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For a graph G (V , E) and ℓ ∈ ℕ , an ℓ distance coloring is a coloring f : V → { 1 , 2 , ... , t } of V with t colors such that ∀ u , v ∈ V , u ≠ v , f (u) ≠ f (v) when d (u , v) ≤ ℓ. Here d (u , v) is the distance between u and v and is equal to the minimum number of edges that connect u and v in G. The span of ℓ distance coloring of G , χ ℓ (G) , is the minimum t among all ℓ distance coloring of G. A class of channel assignment problem in cellular network can be formulated as a distance graph coloring problem in regular grid graphs. The cellular network is often modelled as an infinite hexagonal grid H , and hence determining χ ℓ (H) has relevance from practical point of view. Jacko and Jendrol [Discussiones Mathematicae Graph Theory, 2005] determined the exact value of χ ℓ (H) for any odd ℓ and for even ℓ ≥ 8 , it is conjectured that χ ℓ (H) = 3 8 (ℓ + 4 3) 2 where [ x ] is an integer, x ∈ ℝ and x − 1 2 < [ x ] ≤ x + 1 2 . For ℓ = 8 , the conjecture has been proved by Ghosh and Koley [ 2 2 nd Italian Conference on Theoretical Computer Science, 2021]. In this paper, we prove the conjecture for any even ℓ ≥ 1 0. [ABSTRACT FROM AUTHOR]

Soft set theory, introduced by D. Molodtsov, is a mathematical approach that tackles uncertain data. Semigraph is a generalization of the concept of a graph. By incorporating soft sets into semigraphs, a new concept called soft semigraphs has emerged. The study of soft semigraphs has gained significance in graph theory due to their valuable applications in parametrization. This paper explores the properties of various vertex degrees in soft semigraphs, including degrees, end degrees, edge degrees, adjacent degrees, and consecutive adjacent degrees. [ABSTRACT FROM AUTHOR]

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 π-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. [ABSTRACT FROM AUTHOR]

The graph parameter domination is considered perhaps one of the fastest-growing areas in graph theory since the fundamental efforts of Berge and Ore in 1962. The domination number of a graph may be determined using the parameter neighbor-component order connectivity which is defined as the minimum number of nodes such that removing these nodes and their adjacent neighbors ensure all remaining components have order less than some given threshold value. Given a threshold value of one, the neighbor-component order connectivity of a graph is equivalent to the well-known parameter domination number of the graph. The problem of computing the domination number of an arbitrary graph is NP-hard, and computing neighbor-component order connectivity of a graph for an arbitrary threshold value is also NP-hard. We will look at the organizational structure of a unicycle graph which aids in modeling the vulnerability of a communication network. In order to assess the vulnerability of this network, we present a polynomial algorithm for computing the neighbor-component order connectivity of a unicycle. We also note that this algorithm covers the case of a threshold value of one and offers an algorithm for the domination number of the unicycle. [ABSTRACT FROM AUTHOR]

Recently, Graph Neural Networks (GNNs) have gained widespread application in automatic brain network classification tasks, owing to their ability to directly capture crucial information in non-Euclidean structures. However, two primary challenges persist in this domain. First, within the realm of clinical neuro-medicine, signals from cerebral regions are inevitably contaminated with noise stemming from physiological or external factors. The construction of brain networks heavily relies on set thresholds and feature information within brain regions, making it susceptible to the incorporation of such noises into the brain topology. Additionally, the static nature of the artificially constructed brain network's adjacent structure restricts real-time changes in brain topology. Second, mainstream GNN-based approaches tend to focus solely on capturing information interactions of nearest neighbor nodes, overlooking high-order topology features. In response to these challenges, we propose an adaptive unsupervised Spatial-Temporal Dynamic Hypergraph Information Bottleneck (ST-DHIB) framework for dynamically optimizing brain networks. Specifically, adopting an information theory perspective, Graph Information Bottleneck (GIB) is employed for purifying graph structure, and dynamically updating the processed input brain signals. From a graph theory standpoint, we utilize the designed Hypergraph Neural Network (HGNN) and Bi-LSTM to capture higher-order spatial-temporal context associations among brain channels. Comprehensive patient-specific and cross-patient experiments have been conducted on two available datasets. The results demonstrate the advancement and generalization of the proposed framework. [ABSTRACT FROM AUTHOR]

