Complement Of Graph Research Articles

Fermat Principle, Ramsey Theory and Metamaterials.

Reinterpretation of the Fermat principle governing the propagation of light in media within the Ramsey theory is suggested. Complete bi-colored graphs corresponding to light propagation in media are considered. The vertices of the graphs correspond to the points in real physical space in which the light sources or sensors are placed. Red links in the graphs correspond to the actual optical paths, emerging from the Fermat principle. A variety of optical events, such as refraction and reflection, may be involved in light propagation. Green links, in turn, denote the trial/virtual optical paths, which actually do not occur. The Ramsey theorem states that within the graph containing six points, inevitably, the actual or virtual optical cycle will be present. The implementation of the Ramsey theorem with regard to light propagation in metamaterials is discussed. The Fermat principle states that in metamaterials, a light ray, in going from point S to point P, must traverse an optical path length L that is stationary with respect to variations of this path. Thus, bi-colored graphs consisting of links corresponding to maxima or minima of the optical paths become possible. The graphs, comprising six vertices, will inevitably demonstrate optical cycles consisting of the mono-colored links corresponding to the maxima or minima of the optical path. The notion of the "inverse graph" is introduced and discussed. The total number of triangles in the "direct" (source) and "inverse" Ramsey optical graphs is the same. The applications of "Ramsey optics" are discussed, and an optical interpretation of the infinite Ramsey theorem is suggested.

Computing Topological Descriptors of Prime Ideal Sum Graphs of Commutative Rings

Let n≥1 be a fixed integer. The main objective of this paper is to compute some topological indices and coindices that are related to the graph complement of the prime ideal sum (PIS) graph of Zn, where n=pα,p2q,p2q2,pqr,p3q,p2qr, and pqrs for the different prime integers p,q,r, and s. Moreover, we construct M-polynomials and CoM-polynomials using the PIS-graph structure of Zn to avoid the difficulty of computing the descriptors via formulas directly. Furthermore, we present a geometric comparison for representations of each surface obtained by M-polynomials and CoM-polynomials. Finally, we discuss the applicability of algebraic graphs to chemical graph theory.

BatmanNet: bi-branch masked graph transformer autoencoder for molecular representation.

Although substantial efforts have been made using graph neural networks (GNNs) for artificial intelligence (AI)-driven drug discovery, effective molecular representation learning remains an open challenge, especially in the case of insufficient labeled molecules. Recent studies suggest that big GNN models pre-trained by self-supervised learning on unlabeled datasets enable better transfer performance in downstream molecular property prediction tasks. However, the approaches in these studies require multiple complex self-supervised tasks and large-scale datasets , which are time-consuming, computationally expensive and difficult to pre-train end-to-end. Here, we design a simple yet effective self-supervised strategy to simultaneously learn local and global information about molecules, and further propose a novel bi-branch masked graph transformer autoencoder (BatmanNet) to learn molecular representations. BatmanNet features two tailored complementary and asymmetric graph autoencoders to reconstruct the missing nodes and edges, respectively, from a masked molecular graph. With this design, BatmanNet can effectively capture the underlying structure and semantic information of molecules, thus improving the performance of molecular representation. BatmanNet achieves state-of-the-art results for multiple drug discovery tasks, including molecular properties prediction, drug-drug interaction and drug-target interaction, on 13 benchmark datasets, demonstrating its great potential and superiority in molecular representation learning.

Constraining MC-numbers by the connectivity of complement graphs

An edge-coloring of a graph G is called a monochromatic connection coloring (or MC-coloring for short) if any two vertices of G are joined by a monochromatic path (a path whose edges are colored with the same color). The monochromatic connection number (or MC-number for short) of G, denoted by mc(G), is the maximum number of colors used in an MC-coloring of G. In this paper we main explore the relationships between mc(G) and the connectivity of its complement graph G‾. We give a method for computing mc(G) if G‾ is disconnected and give the sharp upper bounds of mc(G) when G‾ is k-connected, where 1≤k≤3. If G is a graph with G‾k-connected and k≥4, then mc(G)=m−n+2 (this result has been proved by Caro and Yuster) and we further talk about the characteristics of MC-colorings of G. In addition, we prove that any graph G with mc(G)≥m−n+t contains a Kt.

