Graph Parameters Research Articles

Total Regular Domination on a Litact Graph

Objectives: This work's primary goal is to ascertain the new domination parameter known as total regular domination on a litact graph. Methods: Several results on total regular domination on a litact graph are obtained in the present study, both in terms of different G parameters (vertices, edges, diameter, maximum degree, and many more) and domination parameters (connected domination, total domination, edge domination, and many more). A few fundamental definitions, findings, and the ideas of several dominations parameters have been used in this. Findings: The current study aims to determine some relations between total regular domination on a litact graph and graph s different parameters, and domination parameters. Furthermore, results comparable to Nordhaus-Gaddum's were also established. Novelty: By merging the two distinct parameters total domination and regular domination, a new domination parameter known as total regular domination has been introduced. Using total regular domination on a litact graph, certain results were produced in this study, with respect to various parameters and domination parameters of the original graph G, but not into itself. This type of task has never been completed before. Keywords: Graph, Litact Graph, Total Domination Number, Regular Domination Number, Total Regular Domination Number

The Weisfeiler-Leman Dimension of Conjunctive Queries

A graph parameter is a function f on graphs with the property that, for any pair of isomorphic graphs G 1 and G 2 , f(G 1 )=f(G 2 ). The Weisfeiler--Leman (WL) dimension of f is the minimum k such that, if G 1 and G 2 are indistinguishable by the k-dimensional WL-algorithm then f(G 1 )=f(G 2 ). The WL-dimension of f is ∞ if no such k exists. We study the WL-dimension of graph parameters characterised by the number of answers from a fixed conjunctive query to the graph. Given a conjunctive query φ, we quantify the WL-dimension of the function that maps every graph G to the number of answers of φ in G. The works of Dvorak (J. Graph Theory 2010), Dell, Grohe, and Rattan (ICALP 2018), and Neuen (ArXiv 2023) have answered this question for full conjunctive queries, which are conjunctive queries without existentially quantified variables. For such queries φ, the WL-dimension is equal to the treewidth of the Gaifman graph of φ. In this work, we give a characterisation that applies to all conjunctive queries. Given any conjunctive query φ, we prove that its WL-dimension is equal to the semantic extension width sew(φ), a novel width measure that can be thought of as a combination of the treewidth of φ and its quantified star size, an invariant introduced by Durand and Mengel (ICDT 2013) describing how the existentially quantified variables of φ are connected with the free variables. Using the recently established equivalence between the WL-algorithm and higher-order Graph Neural Networks (GNNs) due to Morris et al. (AAAI 2019), we obtain as a consequence that the function counting answers to a conjunctive query φ cannot be computed by GNNs of order smaller than sew(φ). The majority of the paper is concerned with establishing a lower bound of the WL-dimension of a query. Given any conjunctive query φ with semantic extension width k, we consider a graph F of treewidth k obtained from the Gaifman graph of φ by repeatedly cloning the vertices corresponding to existentially quantified variables. Using a modification due to Furer (ICALP 2001) of the Cai-Fürer-Immerman construction (Combinatorica 1992), we then obtain a pair of graphs χ(F) and ^χ(F) that are indistinguishable by the (k-1)-dimensional WL-algorithm since F has treewidth k. Finally, in the technical heart of the paper, we show that φ has a different number of answers in χ(F) and ^χ(F). Thus, φ can distinguish two graphs that cannot be distinguished by the (k-1)-dimensional WL-algorithm, so the WL-dimension of φ is at least k.

Gated independence in graphs

If G=(V,E) is a (finite and simple) graph, we call an independent set X a gated independent set in G if for each x∈X, there exists a neighbor y of x such that (X∖{x})∪{y} is an independent set in G. We define the gated independence numbergi(G) of G to be the maximum cardinality of a gated independent set in G. We demonstrate that the gated independence number is closely related to both matching and domination parameters of graphs. We prove that the inequalities im(G)⩽gi(G)⩽mur(G) hold for every graph G, where im(G) and mur(G) denote the induced and uniquely restricted matching numbers of G. On the other hand, we show that γi(G)⩽gi(G) and γpr(G)⩽2gi(G) for every graph G without any isolated vertex, where γi(G) and γpr(G) denote the independence and paired domination numbers. Furthermore, we provide bounds on the gated independence number involving the order, size and maximum degree. In particular, we prove that gi(G)⩾n5 for every n-vertex subcubic graph G without any isolated vertex or any component isomorphic to K3,3, and gi(B)⩽3n8 for every n-vertex connected cubic bipartite graph B.