The concepts of domination and topological index hold great significance within the realm of graph theory. Therefore, it is pertinent to merge these concepts to derive the domination index of a graph. A novel concept of the domination index is introduced, which utilizes the domination degree of a vertex. The domination degree of a vertex a is defined as the minimum cardinality of a minimal dominating set (MDS) that includes a. Methods to find a MDS containing a particular vertex is also discussed in the study. The notion of domination degree and domination index are studied for graphs like complete graphs, complete bipartite, r − partite graphs, cycles, wheels, paths, book graphs, windmill graphs, Kragujevac trees. The study is extended to operation in graphs. Inequalities involving domination degree and already established graph parameters are discussed. An application of domination degree is discussed in facility allocation in a city. [ABSTRACT FROM AUTHOR]

The investigation of a graph’s topology using numerical quantities known as “Topological indices” is a developing topic in graph theory. The primary goal is to estimate the Weighted PI Index of Edge Corona Product, one of the topological indices, for a variety of simple graphs. The study of the molecular structure of chemical compounds is one of the most recent areas of interest in graph theory, along with the visualization of graphs in Google maps, nanochemistry, network design in electrical and electronic engineering, and computer science. Topological indices are estimated to analyze chemical substances and their physio-chemical characteristics. Some exceptional instances of WPI of those graphs are assessed based on the computed results. [ABSTRACT FROM AUTHOR]

The topological index is just one of several very useful tools that graph theory has made available to chemists. Topological indices are invariants of real numbers under graph isomorphisms. Several topological indices have been defined. Some of them are used to model chemical, pharmaceutical and other properties of molecules. The eccentric connectivity index (eci) is also a topological index. The eci of G, denoted by 휀c(G), is defined as 휀c(G) =∑ v∈V (G)deg(v)e(v), where deg(v) represents the degree of a vertex v and e(v) is its eccentricity. In this paper, exact formula for the eci of complementary prisms is derived. [ABSTRACT FROM AUTHOR]

Graph theory has many applications in chemistry and is used to analyze molecular structures. Topological descriptors are numerical numbers that contain chemical information and provide structural features of a compound associated with a chemical approach. The purpose of the topological index is to study the physicochemical properties of molecular structures. This paper investigates the molecular graph of 2D silicon carbide structures. The scope of this paper is to determine the highest thermal stability property among silicon carbide structures using topological indices. [ABSTRACT FROM AUTHOR]

Graph theory is employed in this study to examine the Paate clan of the Nyishi tribe’s unique naming system. The network represented by the Parental graph is a directed acyclic graph with 1388 nodes which represent individuals of the clan, and 1387 arcs indicate father–son relationships and the transfer of the naming system. Centrality measures show that individual Kadu is the most essential node for linking others, while individual Nina is capable of communicating more efficiently with other nodes in the network. No clusters or cliques were observed in the network. The network exhibits characteristics of a scale-free network. No relinking marriages were observed in the genealogical network. [ABSTRACT FROM AUTHOR]