A Knowledge Graph Embedding Model Based on Cyclic Consistency—Cyclic_CKGE

Most of the existing medical knowledge maps are incomplete and need to be completed/predicted to obtain a complete knowledge map. To solve this problem, we propose a knowledge graph embedding model (Cyclic_CKGE) based on cyclic consistency. The model first uses the “graph” constructed with the head entity and relationship to predict the tail entity, and then uses the “inverse graph” constructed with the tail entity and relationship to predict the head entity. Finally, the semantic space distance between the head entity and the original head entity should be very close, which solves the reversibility problem of the network. The Cyclic_CKGE model with a parameter of 0.46 M has the best results on FB15k-237, reaching 0.425 Hits@10. Compared with the best model R-GCN, its parameter exceeds 8 M and reaches 0.417 Hits@10. Overall, Cyclic_CKGE’s parametric efficiency is more than 17 times that of R-GCNs and more than 8 times that of DistMult. In order to better show the practical application of the model, we construct a visual medical information platform based on a medical knowledge map. The platform has three kinds of disease information retrieval methods: conditional query, path query and multi-symptom disease inference. This provides a theoretical method and a practical example for realizing knowledge graph visualization.

Conjugacy classes and conjugate graph of a K-metacyclic group

The relationship between algebraic structures and graphs has become an interesting topic of research nowadays. In this paper, we have considered the conjugate graph related to the conjugacy relation of a group. The vertices of the said graph are the noncentral elements of the group, and two vertices are adjacent if they are conjugate. For this particular study, we focused on the conjugate graph of a K-metacyclic group of order p(p-1). We first determine the conjugacy classes of this group and then obtain its conjugate graph. Various graph properties such as planarity, line graph, complement graph, clique number, dominating number, spectrum, and Laplacian are also studied in this paper.

On the eccentricity energy and eccentricity spectral radius of graphs with odd diameter

The eccentricity matrix of a graph is defined as the matrix obtained from its distance matrix by retaining the largest elements in each row and column, while the rest elements are set to be zero. The eccentricity eigenvalues of a graph are the eigenvalues of its eccentricity matrix, the eccentricity energy of a graph is the sum of the absolute values of its eccentricity eigenvalues, and the eccentricity spectral radius of a graph is its largest eccentricity eigenvalue. Let gn,d be the set of n-vertex connected graphs with odd diameter d, where each graph G in gn,d has a diametrical path whose center edge is a cut edge of G. For any graph G in gn,d, we construct a weighted graph Hω such that its adjacency matrix is just the eccentricity matrix of G, where H is the sequential join graph of the complement graphs of four disjoint complete graphs. In terms of the energy and spectral radius of the weighted graphs, we determine the graphs with minimum eccentricity energy, minimum and maximum eccentricity spectral radius in gn,d, respectively. As corollaries, we determine the trees with minimum eccentricity energy, minimum and maximum eccentricity spectral radius among all trees with odd diameter, respectively.

Finding biclique partitions of co-chordal graphs

The biclique partition number (bp) of a graph G is referred to as the least number of complete bipartite (biclique) subgraphs that are required to cover the edges of the graph exactly once. In this paper, we show that the biclique partition number of a co-chordal (complementary graph of chordal) graph G=(V,E) is less than the number of maximal cliques (mc) of its complementary graph: A chordal graph Gc=(V,Ec). We first provide a general framework of the “divide and conquer” heuristic of finding minimum biclique partitions of co-chordal graphs based on clique trees. Furthermore, a heuristic of complexity O(|V|(|V|+|Ec|)) is proposed by applying a lexicographic breadth-first search to find structures called moplexes. Either heuristic gives us a biclique partition of G with a size of mc(Gc)−1. In addition, we prove that both of our heuristics can solve the minimum biclique partition problem on G exactly if its complement Gc is chordal and clique vertex irreducible. We also show that mc(Gc)−2≤bp(G)≤mc(Gc)−1 if G is a split graph.

Local Antisymmetric Connectedness in Asymmetrically Normed Real Vector Spaces

In this paper, some properties of locally antisymmetrically connected spaces which are the localized version of the antisymmetrically connected $T_0$-quasi-metric spaces constructed as the natural counterparts of connected complementary graphs, are presented in terms of asymmetric norms. According to that, we investigated some different aspects and examples of local antisymmetric connectedness in the framework of asymmetrically normed real vector spaces. Specifically, it is proved that the structures of antisymmetric connectedness and local antisymmetric connectedness coincide for the $T_0$-quasi-metrics induced by the asymmetric norms which associate the theory of quasi-metrics with functional analysis.

Combining different points of view on plant descriptions: mapping agricultural plant roles and biological taxa.