Omega Invariant of Complement Graphs and Nordhaus-Gaddum Type Results.

The study aimed to obtain relationships between the omega invariants of a graph and its complement. We used some graph parameters, including the cyclomatic numbers, number of components, maximum number of components, order, and size of both graphs G and G. Also, we used triangular numbers to obtain our results related to the cyclomatic numbers and omega invariants of G and G. Several bounds for the above graph parameters have been obtained by the direct application of the omega invariant. We used combinatorial and graph theoretical methods to study formulae, relations, and bounds on the omega invariant, the number of faces, and the number of compo-nents of all realizations of a given degree sequence. Especially so-called Nordhaus-Gaddum type resulted in our calculations. In these calculations, the triangular numbers less than a given number play an important role. Quadratic equations and inequalities are intensively used. Several relations between the size and order of the graph have been utilized in this study. In this paper, we have obtained relationships between the omega invariants of a graph and its complement in terms of several graph parameters, such as the cyclomatic numbers, number of components, maximum number of components, order, and size of G and G, and triangular numbers. Some relationships between the omega invariants of a graph and its complement have been obtained.

Topological numbers of fuzzy soft graphs and their applications in globalizing the world by mutual trade

In this research draft, by keeping in view, the importance of Zagreb numbers, first we defined some familiar graph families and their degrees in fuzzy soft environment, and then by converting these two novel topological numbers in fuzzy soft framework, their formulae are calculated for these graphical networks. An application of international trade using two kinds of routes is also discussed in the article, and by comparing the results efficiency of both the numbers is compared and analyzed how topological descriptors are useful in examining the progress and profit in the countries, that are a part of the project. Graphs are used in the modeling of links and processes within physical, biological, social, and information systems. A fuzzy graph is the most adaptable tool in mathematics, which describes the fuzzy relationship between objects. Fuzzy graphs are very helpful to give more accuracy to the system as compared to crisp graphs. A classification of the universe with respect to certain specified parameters is known as a soft set. It is a new method for simulating ambiguity and uncertainty. Soft set theory was created by Molodtsov in 1999 as a method for simulating vagueness and uncertainty. Soft set theory is presently used widely in academia to address decision-making issues. The concept of soft graphs is used to provide a parameterized point of view for graphs. In many areas of mathematics and chemistry, as well as in network theory, spectral graph theory, and molecular chemistry, the Zagreb numbers are crucial graph parameters. Due to the importance and significance of Zagreb numbers, in this research article we define these graph parameters in fuzzy soft environment and then derived the fundamental formula for first and second Zagreb numbers of some popular graph families. Also by keeping in view the importance of graph theory in solving decision making problems in business industry, and the most generic behavior of fuzzy soft graph due to its parameterized nature we use our work in application of decision making in trade among different nations.

Chordal graphs and distinguishability of quantum product states

We investigate a graph-theoretic approach to the problem of distinguishing quantum product states in the fundamental quantum communication framework called local operations and classical communication (LOCC). We identify chordality as the key graph structure that drives distinguishability in one-way LOCC, and we derive a one-way LOCC characterization for chordal graphs that establishes a connection with the theory of matrix completions. We also derive minimality conditions on graph parameters that allow for the determination of indistinguishability of states. We present a number of applications and examples built on these results.

Fractional Domination Number of the Fractal Graphs

Background/Objectives: A fractal is a collection of clearly defined structures that are uneven or irregular and can be divided into smaller pieces. Each of these smaller pieces has a relationship to the overall structure at random ranges that are analogous or identical. Fractal graphs are a highly effective way to construct well-defined patterns. The fractional domination number in a graph refers to the weight of a minimum fractional dominating function. The study is to find the bounds of the fractional domination and their related parameters in the fractal graphs. Methods: The sharp bounds and the constant ratio of fractional domination and their related parameters were calculated using an iterative process in all iterations of a self-similarity fractal graph. Findings: The study established the relation between the fractional domination number and other relevant parameters as fractional independence domination numbers for each iteration in the fractal graphs and explained how the concept of fractional domination works on fractal graphs such as the Koch snowflake and the Sierpinski triangle. Novelty: Previous researchers have studied bounds on maximum matching and dominating sets in fractal graphs such as the Koch snowflake and the Sierpinski triangle. The present study enhances the idea of finding the bounds of fractional domination parameters in a fractal graph. The construction of new beautiful ornaments, locating damaged cells for drug injection in medical science, placing stones in jewels, plumping work, and designing fractal antennas all can be made possible by this work in self-similarity fractal graphs. Keywords: Fractals, Fractional Dominating Functions, Fractional Domination Number, Fractional Independence Domination Number, Fractional Domination Chain

