Characterisation of Certain Types of Soft Graphs

Soft graph theory has emerged as a robust mathematical framework for modeling uncertainty through parameterized structures. In recent years, significant advancements have been made by integrating soft sets with neutrosophic, fuzzy, and rough frameworks. This paper presents a comprehensive study of modern innovations in soft graph theory, including neutrosophic soft graphs, soft rough hybrid graphs, and multi-layer parameterized models. A novel structure, the Parameterized Multi-Layer Soft Graph (PMLSG), is proposed to address limitations in existing models. Theoretical properties, algebraic operations, and application perspectives are discussed. A detailed and plagiarism-safe literature review highlights research gaps and motivates future directions in uncertain graph modeling.

A Pythagorean fuzzy set is very effective mathematical framework to represent parameter-wise imprecision which is the property of linguistic communication. A Pythagorean fuzzy soft graph is more potent than the intuitionistic fuzzy soft as well as the fuzzy soft graph as it depicts the interactions among the objects of a system using Pythagorean membership grades with respect to different parameters. This article addresses the content of competition graphs as well as economic competition graphs like k-competition graphs, m-step competition graphs and p-competition graphs in Pythagorean fuzzy soft environment. All these concepts are illustrated with examples and fascinating results. Furthermore, an application which describes the competition among distinct forest trees, that grow together in the mixed conifer forests of California, for plant resources is elaborated graphically. An algorithm is also designed for the construction of Pythagorean fuzzy soft competition graphs. It is worthwhile to express the competing and non-competing interactions in various networks with the help of Pythagorean fuzzy soft competition graphs wherein a variation in competition relative to different attributes is visible.

The theory of soft set offers a mathematical tool to deal with uncertainty. Nowadays, work on soft graph theory is progressing rapidly. In this paper, we define connected soft graph and derive some results. We also define cut vertex and bridge of soft graph along with some results on it.

This dissertation solves two problems relating to the structure of graphs. The first of these is motivated by Kuratowski's Theorem, perhaps the most famous result in graph theory. This theorem states the $K\sb5$ and K$\sb{3,3}$ are the only non-planar graphs for which both $G\\ e$, the deletion of the edge e, and G/e, the contraction of the edge e, are planar for all edges e of G. We characterize the non-planar graphs for which $G\\ e$ or G/e is planar for all edges e of G. The second problem we solve is motivated by Tutte's wheels theorem. An immediate consequence of this theorem is that the wheel graphs are the basic building blocks for the collection of simple, 3-connected graphs. Therefore it is of interest to examine the structure of the graphs that do not have a minor isomorphic to the k-spoked wheel, $W\sb{k}$, for small values of k. Dirac determined that the graphs having no $W\sb3$-minor are the series-parallel networks. It follows easily from Tutte's wheels theorem is that $W\sb3$ is the only graph that has a $W\sb3$-minor and no $W\sb4$-minor. Oxley characterized the graphs that have a $W\sb4$-minor and no $W\sb5$-minor. We characterize the planar graphs that have a $W\sb5$-minor and no $W\sb6$-minor. We also determine the best-possible upper bound on the number of edges of such a graph.

An edge coloring of a graph \(G\) is called a majestic edge coloring if there is the induced proper vertex coloring. If the colors of edges of the \(G\) graph have 2-elements sets and the \(G\) graph has induced proper vertex coloring then an edge coloring of the \(G\) graph is called the majestic 2-tone coloring. The majestic and the majestic 2-tone chromatic indices for some cycle related graphs which are wheel graph, gear graph, helm graph, web graph, and friendship graph are computed.

The quadratic embedding constant (QE constant) of a graph is a new characteristic value of a graph defined through the distance matrix. We derive formulae for the QE constants of the join of two regular graphs, double graphs and certain lexicographic product graphs. Examples include complete bipartite graphs, wheel graphs, friendship graphs, completely split graph, and some graphs associated to strongly regular graphs.

Let be a graph with p vertices and q edges. We have introduced a new graph labeling method using integers and cordial-related works and investigated some graphs for this labeling technique. Using this labeling concept, we have examined the graphs like path, cycle, star, complete graph, comb, and wheel graph. The first research paper on graph theory was published by Leonhard Euler. However, he did not use the word ‘graph’ in his work. In the early stages of the development of the subject, the vertices of a graph were specified as , and the edges were denoted by, . In recent times, several researchers have attempted to provide different types of labeling to the vertices and edges of a graph by identifying the relevant mathematical properties. The present paper provides a novel method of labeling by employing integers, which may form a foundation for future research work. In this paper, we investigate the pair mean cordial labeling behavior of some cycle-related graphs like the ice cream graph, closed web graph, circulant graph, zig-zag chord graph, pentagonal circular ladder, djembe graph, quadrilateral friendship graph, and origami graph.

In many applications, data is in graph structure, which can be naturally represented as graph-structured XML. Existing queries defined on tree-structured and graph-structured XML data mainly focus on subgraph matching, which can not cover all the requirements of querying on graph. In this paper, a new kind of queries, topological query on graph-structured XML is presented. This kind of queries consider not only the structure of subgraph but also the topological relationship between subgraphs. With existing subgraph query processing algorithms, efficient algorithms for topological query processing are designed. Experimental results show the efficiency of implementation algorithms.

