Class Of Graphs Research Articles

A Short Note for Hypersoft Rough Graphs

Graph theory, a branch of mathematics, explores relationships among entities using vertices and edges. To address the uncertainties inherent in real-world networks, Uncertain Graph Theory has emerged as a significant extension within this field. Hypersoft Graphs advance traditional graphs further by incorporating multi-attribute nodes, allowing each node to represent multiple attribute values, thus capturing more complex, multi-dimensional relationships. The Soft Rough Set and Soft Rough Graph combine the principles of Soft Sets and Rough Sets to manage uncertainty in a more refined manner. In this paper, we introduce the Hypersoft Rough Graph and analyze its relationships with other graph classes.

Conformal novelty detection for multiple metabolic networks.

Graphical representations are useful to model complex data in general and biological interactions in particular. Our main motivation is the comparison of metabolic networks in the wider context of developing noninvasive accurate diagnostic tools. However, comparison and classification of graphs is still extremely challenging, although a number of highly efficient methods such as graph neural networks were developed in the recent decade. Important aspects are still lacking in graph classification: interpretability and guarantees on classification quality, i.e., control of the risk level or false discovery rate control. In our contribution, we introduce a statistically sound approach to control the false discovery rate in a classification task for graphs in a semi-supervised setting. Our procedure identifies novelties in a dataset, where a graph is considered to be a novelty when its topology is significantly different from those in the reference class. It is noteworthy that the procedure is a conformal prediction approach, which does not make any distributional assumptions on the data and that can be seen as a wrapper around traditional machine learning models, so that it takes full advantage of existing methods. The performance of the proposed method is assessed on several standard benchmarks. It is also adapted and applied to the difficult task of classifying metabolic networks, where each graph is a representation of all metabolic reactions of a bacterium and to real task from a cancer data repository. Our approach efficiently controls - in highly complex data - the false discovery rate, while maximizing the true discovery rate to get the most reasonable predictive performance. This contribution is focused on confident classification of complex data, what can be further used to explore complex human pathologies and their mechanisms.

On the v-number of binomial edge ideals of some classes of graphs

On the v-number of binomial edge ideals of some classes of graphs

Odd chromatic number of graph classes

Odd chromatic number of graph classes

Difference distance magic oriented graphs

This article introduces difference distance magic (DDM) labelings, a new labeling on oriented graphs. We discuss basic properties of oriented graphs that have a DDM labeling and methods for constructing DDM labelings in a wide variety of oriented graph classes. We also give a variety of oriented graph classes that fail to produce a DDM labeling.

Making It Tractable to Detect and Correct Errors in Graphs

This paper develops \(\mathsf {Hercules} \) , a system for entity resolution (ER), conflict resolution (CR), timeliness deduction (TD) and missing value/link imputation (MI) in graphs. It proposes \(\mathsf {GCR^{+}\!s} \) , a class of graph cleaning rules that support not only predicates for ER and CR, but also temporal orders to deduce timeliness and data extraction to impute missing data. As opposed to previous graph rules, \(\mathsf {GCR^{+}\!s} \) are defined with a dual graph pattern to accommodate irregular structures of schemaless graphs, and adopt patterns of a star form to reduce the complexity. We show that while the implication and satisfiability problems are intractable for \(\mathsf {GCR^{+}\!s} \) , it is in PTIME to detect and correct errors with \(\mathsf {GCR^{+}\!s} \) . Underlying \(\mathsf {Hercules} \) , we train a ranking model to predict the temporal orders on attributes, and embed it as a predicate of \(\mathsf {GCR^{+}\!s} \) . We provide an algorithm for discovering \(\mathsf {GCR^{+}\!s} \) by combining the generations of patterns and predicates. We also develop a method for conducting ER, CR, TD and MI in the same process to improve the overall quality of graphs, by leveraging their interactions and chasing with \(\mathsf {GCR^{+}\!s} \) ; we show that the method has the Church-Rosser property under certain conditions. Using real-life and synthetic graphs, we empirically verify that \(\mathsf {Hercules} \) is 53% more accurate than the state-of-the-art graph cleaning systems, and performs comparably in efficiency and scalability.