Molodtsov’s introduction of soft set theory marked a significant advancement in managing uncertainty, addressing the limitations of classical set theory in handling imprecise and vague information. This versatile framework has since been extensively explored, leading to sophisticated models for decision-making and medical diagnosis. Building on the principles of soft set theory, the concept of soft graphs was introduced to enhance the representation and analysis of uncertain information. Soft graphs incorporate parametrization, enabling dynamic modeling of graph-based relationships across various contexts and applications. This innovative approach creates multiple, adaptable representations that better reflect real-world complexities. This survey paper provides a comprehensive guide to the evolution of soft graphs, starting with an overview of the origins and basic concepts of soft set theory, highlighting its strengths and applications. It then delves into the development of soft graphs, explaining their theoretical foundations and methodologies. The paper explores how soft graphs have been used to model and solve problems in different domains, emphasizing their flexibility and effectiveness. By examining both foundational concepts and contemporary research advancements, this paper serves as a valuable resource for researchers interested in the applications of soft set theory and soft graphs in managing uncertainty and imprecision. [ABSTRACT FROM AUTHOR]

The Maximum Independent Set problem is well-studied in graph theory and related areas. An independent set of a graph is a subset of non-adjacent vertices of the graph. A maximum independent set is an independent set of maximum size. This paper studies the Maximum Independent Set problem in some classes of geometric intersection graphs in a distributed setting. More precisely, we study the Maximum Independent Set problem on two geometric intersection graphs, interval and axis-parallel segment intersection graphs, and present deterministic distributed algorithms in a model that is similar but a little weaker than the local communication model. We compute the maximum independent set on interval graphs in O(k) rounds and O(n) messages, where k is the size of the maximum independent set and n is the number of nodes in the graph. We provide a matching lower bound of Ω(k) on the number of rounds, whereas Ω(n) is a trivial lower bound on message complexity. Thus, our algorithm is both time and message-optimal. We also study the Maximum Independent Set problem in interval count l graphs, a special case of the interval graphs where the intervals have exactly l different lengths. We propose an 1 2-approximation algorithm that runs in O(l) round. For axis-parallel segment intersection graphs, we design an 1 2-approximation algorithm that obtains a solution in O(D) rounds. The results in this paper extend the results of Molla et al. [J. Parallel Distrib. Comput. 2019]. [ABSTRACT FROM AUTHOR]

The edge-Wiener index is an important topological index in Chemical Graph Theory, defined as the sum of distances among all pairs of edges. Fractal structures have received much attention from scientists because of their philosophical and aesthetic significance, and chemists have even attempted to synthesize various types of molecular fractal structures. The level-3 Sierpinski triangle is constructed similarly to the Sierpinski triangle and its skeleton networks have self-similarity. In this paper, by using the method of finite pattern, we obtain the edge-Wiener index of skeleton networks according to level-3 Sierpinski triangle. This provides insights for a better understanding of molecular fractal structures. [ABSTRACT FROM AUTHOR]

Numerous graph-theoretic methods can be used to investigate the efficiency and reliability of networks. The reliability of a network is determined based on its connectivity. At the very least, the system must continue to function in the event of part of its component (node or link) failures. An ideal system remains reliable even when errors occur. Diameter is a measure of network efficiency used to study the effects of link failures in networks. In this paper, we look into the eliminating edges affect to the hypercube graph’s diameter (call it by d) and calculate the maximum diameter that is denoted by f(t,d) after deleting t edges from hypercube graphs Qn for some n. [ABSTRACT FROM AUTHOR]

Two mathematical fields that concentrate on creating and analyzing structures are algebra and graph theory. There are numerous studies linking algebraic structures like groups, rings, fields and vector spaces with graph theory. Several algebraic graphs have been defined based on the properties of the order of the group and its elements. In this paper, we systematically review the literature on such graphs to understand the research dynamics in the field. [ABSTRACT FROM AUTHOR]

One of the well-known NP-hard optimization problems in graph theory is finding the longest path in a graph. This problem remains NP-hard for general grid graphs, and its complexity is open for grid graphs that have a limited number of holes. In this paper, we study this problem for odd-sized O -shaped grid graphs, i.e. a rectangular grid graph with a rectangular hole. We show that this problem can be solved in linear time. [ABSTRACT FROM AUTHOR]

