G→ R+ which maps each durg molecular structure to certain real numbers. Graphene sheets are composed of carbon atoms linked in hexagonal shapes with each carbon atom covalently bonded to three other carbon atoms. Each sheet of graphene is only one atom thick and each graphene sheet is considered a single molecule. Graphene has the same structure of carbon atoms linked in hexagonal shapes to form carbon nanotubes, but graphene is flat rather than cylindrical.. This paper addresses the problem of computing the Wiener , First Zagreb index and Forgotten index of Complementary graphs of graphene sheets, triangular benzenoid graph, circumcoronene molecular graph and nanostar dendrimers. Conclusion: The line graphs were used for modeling amino acid sequences of proteins and of the genetic code. The connected graphs are isomorphic to self complementary graphs. Recently, molecular graphs have proved to be highly useful for drugs activity. Non empirical parameters of chemical structures derived from graph theoretic formalisms are being widely used by many researchers in studies pertaining to molecular design, pharmaceutical drug-design, and environmental hazard assessment of chemicals.

EPTAS and Subexponential Algorithm for Maximum Clique on Disk and Unit Ball Graphs

A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for M AXIMUM C LIQUE on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics ’90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show that the disjoint union of two odd cycles is never the complement of a disk graph nor of a unit (3-dimensional) ball graph. From that fact and existing results, we derive a simple QPTAS and a subexponential algorithm running in time 2 Õ( n 2/3 ) for M AXIMUM C LIQUE on disk and unit ball graphs. We then obtain a randomized EPTAS for computing the independence number on graphs having no disjoint union of two odd cycles as an induced subgraph, bounded VC-dimension, and linear independence number. This, in combination with our structural results, yields a randomized EPTAS for M AX C LIQUE on disk and unit ball graphs. M AX C LIQUE on unit ball graphs is equivalent to finding, given a collection of points in R 3 , a maximum subset of points with diameter at most some fixed value. In stark contrast, M AXIMUM C LIQUE on ball graphs and unit 4-dimensional ball graphs, as well as intersection graphs of filled ellipses (even close to unit disks) or filled triangles is unlikely to have such algorithms. Indeed, we show that, for all those problems, there is a constant ratio of approximation that cannot be attained even in time 2 n 1−ɛ , unless the Exponential Time Hypothesis fails.

Remarks on k-Clique, k-Independent Set and 2-Contamination in Complementary Prisms

Complementary prism graphs arise from the disjoint union of a graph [Formula: see text] and its complement [Formula: see text] by adding the edges of a perfect matching joining pairs of corresponding vertices of [Formula: see text] and [Formula: see text]. Classical graph problems such as Clique and Independent Set were proved to be NP-complete on such a class of graphs. In this work, we study the complexity of both problems on complementary prism graphs from the parameterized complexity point of view. First, we prove that both problems admit a kernel and therefore are fixed-parameter tractable (FPT) when parameterized by the size of the solution, [Formula: see text]. Then, we show that [Formula: see text]-Clique and [Formula: see text]-Independent Set on complementary prisms do not admit polynomial kernel when parameterized by [Formula: see text], unless [Formula: see text]. Furthermore, we address the [Formula: see text]-Contamination problem in the context of complementary prisms. This problem consists in completely contaminating a given graph [Formula: see text] using a minimum set of initially infected vertices. For a vertex to be contaminated, it is enough that at least two of its neighbors are contaminated. The propagation of the contamination follows this rule until no more vertex can be contaminated. It is known that the minimum set of initially contaminated vertices necessary to contaminate a complementary prism of connected graphs [Formula: see text] and [Formula: see text] has cardinality at most [Formula: see text]. In this paper, we show that the tight upper bound for this invariant on complementary prisms is [Formula: see text], improving a result of Duarte et al. (2017).

On the Generalized Complement of Some Graphs

On the Generalized Complement of Some Graphs

Some Operations on Complex Vague Graphs with Application

Vague graphs has found its importance in most recent years. It has played the vital role in many real life applications. The mathematical approach of blending different aspects of vague graphs gives a more generalized approach. As a result of it, we introduce the concepts of Complex vague graphs (CVG). In this chapter, we introduce certain notions including union, join, complement and composition of complex vague graphs. This helps in manipulating the real life applications in decision making problems. We also investigate on the ideas of homomorphisms of complex vague graphs. Finally we provide an application of CVG in launching a water reservoir in different locations.

Graphs with large diameter and their two distance forcing number

For a given simple graph G = (V,E), the two distance forcing number Z 2d (G) is defined as the minimum cardinality among all Z 2d -sets in G. This paper examine about Z 2d (G) of some graphs with large diameter. Also we determine the 2-distance forcing number of some complement graphs.

On the Lanzhou Indices of Trees under Graph Decoration

The Lanzhou index of a graph G is defined as the sum of the product between and square of du over all vertices u of G, where du and are respectively the degree of u in G and the degree of u in the complement graph of G. R(G) is obtained from G by adding a new vertex corresponding to each edge of G, then joining each new vertex to the end vertices of the corresponding edge. Lanzhou index is an important topological index. It is closely related to the forgotten index and first Zagreb index of graphs. In this note, we characterize the bound of Lanzhou index of R(T) of a tree T. And the corresponding extremal graphs are also determined.