Structure connectivity of a class of Cayley graph

As a generalization of connectivity, structure connectivity was proposed. For connected graphs G and T, the T-structure connectivity κ ( G ; T ) (resp. T-substructure connectivity κ s ( G ; T ) ) of G is the minimum cardinality of a set of subgraphs F of G that each is isomorphic to T (resp. a connected subgraph of T) such that G − F is disconnected or the singleton. n-dimensional graph Q n + is a class of Cayley graph, which obtained by adding edges to n-dimensional hypercubes. In this paper, we study star, path and cycle structure connectivity and substructure connectivity of graph Q n + for n ≥ 5 . For star ( K 1 , m ) structure, K 1 , m -structure (substructure) connectivity of Q n + is ⌈ n + 1 2 ⌉ for 2 ≤ m ≤ n − 1 . For path ( P k ) and cycle ( C k ) structures, we show that if 3 ≤ k ≤ 2 n , then P k -structure (substructure) connectivity is ⌈ 2 ( n + 1 ) k + 1 ⌉ and ⌈ 2 ( n + 1 ) k ⌉ for odd k and even k, respectively. We also obtain that P k -structure connectivity is equal to C k -substructure connectivity. Further, we calculate κ ( Q n + ; C 2 k ) = ⌈ n + 1 k ⌉ for 3 ≤ k ≤ n .

Method for Locating Secondary Circuit Faults in Substations Based on Graph Neural Networks

To improve the accuracy and portability of fault location in secondary circuits of smart substations, a fault location method for secondary circuits based on graph neural networks is proposed. First, a fault analysis of the secondary circuits in smart substations is conducted, and a graph database model of the secondary circuits is constructed, revealing the connection relationships between the secondary devices. Then, the fault location problem in secondary circuits is defined as a graph classification problem, and a fault graph generation model is proposed based on a representation method for fault information in secondary circuits. Finally, a fault location model for secondary circuits based on graph neural networks is constructed, and a performance optimization method based on binary cross-entropy is proposed to achieve accurate fault location in substation secondary circuits. Using the secondary circuits of a 220 kV smart substation as a reference, different case studies of secondary circuit faults are generated by altering the network topology, connection relationships, and network configurations through the fault graph generation model. Experiments comparing the proposed location method with other models show that this method has higher location accuracy and robustness.

Elegant Fuzzy Labeling of Graphs

Fuzzy graphs and fuzzy labelings are crucial in modeling real-time systems by accommodating varying degrees of information precision. In this paper, we introduce a novel fuzzy labeling called Elegant Fuzzy Labeling. We prove that a simple graph admits Elegant Fuzzy Labeling if and only if it admits elegant labeling. Furthermore, we prove that while a simple graph that admits Elegant Fuzzy Labeling also admits fuzzy labeling, but not conversely. We identify specific classes of simple graphs that admit Elegant Fuzzy Labeling and explore practical application of this new labeling approach.

From communities to interpretable network and word embedding: an unified approach

Abstract Modelling information from complex systems such as humans social interaction or words co-occurrences in our languages can help to understand how these systems are organized and function. Such systems can be modelled by networks, and network theory provides a useful set of methods to analyze them. Among these methods, graph embedding is a powerful tool to summarize the interactions and topology of a network in a vectorized feature space. When used in input of machine learning algorithms, embedding vectors help with common graph problems such as link prediction, graph matching, etc In Natural Language Processing (NLP), such a vectorization process is also employed. Word embedding has the goal of representing the sense of words, extracting it from large text corpora. Despite differences in the structure of information in input of embedding algorithms, many graph embedding approaches are adapted and inspired from methods in NLP. Limits of these methods are observed in both domains. Most of these methods require long and resource greedy training. Another downside to most methods is that they are black-box, from which understanding how the information is structured is rather complex. Interpretability of a model allows understanding how the vector space is structured without the need for external information, and thus can be audited more easily. With both these limitations in mind, we propose a novel framework to efficiently embed network vertices in an interpretable vector space. Our Lower Dimension Bipartite Framework (LDBGF) leverages the bipartite projection of a network using cliques to reduce dimensionality. Along with LDBGF, we introduce two implementations of this framework that rely on communities instead of cliques: SINr-NR and SINr-MF. We show that SINr-MF can perform well on classical graphs and SINr-NR can produce high-quality graph and word embeddings that are interpretable and stable across runs.