Soft set theory is a general mathematical method for dealing with uncertain data. It is a classification of elements of the universe with respect to some given set of parameters. Semigraph is a generalization of graph which is different from hypergraph. Soft semigraph was introduced by applying the concept of soft set in semigraph. Connectedness is one of the basic concepts of graph theory. In this paper, we introduce the concepts of s -walk, s -trail, s -path and s -cycle in soft semigraphs and we define an s -connected soft semigraph. We explain the procedures of vertex and f p -edge deletions from a soft semigraph. Also, we introduce p -part cut vertex, s -cut vertex, p -part bridge and s -bridge of a soft semigraph and investigate some of their properties. [ABSTRACT FROM AUTHOR]

The soft set theory proposed by D. Molodtsov in 1999 is a general mathematical method for dealing with uncertain data. Now many researchers are applying soft set theory in decision making problems. Graph theory is the mathematical study of objects and their pairwise relationships, known as vertices and edges, respectively. The concept of soft graphs is used to provide a parameterized point of view for graphs. Directed graphs can be used to analyze and resolve problems with electrical circuits, project timelines, shortest routes, social links and many other issues. We introduced the notion of the soft directed graph by applying the concepts of soft set in a directed graph. In this paper, we introduce the concept of soft subdigraph and some soft directed graph operations like AND operation, OR operation, soft union, extended union, extended intersection, restricted union and restricted intersection and investigate some of their properties. [ABSTRACT FROM AUTHOR]

Investigating the time–frequency-based phase synchronization between nonstationary time series of the cryptocurrencies' prices can be a suitable tool to reveal their complicated interactions at different frequencies. In this work, the phase synchronization between 25 cryptocurrencies with the highest capitalization from December 1, 2021 to June 1, 2022 is calculated using the phase-locking value method based on wavelet transform. Then, utilizing the graph theory, the cryptocurrency networks are constructed, and their topological features like path length (PL), clustering coefficient (CC) and node strength are evaluated in various frequencies. This research indicates a strong phase synchronization between the investigated cryptocurrencies, especially in low frequencies. Also, the networks' high average CC and short PL compared to their equivalent regular and random networks display the small-worldness of the networks. We observe from the obtained results that Bitcoin, Ethereum and Binance currencies, among the most popular cryptocurrencies, have the highest average node strengths at different frequencies. Also, TRON shows the lowest CC and node strength among all currencies, representing its limited phase interaction with other currencies. [ABSTRACT FROM AUTHOR]

The hyper-Wiener index plays an important role in chemical graph theory. In this paper, using the technique named finite pattern, we discuss the hyper-Wiener index on level-3 Sierpinski gasket which is a self-similar fractal. [ABSTRACT FROM AUTHOR]

Most of our conventional tools for formal reasoning, computing, and modeling are precise, deterministic, and crisp. However, many complicated problems in the domains of economics, medicine, engineering, the environment, social science, and other disciplines demand data that is not always precise. We cannot always use traditional approaches because there are so many different types of uncertainty present in these problems. This difficulty could be caused by the parameterization tool's insufficiency. Soft set theory, which Molodtsov presented in 1999, is a generic mathematical approach for handling uncertain data. Many researchers are currently using soft set theory to solve problems involving decision-making. The concept of soft graphs is used to provide a parameterized point of view for graphs. The topic of graph products has received a lot of interest in graph theory. It is a binary operation on graphs with numerous combinatorial uses. On soft graphs, we can define product operations in a manner similar to how graph products are defined. The co-normal product, the restricted co-normal product, the modular product, and the restricted modular product of soft graphs are all introduced in this study. We prove that these products of soft graphs are again soft graphs and derive methods for computing their vertex count, edge count, and the sum of part degrees. [ABSTRACT FROM AUTHOR]