Graph complement conjecture for classes of shadow graphs

Graph complement conjecture for classes of shadow graphs

Distributed Nonlinear Polynomial Graph Filter and Its Output Graph Spectrum: Filter Analysis and Design

While frequency-domain algorithms have been demonstrated to be powerful for conventional nonlinear signal processing, there is still not much progress in literature dedicated to nonlinear graph signal processing in frequency domain. The difficulties come from the irregular structure of graph signals and the lack of a straightforward description for the nonlinear relationship between input and output graph signals. In this paper, a distributed nonlinear polynomial graph filter (NPGF) is proposed to characterize the nonlinear input-output relationship by employing polynomial-type nonlinearity based on Hadamard product. Then, the nonlinear output graph signal is reformulated based on the multi-dimensional inverse graph Fourier transform (GFT), where the m-dimensional GFT coefficients are defined as explicit functions of input graph spectrum and filter structural factors (filter parameters, graph shift matrix, nonlinear degree, and graph shift order). The effects of filter structural factors on the output graph spectrum are further evaluated by proposing graph frequency response (GFR) coefficients, which inherently characterize the irregular structural behaviors of a distributed NPGF in frequency domain. The nonlinear output graph spectrum is then developed based on GFR vectors by mapping the GFR coefficients in the tensor formulation to the output GFT coefficients in the vector expression. This can greatly improve the efficiency in calculating the output graph signal if the input signal only involves a few frequency components. Finally, algorithms are proposed to design the filter parameters for the desired output signal. Numerical studies are presented to illustrate and validate the proposed frequency-domain framework for nonlinear graph signal processing.

OPEN PACKING NUMBER FOR SOME CLASSES OF PERFECT GRAPHS

Let \(G\) be a graph with the vertex set \(V(G)\). A subset \(S\) of \(V(G)\) is an open packing set of \(G\) if every pair of vertices in \(S\) has no common neighbor in \(G.\) The maximum cardinality of an open packing set of \(G\) is the open packing number of \(G\) and it is denoted by \(\rho^o(G)\). In this paper, the exact values of the open packing numbers for some classes of perfect graphs, such as split graphs, \(\{P_4, C_4\}\)-free graphs, the complement of a bipartite graph, the trestled graph of a perfect graph are obtained.

Total pitchfork domination and its inverse in graphs

New two domination types are introduced in this paper. Let [Formula: see text] be a finite, simple, and undirected graph without isolated vertex. A dominating subset [Formula: see text] is a total pitchfork dominating set if [Formula: see text] for every [Formula: see text] and [Formula: see text] has no isolated vertex. [Formula: see text] is an inverse total pitchfork dominating set if [Formula: see text] is a total pitchfork dominating set of [Formula: see text]. The cardinality of a minimum (inverse) total pitchfork dominating set is the (inverse) total pitchfork domination number ([Formula: see text]) [Formula: see text]. Some properties and bounds are studied associated with maximum degree, minimum degree, order, and size of the graph. These modified domination parameters are applied on some standard and complement graphs.

Physical-Virtual Collaboration Modeling for Intra- and Inter-Station Metro Ridership Prediction

Due to the widespread applications in real-world scenarios, metro ridership prediction is a crucial but challenging task in intelligent transportation systems. However, conventional methods either ignore the topological information of metro systems or directly learn on physical topology, and cannot fully explore the patterns of ridership evolution. To address this problem, we model a metro system as graphs with various topologies and propose a unified Physical-Virtual Collaboration Graph Network (PVCGN), which can effectively learn the complex ridership patterns from the tailor-designed graphs. Specifically, a physical graph is directly built based on the realistic topology of the studied metro system, while a similarity graph and a correlation graph are built with virtual topologies under the guidance of the inter-station passenger flow similarity and correlation. These complementary graphs are incorporated into a Graph Convolution Gated Recurrent Unit (GC-GRU) for spatial-temporal representation learning. Further, a Fully-Connected Gated Recurrent Unit (FC-GRU) is also applied to capture the global evolution tendency. Finally, we develop a Seq2Seq model with GC-GRU and FC-GRU to forecast the future metro ridership sequentially. Extensive experiments on two large-scale benchmarks (e.g., Shanghai Metro and Hangzhou Metro) well demonstrate the superiority of our PVCGN for station-level metro ridership prediction. Moreover, we apply the proposed PVCGN to address the online origin-destination (OD) ridership prediction and the experiment results show the universality of our method. Our code and benchmarks are available at <uri>https://github.com/HCPLab-SYSU/PVCGN</uri>.