This article describes our study on the alignment of two complementary knowledge graphs useful in agriculture: the thesaurus of cultivated plants in France named French Crop Usage (FCU) and the French national taxonomic repository TAXREF for fauna, flora, and fungi. FCU describes the usages of plants in agriculture: "tomatoes" are crops used for human food, and "grapevines" are crops used for human beverage. TAXREF describes biological taxa and associated scientific names: for example, a tomato species may be "Solanum lycopersicum" or a grapevine species may be "Vitis vinifera". Both knowledge graphs contain vernacular names of plants but those names are ambiguous. Thus, a group of agricultural experts produced some mappings from FCU crops to TAXREF taxa. Moreover, new RDF properties have been defined to declare those new types of mapping relations between plant descriptions. The metadata for the mappings and the mapping set are encoded with the Simple Standard for Sharing Ontological Mappings (SSSOM), a new model which, among other qualities, offers means to report on provenance of particular interest for this study. The produced mappings are available for download in Recherche Data Gouv, the federated national platform for research data in France.

Semi-equivariant conditional normalizing flows, with applications to target-aware molecule generation

Learning over the domain of 3D graphs has applications in a number of scientific and engineering disciplines, including molecular chemistry, high energy physics, and computer vision. We consider a specific problem in this domain, namely: given one such 3D graph, dubbed the base graph, our goal is to learn a conditional distribution over another such graph, dubbed the complement graph. Due to the three-dimensional nature of the graphs in question, there are certain natural invariances such a distribution should satisfy: it should be invariant to rigid body transformations that act jointly on the base graph and the complement graph, and it should also be invariant to permutations of the vertices of either graph. We propose a general method for learning the conditional probabilistic model, the central part of which is a continuous normalizing flow. We establish semi-equivariance conditions on the flow which guarantee the aforementioned invariance conditions on the conditional distribution. Additionally, we propose a graph neural network architecture which implements this flow, and which is designed to learn effectively despite the typical differences in size between the base graph and the complement graph. We demonstrate the utility of our technique in the molecular setting by training a conditional generative model which, given a receptor, can generate ligands which may successfully bind to that receptor. The resulting model, which has potential applications in drug design, displays high quality performance in the key ΔBinding metric.

Algebraic approach to various chemical structures with new Banhatti coindices

Topological coindices are the numeric values, obtained by the complement graph of a molecular structure, which is used in Quantitative Structure Property/Activity Relationship (QSPR/QSAR) studies to evaluate the physicochemical and biological properties of compounds. In this article, we introduce new distance-based topological indices calculated with the C o M -polynomial approach. We present an application about the compatibility of these indices with some drugs that are candidates for use in the treatment of COVID-19 with QSPR analysis.

Matrix representations of the inverse problem in the graph model for conflict resolution with fuzzy preference

The inverse analysis of the graph model for conflict resolution (GMCR) determines all the preference relationships required by each decision-maker (DM) according to a given stability definition to ensure that the given state is stable. In the paper, a matrix representation approach for the inverse graph model with fuzzy preference relations is proposed to obtain all the preference relations for each DM making the stability result based on the fuzzy stability definition, including fuzzy Nash, fuzzy general metarationality, fuzzy symmetric metarationality, and fuzzy sequential stability definition. For the two-DM graph model, the fuzzy preference relation over states, unilateral movements, and fuzzy unilateral improvements are refreshed by using the matrix forms. Furthermore, for the multiple DMs model, the joint movements and joint improvement for a collection including at least two DM are represented by using matrix representation. Finally, the matrix relationship for specifying the required preference relation in the framework of inverse graph model with fuzzy preferences is derived. To illustrate the usefulness of the matrix representation of inverse graph model with fuzzy preference relations, this paper demonstrates it using a real-life conflict of the Elmira groundwater contamination conflict.

Complete minors in complements of nonseparating planar graphs

Complete minors in complements of nonseparating planar graphs

SOME CONTRIBUTION TO COMPLEMENT OF STRONG NEUTROSOPHIC GRAPHS

The Fundamental operations on Complement of Strong Neutrosophic Graphs. We have applied the concept of strong Neutrosophic Graphs and also some graphs are connected, we explore some Particular Cases of strong Neutrosophic Graphs.

Optimal chromatic bound for (P2+P3,P2+P3¯ ${P}_{2}+{P}_{3},\bar{{P}_{2}+{P}_{3}}$)‐free graphs

AbstractFor a graph , let () denote its chromatic (clique) number. A is the graph obtained by taking the disjoint union of a two‐vertex path and a three‐vertex path . A is the complement graph of a . In this paper, we study the class of ()‐free graphs and show that every such graph with satisfies . Moreover, the bound is tight. Indeed, for any and , there is a ()‐free graph with and .