In this paper, a finite spin system corresponding to magnetic molecules with the wheel geometry is discussed in the Ising-like limit of the Heisenberg Hamiltonian. We put stress on the ground state degeneration, which is determined with the use of some discrete mathematics methods, especially investigating some integer sequences, including the famous Fibonacci one. [ABSTRACT FROM AUTHOR]

D. Molodtsov proposed soft set theory in 1999 as a general mathematical framework for dealing with uncertain data. Many academics are now applying soft set theory in decision-making problems. In graph theory, a directed graph is a graph made up of vertices connected by directed edges, also known as arcs. Electrical circuits, shortest routes, social links and a variety of other problems can all be analyzed and solved using directed graphs. In this paper, we introduce soft directed graphs by applying the concept of soft set to directed graphs. Soft directed graphs provide a parameterized point of view for directed graphs. We define and investigate the degrees and matrices associated with a soft directed graph. We also introduce several product operations in soft directed graphs and analyze some of their features. [ABSTRACT FROM AUTHOR]

Molodtsov developed soft set theory in 1999 as an approach for modeling vagueness and uncertainty. Many academics currently employ soft set theory to solve decision-making problems. A parameterized point of view for graphs is provided using the idea of soft graphs. Soft graph theory is a rapidly growing field in graph theory because of its adaptability to parameterization techniques. In graph theory, the topic of graph products has attracted much attention. It is a binary operation on graphs that has many combinatorial applications. Analogous to the definitions of graph products, we can define product operations on soft graphs. In this paper, we introduce the tensor product, the restricted tensor product, the strong product, and the restricted strong product of soft graphs. We prove that these products of soft graphs are also soft graphs, and we derive formulas for finding vertex count, edge count, and the sum of part degrees in them. [ABSTRACT FROM AUTHOR]

Dominant local metric dimension (DLMD) combines two interesting graph theory concepts: domination and local metric dimension (LMD). Graph invariants are a fantastic tool for analyzing graph abstract structures. Metric dimension and DLMD as graph invariants have a wide range of applications, which includes robot navigation, pharmaceutical chemistry, pattern recognition, and image processing. In this paper, we computed the DLMD of the prism graph Y n , antiprism graph A n , and the sunflower graph S F n . [ABSTRACT FROM AUTHOR]

Generalized splines on a graph G with edge labels in a commutative ring R are vertex labelings such that if two vertices share an edge in G, the difference between the vertex labels lies in the ideal generated by the edge label. When R is an integral domain, the set of all such splines is a finitely generated R-module RG of rank n, the number of vertices of G. We find determinantal conditions on subsets of RG that determine whether RG is a free module, and if so, whether a so-called “flow-up class basis” exists. [ABSTRACT FROM AUTHOR]

Semigraph is a generalization of graph introduced by E. Sampathkumar which is different from hypergraph. In 1999, D. Molodtsov initiated the novel concept of soft set theory. This is an approach for modeling vagueness and uncertainty. It is a classification of elements of the universe with respect to some given set of parameters. The concept of soft graph introduced by Rajesh K. Thumbakara and Bobin George is used to provide a parameterized point of view for graphs. Theory of soft graphs is a fast developing area in graph theory due to its capability to deal with the parameterization tool. In this paper, we introduce the concepts of soft semigraph by applying the concept of soft set in semigraph. Also we introduce some operations on soft semigraphs and investigate some of their properties. [ABSTRACT FROM AUTHOR]

In this work and its sequel, we study the expanding phenomenon of matrices over a finite chain ring of large residue field. A sum-product estimate is proved. It is shown that x + y z is a moderate expander on n × n matrices with exponent n + 1 6 . These results generalize the main theorems in the recent work [29] of Xie and Ge. The proofs use spectral graph theory and elementary divisor theory. [ABSTRACT FROM AUTHOR]

The global defensive k -alliance is a very well-studied notion in graph theory, it provides a method of classification of graphs based on relations between members of a particular set of vertices. In this paper, we explore this notion in zero-divisor graph of commutative rings. The established results generalize and improve recent work by Muthana and Mamouni who treated a particular case for k = − 1 known by the global defensive alliance. Various examples are also provided which illustrate and delimit the scope of the established results. [ABSTRACT FROM AUTHOR]