Vertex disjoint copies of [formula omitted] in claw-free graphs

A complete bipartite graph with partite sets X and Y, where |X|=1 and |Y|=r, is denoted by K1,r. A graph G is said to be claw-free if G does not contain K1,3 as an induced subgraph. There are several well-known and important families of graphs that are claw-free such as line graphs and complements of triangle-free graphs. Claw-free graphs have numerous interesting properties and applications. This paper considers vertex disjoint K1,4s in claw-free graphs. Let k be an integer with k≥2 and let G be a claw-free graph with |V(G)|≥10k−9. We prove that if the minimum degree of G is at least 4, then it contains k vertex disjoint K1,4s. This result answers the question in [Jiang, Chiba, Fujita, Yan, Discrete Math. 340 (2017) 649–654].

Подход к пространственному моделированию сетей и групп объектов на основе процедуры построения взвешенного графа

The direction of research on the development of a scientific and methodological tool for the analysis of spatial objects in order to determine their generalized spatial parameters was selected. An approach to the problem of modeling networks and groups of objects based on the synthesis of a weighted graph is proposed. The spatial configuration of objects based on the given conditions is described by a weighted graph, the edge length of which is considered as the weight of the edges. A generalization to the typical structure of a spatial graph is formulated; its essence is representation of nodal elements as two-dimensional (polygonal) objects. To take into account the restrictions on the convergence of the vertices described by the buffer zones, a complementary graph is formed. An algorithm for constructing the implementation of a spatial object based on the sequential determination of vertices that comply with the given conditions is proposed. Using the software implementation of the developed algorithm, an experiment was performed to evaluate the spatial parameters of the simulated objects described by typical graph structures. The following parameters were investigated as spatial ones

A segment-graph algorithm for two-objective wireless spectrum allocation in cognitive networks

A segment-graph algorithm is provided for wireless spectrum allocation in cognitive networks based on the cooperation of the secondary users, which aim to maximize the utilization of the limited spectrum bands and minimize the total cost of all the secondary users to buy (or lease) the spectrum bands. The segment-graph of the complement graph of the original interference graph is defined. The proposed segment-graph algorithm is based on finding the complete subgraph of the segment-graph repeatedly. It is proved that the proposed algorithm always obtains the optimal solution if one of the segment-graphs is complete in each iteration. In general cases, the proposed algorithm dramatically reduces the total cost of all the secondary users. Comparative simulations agree with the theoretical results obtained.

High performance GPU primitives for graph-tensor learning operations

Graph-tensor learning operations extend tensor operations by taking the graph structure into account, which have been applied to diverse domains such as image processing and machine learning. However, the running time of graph-tensor operations increases rapidly with the number of nodes and the dimension of data on nodes, making them impractical for real-time applications. In this paper, we propose a GPU library called cuGraph-Tensor for high-performance graph-tensor learning operations, which consists of eight key operations: graph shift (g-shift), graph Fourier transform (g-FT), inverse graph Fourier transform (inverse g-FT), graph filter (g-filter), graph convolution (g-convolution), graph-tensor product (g-product), graph-tensor SVD (g-SVD) and graph-tensor QR (g-QR). cuGraph-Tensor supports scalar, vector, and matrix data processing on each graph node. We propose optimization techniques on computing, memory accesses, and CPU–GPU communications that significantly improve the performance of the graph-tensor learning operations. Using the optimized operations, cuGraph-Tensor builds a graph data completion application for fast and accurate reconstruction of incomplete graph data. In the experiments, the proposed graph learning operations achieve up to 142.12× speedups versus CPU-based GSPBOX and CPU MATLAB implementations running on two Xeon CPUs. The graph data completion application achieves up to 174.38× speedups over the CPU MATLAB implementation, and up to 3.82× speedups with better accuracy over the GPU-based tensor completion in the cuTensor-tubal library.

The forgotten index of complement graph operations and its applications of molecular graph

A topological index of graph \(G\) is a numerical parameter related to graph which characterizes its molecular topology and is usually graph invariant. Topological indices are widely used to determine the correlation between the specific properties of molecules and the biological activity with their configuration in the study of quantitative structure-activity relationships (QSARs). In this paper some basic mathematical operations for the forgotten index of complement graph operations such as join \(\overline {G_1+G_2}\), tensor product \(\overline {G_1 \otimes G_2}\), Cartesian product \(\overline {G_1\times G_2}\), composition \(\overline {G_1\circ G_2}\), strong product \(\overline {G_1\ast G_2}\), disjunction \(\overline {G_1\vee G_2}\) and symmetric difference \(\overline {G_1\oplus G_2}\) will be explained. The results are applied to molecular graph of nanotorus and titania nanotubes.

A study on picture Dombi fuzzy graph

The picture fuzzy graph is a newly introduced fuzzy graph model to handle with uncertain real scenarios, in which a simple fuzzy graph and intuitionistic fuzzy graph may fail to model those problems properly. The picture fuzzy graph is used efficiently in real-world scenarios which involve several answers to these types: yes, no, abstain, and refusal. In this paper, the new idea of Dombi picture fuzzy graph is introduced. We also describe some operations on Dombi picture graphs, viz. union, join, composition, and cartesian product. In addition, we investigated many interesting results regarding the operations. The concept of complement and isomorphism of Picture dombi fuzzy graph is presented in this paper. Some important results on weak and co-weak isomorphism of Picture dombi fuzzy graphs are derived.