Bounding the Joint Numerical Range of Pauli Strings by Graph Parameters

The relations among a given set of observables on a quantum system are effectively captured by their so-called joint numerical range, which is the set of tuples of jointly attainable expectation values. Here we explore geometric properties of this construct for Pauli strings, whose pairwise commutation and anticommutation relations determine a graph G. We investigate the connection between the parameters of this graph and the structure of minimal ellipsoids encompassing the joint numerical range, and we develop this approach in different directions. As a consequence, we find counterexamples to a conjecture by de Gois [Phys. Rev. A 107, 062211 (2023)], and answer an open question raised by Hastings and O’Donnell [, pp. 776–789], which implies a new graph parameter that we call “β(G).” Furthermore, we provide new insights into the perennial problem of estimating the ground-state energy of a many-body Hamiltonian. Our methods give lower bounds on the ground-state energy, which are typically hard to come by, and might therefore be useful in a variety of related fields. Published by the American Physical Society 2024

Flood Prediction System Using IOT & Artificial Neural Network

Floods pose significant challenges as one of nature's most devastating disasters, making the development of accurate forecast model’s complex. This issue has led to severe consequences such as crop loss, population displacement, damage to infrastructure, and disruption of essential services. Advanced research on flood prediction models has played a crucial role in providing policy recommendations, mitigating risks, reducing human casualties, and minimizing property damage caused by floods. In this context, we propose an Internet of Things (IoT)-based flood prediction and forecasting model that prioritizes energy efficiency. Given the limited battery and memory capacity of IoT sensor nodes, we employ an energy-saving strategy within the fog layer, leveraging data diversity to minimize energy consumption. Additionally, cloud technology offers an effective storage solution. To accurately calibrate flood phases, we investigate climatic factors such as humidity, temperature, rainfall, as well as water body parameters including water flow and elevation. Neural networks are commonly used in constructing forecast systems, as they can replicate the complex calculations involved in flood physical processes, resulting in improved performance and cost-effectiveness. In our approach, the Artificial Neural Network (ANN) technique is employed for flood forecasting, and the effectiveness of different algorithms, such as Logistic Regression and Decision Tree, is assessed by comparing them to ANN. Accuracy values are computed using a classification report assessment, and graph parameters are carefully evaluated. Ultimately, our proposed system utilizes the ANN technique to train a predictive model by examining the dataset. This model generates real-time flood risk forecasts through a user-friendly graphical interface.

Grouped domination parameterized by vertex cover, twin cover, and beyond

A dominating set S of graph G is called an r-grouped dominating set if S can be partitioned into S1,S2,…,Sk such that the size of each unit Si is r and the subgraph of G induced by Si is connected. The concept of r-grouped dominating sets generalizes several well-studied variants of dominating sets with requirements for connected component sizes, such as the ordinary dominating sets (r=1), paired dominating sets (r=2), and connected dominating sets (r is arbitrary and k=1). In this paper, we investigate the computational complexity of r-Grouped Dominating Set, which is the problem of deciding whether a given graph has an r-grouped dominating set with at most k units. For general r, r-Grouped Dominating Set is hard to solve in various senses because the hardness of the connected dominating set is inherited. We thus focus on the case in which r is a constant or a parameter, but we see that r-Grouped Dominating Set for every fixed r>0 is still hard to solve. From the observations about the hardness, we consider the parameterized complexity concerning well-studied graph structural parameters. We first see that r-Grouped Dominating Set is fixed-parameter tractable for r and treewidth, which is derived from the fact that the condition of r-grouped domination for a constant r can be represented as monadic second-order logic (MSO2). This fixed-parameter tractability is good news, but the running time is not practical. We then design an O⁎(min⁡{(2τ(r+1))τ,(2τ)2τ})-time algorithm for general r≥2, where τ is the twin cover number, which is a parameter between vertex cover number and clique-width. For paired dominating set and trio dominating set, i.e., r∈{2,3}, we can speed up the algorithm, whose running time becomes O⁎((r+1)τ). We further argue the relationship between FPT results and graph parameters, which draws the parameterized complexity landscape of r-Grouped Dominating Set.