Alzheimer's Disease (AD) is basically a progressive neurodegenerative disorder associated with abnormal brain networks that affect millions of elderly people and degrades their quality of life. The abnormalities in brain networks are due to the disruption of White Matter (WM) fiber tracts that connect the brain regions. Diffusion-Weighted Imaging (DWI) captures the brain's WM integrity. Here, the correlation betwixt the WM degeneration and also AD is investigated by utilizing graph theory as well as Machine Learning (ML) algorithms. By using the DW image obtained from Alzheimer's Disease Neuroimaging Initiative (ADNI) database, the brain graph of each subject is constructed. The features extracted from the brain graph form the basis to differentiate between Mild Cognitive Impairment (MCI), Control Normal (CN) and AD subjects. Performance evaluation is done using binary and multiclass classification algorithms and obtained an accuracy that outperforms the current top-notch DWI-based studies. [ABSTRACT FROM AUTHOR]

Given a finite simple undirected graph G there is a simplicial complex Ind(G), called the independence complex, whose faces correspond to the independent sets of G. This is a well-studied concept because it provides a fertile ground for interactions between commutative algebra, graph theory and algebraic topology. In this paper, we consider a generalization of independence complex. Given r ≥ 1 , a subset of the vertex set is called r-independent if the connected components of the induced subgraph have cardinality at most r. The collection of all r-independent subsets of G form a simplicial complex called the r-independence complex and is denoted by Indr(G). It is known that when G is a chordal graph the complex Indr(G) has the homotopy type of a wedge of spheres. Hence, it is natural to ask which of these complexes are shellable or even vertex decomposable. We prove, using Woodroofe's chordal hypergraph notion, that these complexes are always shellable when the underlying chordal graph is a tree. Using the notion of vertex splittable ideals we show that for caterpillar graphs the associated r-independence complex is vertex decomposable for all values of r. Further, for any r ≥ 2 we construct chordal graphs on 2 r + 2 vertices such that their r-independence complexes are not sequentially Cohen–Macaulay. [ABSTRACT FROM AUTHOR]

Analogues of Ramsey's Theorem for infinite structures such as the rationals or the Rado graph have been known for some time. In this context, one looks for optimal bounds, called degrees, for the number of colors in an isomorphic substructure rather than one color, as that is often impossible. Such theorems for Henson graphs however remained elusive, due to lack of techniques for handling forbidden cliques. Building on the author's recent result for the triangle-free Henson graph, we prove that for each k ≥ 4 , the k-clique-free Henson graph has finite big Ramsey degrees, the appropriate analogue of Ramsey's Theorem. We develop a method for coding copies of Henson graphs into a new class of trees, called strong coding trees, and prove Ramsey theorems for these trees which are applied to deduce finite big Ramsey degrees. The approach here provides a general methodology opening further study of big Ramsey degrees for ultrahomogeneous structures. The results have bearing on topological dynamics via work of Kechris, Pestov, and Todorcevic and of Zucker. [ABSTRACT FROM AUTHOR]

Directed cycles in digraphs are useful in embedding linear arrays and rings, and are suitable for designing simple algorithm with low communication costs in parallel computer systems, thus the existence of directed cycles on digraphs has been largely investigated. Let n , k ≥ 2 be integers. Bermond et al. [Journal of Graph Theory 4(3) (1980) 337–341] proved that if the size of a strong digraph D with order n is at least n − k + 2 2 + k − 1 , then the girth of D is no more than k. Consequently, when D is a 4-free strong digraph with order n , which means that every directed cycle in D has length at least 5 , then the maximum size of D is n − 2 2 + 2. In this paper, we mainly give the structural characterizations for all 4-free strong digraphs of order n whose arc number exactly is n − 2 2 + 2. [ABSTRACT FROM AUTHOR]