Inverse Graphs in m-Polar Fuzzy Environments and Their Application in Robotics Manufacturing Allocation Problems with New Techniques of Resolvability

The inverse in crisp graph theory is a well-known topic. However, the inverse concept for fuzzy graphs has recently been created, and its numerous characteristics are being examined. Each node and edge in m-polar fuzzy graphs (mPFG) include m components, which are interlinked through a minimum relationship. However, if one wants to maximize the relationship between nodes and edges, then the m-polar fuzzy graph concept is inappropriate. Considering everything we wish to obtain here, we present an inverse graph under an m-polar fuzzy environment. An inverse mPFG is one in which each component’s membership value (MV) is greater than or equal to that of each component of the incidence nodes. This is in contrast to an mPFG, where each component’s MV is less than or equal to the MV of each component’s incidence nodes. An inverse mPFG’s characteristics and some of its isomorphic features are introduced. The α-cut concept is also studied here. Here, we also define the composition and decomposition of an inverse mPFG uniquely with a proper explanation. The connectivity concept, that is, the strength of connectedness, cut nodes, bridges, etc., is also developed on an inverse mPF environment, and some of the properties of this concept are also discussed in detail. Lastly, a real-life application based on the robotics manufacturing allocation problem is solved with the help of an inverse mPFG.

A graph related to the Euler ø function

In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph G is a pair G = (V, E), where V and E are the vertex set and the edge set of G, respectively. The order and size of G is the number of vertices and edges of G, respectively. The degree or valency of a vertex u in a graph G (loopless), denoted by deg (u), is the number of edges meeting at u. If, for every vertex ν in G, deg (ν) = k, we say that G is a k-regular graph. The cycle of order n is denoted by Cn and is a connected 2-regular graph. The path graph of order n is denoted by Pn and obtain by deleting an edge of Cn. A tree is an undirected graph in which any two vertices are connected by exactly one path, or equivalently a connected undirected graph without cycle. A leaf (or pendant vertex) of a tree is a vertex of the tree of degree 1. An edge of a graph is said to be pendant if one of its vertices is a pendant vertex. A complete bipartite graph is a graph G with and such that every vertex of the set (part) X is connected to every vertex of the set (part) Y. If , then this graph is denoted by Km,n. The complete bipartite graph K1,n is called the star graph which has n + 1 vertices. The distance between two vertices u and ν of G, denoted by d (u, ν), is defined as the minimum number of edges of the walks between them. The complement of graph G is denoted by and is a graph with the same vertices such that two distinct vertices of are adjacent if, and only if, they are not adjacent in G. For more information on graphs, refer to [1].

Various results on some asymmetric types of density

The structures of symmetric connectedness and dually, antisymmetric connectedness were described and studied before, especially in terms of graph theory as the corresponding counterparts of the connectedness of a graph and the connectedness of its complementary graph. By taking into consideration the deficiencies of topological density in the context of symmetric and antisymmetric connectedness, two special kinds of density in the theory of non-metric $T_0$-quasi-metrics were introduced in the previous studies under the names symmetric density and antisymmetric density. In this paper, some crucial and useful properties of these two types of density are investigated with the help of the major results and (counter)examples peculiar to the asymmetric environment. Besides these, many further observations about the structures of symmetric and antisymmetric-density are dealt with, especially in the sense of their combinations such as products and unions through various theorems in the context of $T_0$-quasi-metrics. Also, we examine the question of under what kind of quasi-metric mapping these structures will be preserved.

A novel grey-inverse graph model for conflict resolution approach for resolving water resources conflicts in the Poyang Lake Basin, China

With the continuous urbanization and industrialization in recent years, different stakeholders in the same basin meet local water needs and compete for water resources, which hinders economic development and ecological protection of the basin. Different stakeholders conceal information about their own strategies and preferences to protect their vested interests, making it difficult for third-party regulators to develop optimal operational policies for the joint use of water resources in the basin. Therefore, the research objective of this study is to propose a new grey inverse Graph Model for Conflict Resolution (grey inverse GMCR), and apply the new method to the complex problem of water allocation and environmental governance with the Poyang Lake basin conflict. The new GMCR model proposed in this study can solve the cost minimization problem of a third-party mediator that manipulates decision makers (DMs)’ grey preferences to achieve desired outcomes. The results of the equilibrium analysis show that upstream areas' preferences mainly affect the result of water resource conflicts in the PLB, and establishing a scientific mechanism to compensate the water resources protection cost borne by the upstream areas would be a cost-effective way to realize the sustainable development of the basin as a whole.