A new spectral-based index for graphs and its application to polyaromatic compounds

In this study, we introduce a novel index called the modified inverse sum indeg (MISI) energy as an extension of the traditional inverse sum indeg (ISI) energy and neighbor degree sum energy. We devised an algorithm that employs the simplified molecular input line entry system (SMILES) formula of compounds to compute the MISI energy of polycyclic aromatic hydrocarbon (PAH). Additionally, we established the bounds of the MISI energy for certain classes of graphs with fixed number of vertices. A strong correlation between the MISI energy and the total [Formula: see text]-electron energy of PAHs is observed through regression analysis. Remarkably, this correlation outperforms that achieved by the classical ISI energy. These findings highlight the enhanced effectiveness of the MISI energy as a descriptor for capturing significant molecular properties. Moreover, our study establishes a foundation for further theoretical extensions and practical applications in the field of chemical graph theory.

A recurrence for the surface area of (n, k)-arrangement graphs

An interesting property of an interconnected network is the number of nodes at distance i from an arbitrary processor, namely the node-centred surface area. This is an important property due to its applications in various fields of study. The ( n , k ) -arrangement graphs, denoted as A ( n , k ) are important class of graphs as they address the scalability issue in simpler topologies such as the hypercube and the star graph. Much work has been done to obtain a closed-form formula for the surface area of this class of graphs, but generally, it is not trivial to find an algorithm to compute the surface area of such graphs in polynomial time or to find an explicit formula with polynomially many terms in regard to the graph's parameters. In this paper, we present a simple recurrence that has linear computational complexity for A ( n , k ) for any arbitrary n and k in their defined range.

Decreasing graph complexity with transitive reduction to improve temporal graph classification

Decreasing graph complexity with transitive reduction to improve temporal graph classification

Weak Randomness in Graphons and Theons

ABSTRACTCall a hereditary family of graphs strongly persistent if there exists a graphon such that in all subgraphons of , is precisely the class of finite graphs that have positive density in . Our first result is a complete characterization of the hereditary families of graphs that are strongly persistent as precisely those that are closed under substitutions. We call graphons with the self‐similarity property above weakly random. A hereditary family is said to have the weakly random Erdős–Hajnal property () if every graphon that is a limit of graphs in has a weakly random subgraphon. Among families of graphs that are closed under substitutions, we completely characterize the families that belong to as those with “few” prime graphs. We also extend some of the results above to structures in finite relational languages by using the theory of theons.

Dynamic Random Intersection Graph: Dynamic Local Convergence and Giant Structure

ABSTRACTRandom intersection graphs containing an underlying community structure are a popular choice for modeling real‐world networks. Given the group memberships, the classical random intersection graph is obtained by connecting individuals when they share at least one group. We extend this approach and make the communities dynamic by letting them alternate between an active and inactive phase. We analyse the new model, delivering results on degree distribution, local convergence, largest connected component, and maximum group size, paying particular attention to the dynamic description of these properties. We also describe the connection between our model and the bipartite configuration model, which is of independent interest.