Broadcasting is disseminating information in a network where a specific message must spread to all network vertices as quickly as possible. Finding the minimum broadcast time of a vertex in an arbitrary network is proven to be NP-complete. However, this problem is solvable for a few families of networks. In this paper, we present an optimal algorithm for finding the broadcast time of any vertex in a fully connected tree ( FCT n ) in O (| V | log log n) time. An FCT n is formed by attaching arbitrary trees to vertices of a complete graph of size n where | V | is the total number of vertices in the graph. [ABSTRACT FROM AUTHOR]

Graph computation is a core technique for solving realistic problems of graph representations. In solving the shortest path problem (SPP), the current classical methods are encountering a huge performance bottleneck. Attempting to solve this dilemma, we try to solve the SPP with a Quantum Approximate Optimal Algorithm (QAOA)-based quantum method. In this paper, we propose a QAOA-based shortest path algorithm (SPA) by constructing a suitable Hamiltonian quantity and using the idea of variational quantum computing, and verify the algorithm using a quantum simulator and an International Business Machines cloud quantum computer. The proposed algorithm is able to achieve a near-optimal solution with a correct rate that significantly exceeds the invalid solutions, reaching a good preliminary result. Furthermore, the proposed algorithm is expected to achieve a huge advantage over the classical algorithm and the SPA based on Grover's algorithm with a suitable selection of parameters and number of steps. In addition, the proposed algorithm requires fewer quantum bits than other quantum algorithms, thus promising quantum computing superiority on current noisy intermediate-scale quantum (NISQ) quantum computing devices. [ABSTRACT FROM AUTHOR]

The basic intention of this paper is to impart the importance of Nano topology induced by graph. Nano topology induced by graph paves us a method to find the well-known parameter of a graph, namely its domination number as well as all the possible dominating sets of a graph. In general, each paper, depending upon the author's axe (or the pen), has its own point of emphasis. Accordingly, in this paper, the computational and algorithmic aspects of graph are emphasized. By experience, we know that to solve a problem using graph theory leads to a large graph virtually impossible to analyze without the aid of a computer. Hence, to simplify the problem we have used a Java program code that suits even for all graphs of finite number of vertices. We have made an attempt to find the proper location of guards and minimize the number of guards to observe all the warning lights in a nuclear power plant using such a Java program code. [ABSTRACT FROM AUTHOR]

The metric dimension of graphs has been extended in some types and variations such as local metric dimension and complement metric dimension of graphs. Merging two concepts is one way of developing the concept of graph theory as a branch of mathematics. These two variations motivated us to construct a new concept of metric dimension so called local complement metric dimension. We apply this new concept to some particular classes of graphs and the corona product of two graphs. We also characterize the local complement metric dimension of graphs with certain properties, namely, bipartite graphs and odd cycle graphs. Furthermore, we discover the local complement metric dimension of the corona product of two particular graphs as well as the corona product of two general graphs. [ABSTRACT FROM AUTHOR]

Traffic forecasting has emerged as an important task for developing intelligent transportation systems. Recent works focus on representing traffic as graph operation and using graph neural networks for spatial–temporal prediction. Most of the approaches assume a predefined graph structure based on node distances. However, spatial dependencies change over time in many scenarios of traffic flow. In this regard, this study takes an investigation capturing the spatial and temporal dependencies with no prior knowledge structure of traffic road networks. Specifically, we propose a multi-step prediction model named Dynamic Spatial Transformer WaveNet Network (DSTWN) to capture the dynamic conditions and directions of traffic flow in which a temporal convolution layer is adopted for the long time sequence and a spatial transformer layer is proposed to capture the dynamic spatial dependencies. Furthermore, we introduce a new traffic dataset, which is collected from the vehicle detection system in an urban area (UVDS). In particular, compared with existing benchmark traffic data, UVDS contains more complicated spatial information, which is similar to many real-world scenarios of traffic flow. Experiments on both benchmark traffic datasets indicate the promising results of DSTWN compared with state-of-the-art models in this research field. [ABSTRACT FROM AUTHOR]