Semidefinite Approximations for Bicliques and Bi-Independent Pairs

We investigate some graph parameters dealing with bi-independent pairs (A, B) in a bipartite graph [Formula: see text], that is, pairs (A, B) where [Formula: see text], and [Formula: see text] are independent. These parameters also allow us to study bicliques in general graphs. When maximizing the cardinality [Formula: see text], one finds the stability number [Formula: see text], well-known to be polynomial-time computable. When maximizing the product [Formula: see text], one finds the parameter g(G), shown to be NP-hard by Peeters in 2003, and when maximizing the ratio [Formula: see text], one finds h(G), introduced by Vallentin in 2020 for bounding product-free sets in finite groups. We show that h(G) is an NP-hard parameter and, as a crucial ingredient, that it is NP-complete to decide whether a bipartite graph G has a balanced maximum independent set. These hardness results motivate introducing semidefinite programming (SDP) bounds for g(G), h(G), and [Formula: see text] (the maximum cardinality of a balanced independent set). We show that these bounds can be seen as natural variations of the Lovász ϑ-number, a well-known semidefinite bound on [Formula: see text]. In addition, we formulate closed-form eigenvalue bounds, and we show relationships among them as well as with earlier spectral parameters by Hoffman and Haemers in 2001 and Vallentin in 2020. Funding: This work was supported by H2020 Marie Skłodowska-Curie Actions [Grant 813211 (POEMA)].

On the metric-based resolving parameter of the line graph of certain structures

Let G be a graph and R = {r1, r2, …, rk} be an ordered subset of vertices of G, if every two vertices of G have different representation r (v|R) = (d (v, r1) , d (v, r2) , …, d (v, rk)) with respect to R, then R is said to be a metric-based resolving parameter or resolving set of G and its minimum cardinality is called the metric dimension of graph G . Metric dimension is considered as an important applied concept of graph theory especially in the localization of a network and also in the chemical graph theoretical study of molecular compounds. Therefore, it is hot topic to study for different families of graphs as well. Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming. In this paper, we determine the metric-based resolving parameter of line graph of a convex polytope Sn, and conclude that it has constant metric dimension but vary with the parity of n . This article presents a measurement of the line graph of a convex polytope, denoted as ( S n ) . The subsequent section provides the metric dimension of the resulting graph. There are two scenarios pertaining to the metric dimension of a selected graph with respect to the metric dimension. The metric dimension of even cycle-based convex polytopes is three, whereas for other values, the metric dimension is four.

Sex differences in fetal brain functional network topology.

The fetal period is a critical stage in brain development, and understanding the characteristics of the fetal brain is crucial. Although some studies have explored aspects of fetal brain functional networks, few have specifically focused on sex differences in brain network characteristics. We adopted the graph theory method to calculate brain network functional connectivity and topology properties (including global and nodal properties), and further compared the differences in these parameters between male and female fetuses. We found that male fetuses showed an increased clustering coefficient and local efficiency than female fetuses, but no significant group differences concerning other graph parameters and the functional connectivity matrix. Our study suggests the existence of sex-related distinctions in the topological properties of the brain network at the fetal stage of development and demonstrates an increase in brain network separation in male fetuses compared with female fetuses.

Dominance Parameters in Prism Graphs: A Comparative Study of Minimum Dominating Sets

Dominance Parameters in Prism Graphs: A Comparative Study of Minimum Dominating Sets

On the Eigenvalues and Energy of the Seidel and Seidel Laplacian Matrices of Graphs

Let S(Γ) be a Seidel matrix of a graph Γ of order n and let D(Γ) = diag(n − 1 − 2d1, n − 1 − 2d2, …, n − 1 − 2dn) be a diagonal matrix with di denoting the degree of a vertex vi in Γ. The Seidel Laplacian matrix of Γ is defined as SL(Γ) = D(Γ) − S(Γ). In this paper, we obtain an upper bound, and a lower bound on the Seidel Laplacian Estrada index of graphs. Moreover, we find a relation between Seidel energy and Seidel Laplacian energy of graphs. We establish some lower bounds on the Seidel Laplacian energy in terms of different graph parameters. Finally, we present a relation between Seidel Laplacian Estrada index and Seidel Laplacian energy of graphs.