BLoG: Bootstrapped graph representation learning with local and global regularization for recommendation

The use of graph‐theoretic parameters is essential in interpreting the structural robustness, efficiency, and functionality of complex networks. Specifically, edge‐based metrics are more useful in interpreting the susceptibility of links and the concept of upstream impact, as well as structural robustness. Such metrics are essential in a variety of fields such as network security, communication, and optimization theory. In the same way, a new edge‐based invariant is introduced in this research paper, which is known as the b -edge differential . The b‐edge differential is essentially linked to b ‐differential of a graph. Let G = ( V , E ) be a simple connected graph. For any subset of edges X ⊆ E , we associate the set B ( X ) = { e ∈ E \ X : e shares a common vertex with at least one edge in X }, that is, B ( X ) captures all edges outside X which are adjacent to X . The quantity ∂ B E ( X ) = | B ( X )| is called the b ‐edge differential of X . We then define the b ‐edge differential of a graph G by ∂ B E ( G ) = max X ⊆ E ∂ B E ( X ). This parameter calculates the maximum value of the indirect dominances, or coverage, which can be achieved by a given set of edges. It serves as a useful tool in the identification of critical edges as well as the maximization of influence in networks. The problem can also be used in the applications involving the maximization of profit, traffic routing, and vulnerability evaluation for communication networks. We carry out the explicit computation of the b ‐edge differential for some classical types of graphs, such as paths, cyclic graphs, wheel graphs, complete graphs, star graphs, double‐star graphs, comb graphs, complete bipartite graphs, ladder graphs, and triangular ladder graphs. The work described in this article expands the relevance of differential parameters in the theory of graphs, with promising research directions for theoretical and application aspects of network science.

This paper presents first an analysis of the topology of mechanisms via Graph Theory and Screw Theory and next the principle of dual mechanisms in terms of their mobility and overconstraints. Using dual graphs, the graph representations of the mechanisms that are dual to hybrid and Delta mechanisms are revealed. The concept of parallel mechanisms with configurable platforms (PMCPs) is introduced and it is shown that the graph reduction of PMCPs always results in a wheel graph, which has the interesting property of being self-dual. In case of self dual topology, it is then possible to directly convert any method developed for their mobilty analysis into an overconstraint analysis method and vice versa. This self dual topology property can also be exploited to create new PMCPs and is an important aspect in the future development of a type synthesis method that includes PMCPs.KeywordsParallel mechanismsConfigurable platformTopologyMobilityOverconstraintsDuality

ABSTRACTAnalyzing complex multivalued datasets, such as boiling points and molecular Platt numbers, poses significant challenges due to the intricate relationships within the data, which traditional methods struggle to capture effectively. To address this, the study introduces Edge Neighborhood Graphs Ne(G), a novel graph‐based approach where the neighborhoods of edges in a graph G = (V, E) are treated as vertices, and an edge is drawn between two vertices if their corresponding neighborhoods intersect. This method is explored using complete, cycle, and wheel graphs, with formulas derived for vertex degrees in Ne(G) to uncover structural patterns. Polynomial approximations are employed to apply Ne(G) to multivalued data, demonstrating its effectiveness in modeling complex relationships and providing accurate predictions. By offering a powerful new tool for analyzing intricate datasets, this study makes a unique contribution to graph theory and data modeling, with promising potential for application to other graph types and diverse datasets in the future.

Graph domination is a key concept in graph theory, with wide-ranging applications in network security, social networks, and biological systems. This paper presents a comprehensive study of domination in graph structures, exploring various types of domination such as total domination, independent domination, and connected domination. The study delves into the theoretical foundations of domination, including critical bounds, domination numbers, and their computational complexity. The paper also examines real-world applications, highlighting how domination plays a crucial role in optimizing network communication, sensor placement, and resource management. Additionally, we analyze algorithmic approaches for calculating domination numbers and explore how domination impacts graph connectivity and stability in practical scenarios. The results provide insights into how domination theory can be leveraged to enhance the performance and efficiency of complex networks.

Graph convolutional autoencoders with co-learning of graph structure and node attributes

In this paper, we design and evaluate a new substructure-aware Graph Representation Learning (GRL) approach. GRL aims to map graph structure information into low-dimensional representations. While extensive efforts have been made for modeling global and/or local structure information, GRL can be improved by substructure information. Some recent studies exploit adversarial learning to incorporate substructure awareness, but hindered by unstable convergence. This study will address the major research question: is there a better way to integrate substructure awareness into GRL? As subsets of the graph structure, interested substructures (i.e., subgraph) are unique and representative for differentiating graphs, leading to the high correlation between the representation of the graph-level structure and substructures. Since mutual information (MI) is to evaluate the mutual dependence between two variables, we develop a MI inducted substructure-aware GRL method. We decompose the GRL pipeline into two stages: (1) node-level, where we introduce to maximize MI between the original and learned representation by the intuition that the original and learned representation should be highly correlated; (2) graph-level, where we preserve substructures by maximizing MI between the graph-level structure and substructure representation. Finally, we present extensive experimental results to demonstrate the improved performances of our method with real-world data.