Matrix expressions of symmetric n-player games

The symmetric property in n-player games is explored in this paper. First of all, some algebraically verifiable conditions are provided respectively for the totally symmetric, weakly symmetric and anonymous games, which are based on the construction of permutation matrix, and some interesting properties are revealed. Then we investigate how network structures affect the symmetry in a networked game. Some sufficient conditions are proposed and several nice properties are preserved when traditional networked games are extended to generalized networked games, in which the resulting network structure is more complex than a classical network graph.

A Method for Single-Phase Ground Fault Section Location in Distribution Networks Based on Improved Empirical Wavelet Transform and Graph Isomorphic Networks

When single-phase ground faults occur in distribution systems, the fault characteristics of zero-sequence current signals are not prominent. They are quickly submerged in noise, leading to difficulties in fault section location. This paper proposes a method for fault section location in distribution networks based on improved empirical wavelet transform (IEWT) and GINs to address this issue. Firstly, based on kurtosis, EWT is optimized using the N-point search method to decompose the zero-sequence current signal into modal components. Noise is filtered out through weighted permutation entropy (WPE), and signal reconstruction is performed to obtain the denoised zero-sequence current signal. Subsequently, GINs are employed for graph classification tasks. According to the topology of the distribution network, the corresponding graph is constructed as the input to the GIN. The denoised zero-sequence current signal is the node input for the GIN. The GIN autonomously explores the features of each graph structure to achieve fault section location. The experimental results demonstrate that this method has strong noise resistance, with a fault section location accuracy of up to 99.95%, effectively completing fault section location in distribution networks.

Vulnerability assessment of a new class of Cayley graph

Vulnerability assessment of a new class of Cayley graph

Approximation of subgraph counts in the uniform attachment model

Abstract We use Stein’s method to obtain distributional approximations of subgraph counts in the uniform attachment model or random directed acyclic graph; we provide also estimates of rates of convergence. In particular, we give uni- and multi-variate Poisson approximations to the counts of cycles and normal approximations to the counts of unicyclic subgraphs; we also give a partial result for the counts of trees. We further find a class of multicyclic graphs whose subgraph counts are a.s. bounded as $n\to \infty$ .

Some Reverse Topological Indices of Comet and Double Comet Graphs

Introduction: Chemical Graph Theory or CGT for short, is a transdisciplinary field of Mathematics wherein graphs are used to represent chemical compounds. Under this representation, the atoms of a chemical compound are expressed as vertices, while the bonds connecting these atoms are expressed as edges. Once the graph representation of a chemical compound has been determined, graph-theoretic techniques can now be used to determine various topological indices associated with the chemical compound. Topological indices are invariants of a graph that have predictive power for the chemical properties of a chemical compound. In CGT, trees, being connected and acyclic, have been an essential class of graphs. This is because, most compounds have acyclic molecular structures. In the 2023 study by Gowtham and Husin, various reverse topological indices of a family of trees called bistar graphs have been determined. Objectives: Motivated by the work of Gowtham and Husin, we determine some reverse topological indices of another family of trees called comets and double comets. Double comets are natural extension of bistar graphs. We also give a certain computational application of our results on double comet graphs. Finally, we investigate the relationship between the reverse topological indices of bistar graphs and double comets. Methods: Several important graph-theoretic concepts were used to analyze some important properties of comets and double comets, leading to the computation of their reverse vertex degree based topological indices. Results: The following reverse vertex degree-based topological indices for comets and double comets were determined: Reverse sum-connectivity, First reverse Zagreb, Second reverse Zagreb, Reverse arithmetic-geometric, Reverse geometric-arithmetic, Reverse Sombor, and Reverse Nirmala. A computational application of the results to a certain chemical compound (2,2,4,4-Tetramethylpentane) or C9H20 were also demonstrated. Conclusions: The results of the paper provided an extension to an existing result on the reverse vertex degree-based topological indices of bistar graphs by considering double comet graphs. Several topological reverse vertex degree-based topological indices were also computed for comet graphs. The computed topological indices were also applied in determining some invariants of a certain chemical compound. For future studies, we recommend that reverse vertex degree based topological indices of other family of trees will be considered.