ON STYLISTIC DIFFERENTIATION IN UKRAINIAN POETIC SPEECH BASED ON THE SYNTACTIC STRUCTURE OF SENTENCES

Background. The article is devoted to the stylistic differentiation of syntactic structures of Ukrainian-language poetic texts of the Ukrainian language corpus. The goal is to study the idiostyle on models of syntactic structures, graphically presented in the form of dependency trees, with the establishment of parameters for characterizing the author's style. The properties of the linear structure of 7,752 Ukrainian-language sentences in the verse style were studied. An "averaged" dependency graph was built for each author as a generalization of the syntactic parameters of dependency trees. Methods. Applied methods of structural linguistics included quantitative methods and modeling methods, distributive analysis, direct component method, method of constructing dependency trees (DT) and direct components in linguistic analysis modules: automatic morphological (distributive) and automatic syntactic (direct component and dependency tree method). Results. As a result of the study, it was established that the architecture of the phrase is important, that is, a scheme in the form of a graph that reflects the real sequence of syntactic elements of the statement. If the author has developed his individual writing style, this will inevitably affect the originality of the graphic drawings for the structures of his typical phrases. It is in syntactic stylistics that dependence graphs as a research method will find an unlimited field of application. Eight graph parameters are proposed, the choice of which is determined by the fact that they have stylistic distinguishing power. Conclusions. To justify the applied method of statistical analysis, it can be said that in stylistics each analysis gives only what it can give; the overall research picture consists of a comprehensive review of the results of all analyses. Qualitative methods can describe what is perceived from the text through its symbolic character, but only quantitative methods can explain why the text is constructed in the way it is. Qualitative characteristics consist of precisely those quantitative characteristics that cannot be seen by the average reader, and their detection is the task of science.

Enhancing cryptographic robustness through error-correcting codes derived from network graphs

Secure data transmission is essential for securing sensitive data generated in huge volumes in today’s growing digital era. In today’s growing digital world, secure data transfer is essential for securing private information generated in enormous volumes. Data security strategies, called Cyber Security Systems (CSS), are susceptible to errors and other risks that might damage the data’s integrity. While standard Error-Correcting Codes (ECCs) function well for more standard communication errors, they cannot meet the complex rules of cryptographic functions. In order to boost the CSS size, this article provides a detailed approach that develops ECCs from network graph parameters. The error flexibility, computational speed, and use of bandwidth power of the CSS have been designed to be addressed by the graph-based ECCs. The investigation proposes an approach that analyses network graphs and employs information to develop ECCs, which are then integrated into the existing CSS. Compared with standard models, the one recommended performed more successfully when assessed under a range of error scenarios and cryptanalytic attacks.

On Vertex-Edge Resolvability for the Web Graph and Prism Related Graph

Let E ( H ) and V ( H ) denote the edge set and the vertex set of the simple connected graph H , respectively. The mixed metric dimension of the graph H is the graph invariant, which is the mixture of two important graph parameters, the edge metric dimension and the metric dimension. In this article, we compute the mixed metric dimension for the two families of the plane graphs viz., the Web graph W n and the Prism allied graph D t n . We show that the mixed metric dimension is non-constant unbounded for these two families of the plane graph. Moreover, for the Web graph W n and the Prism allied graph D t n , we unveil that the mixed metric basis set M m G is independent.

TOPOLOGICAL ASPECTS ON CORONENE GRAPH USING SOME GRAPH OPERATORS

The Topological index is a numerical parameter of molecular graph which correlates its QSPR(Quantitative Structure Property Relationships) and QSAR(Quantitative Structure Activity Relationships). In this article, we compute topological indices of some graphs obtained from k-Coronene graph using some graph operations.

On the hardness of inclusion-wise minimal separators enumeration

Enumeration problems are often encountered as key subroutines in the exact computation of graph parameters such as chromatic number, treewidth, or treedepth. In the case of treedepth computation, the enumeration of inclusion-wise minimal separators plays a crucial role. However and quite surprisingly, the complexity status of this problem has not been settled since it has been posed as an open direction by Kloks and Kratsch in 1998. Recently at the PACE 2020 competition dedicated to treedepth computation, solvers have been circumventing that by listing all minimal a-b separators and filtering out those that are not inclusion-wise minimal, at the cost of efficiency. Naturally, having an efficient algorithm for listing inclusion-wise minimal separators would drastically improve such practical algorithms. In this note, however, we show that no efficient algorithm is to be expected from an output-sensitive perspective, namely, we prove that there is no output-polynomial time algorithm for inclusion-wise minimal separators enumeration unless P=NP.