Let R and S be commutative rings with identity, J be an ideal of S , and let f : R → S be a ring homomorphism. The amalgamation of R with S along J with respect to f denoted by R ⋈ f J was introduced by D'Anna et al. in 2010. In this paper, we investigate some properties of the comaximal graph of R which are transferred to the comaximal graph of R ⋈ f J , and also we study some algebraic properties of the ring R ⋈ f J by way of graph theory. The comaximal graph of R , Γ (R) , was introduced by Sharma and Bhatwadekar in 1995. The vertices of Γ (R) are all elements of R and two distinct vertices a and b are adjacent if and only if R a + R b = R. Let Γ 2 (R) be the subgraph of Γ (R) generated by non-unit elements, and let J (R) be the Jacobson radical of R. It is shown that the diameter of the graph Γ 2 (R) ∖ J (R) is equal to the diameter of the graph Γ 2 (R ⋈ f J) ∖ J (R ⋈ f J) , and the girth of the graph Γ 2 (R) ∖ J (R) is equal to the girth of the graph Γ 2 (R ⋈ f J) ∖ J (R ⋈ f J) , provided some special conditions. [ABSTRACT FROM AUTHOR]

Topological indices, i.e., numerical invariants suitably associated to graphs and only depending upon their isomorphism type, have important applications in Chemistry. Their computation constitutes an important branch of Chemical Graph Theory. In this paper, we focus on some degree and distance-based invariants related to the Zagreb indices, the Szeged index and the Wiener index, namely, the F-index, the vertex PI index and the hyper-Wiener index. In particular, we find the formulæ to compute these topological invariants for wreath product of graphs. [ABSTRACT FROM AUTHOR]

The eccentric connectivity coindex of a graph G is defined as ECCI (G) = ∑ u v ∉ E (G) [ ecc G (u) + ecc G (v) ] = ∑ v ∈ V (G) ecc G (v) [ | V (G) | − 1 − d G (v) ] , where V (G) is the vertex set of G , E (G) is the edge set of G , ecc G (v) is the eccentricity of v ∈ V (G) and d G (v) is the degree of v in G. We present lower bounds on the eccentric connectivity coindex for bipartite graphs of given order and vertex connectivity, order and matching number, order and odd diameter. The extremal graphs are presented as well. To the best of our knowledge, our extremal graphs of given order and vertex connectivity are not extremal graphs for any other research problem in graph theory. [ABSTRACT FROM AUTHOR]

This paper focuses on the problem of finite-time flocking in the multi-agent nonholonomic system with proximity graph. With the aid of singular communication function and results from graph theory, a novel distributed control protocol is proposed, where each agent merely obtains state information of its neighbors. Based on the theory of finite-time stability, the proposed protocol can achieve flocking within a finite-time and an upper bound on the setting time can be derived. Furthermore, we can deduce several sufficient conditions to the problem under the assumption that the positions and relative distances of agents are unknown. These sufficient conditions show that there are always suitable gains that enable our proposed protocol to perform finite-time flocking control and maintain the connectivity of the graph for any given initial connected graphs. Finally, the theoretical results are validated by numerical simulations. [ABSTRACT FROM AUTHOR]

The topological indexes, such as the Wiener sum and the eccentric distance sum, play important roles in Chemical Graph Theory, where the eccentric distance sum characterizes the geodesic distance of two nodes. In this paper, for a family of self-similar Vicsek networks, we discuss their eccentric distance sums related to the Steiner distance of four nodes. [ABSTRACT FROM AUTHOR]

In this paper, the authors estimate portfolio volatilities and use variance−decomposition techniques and Cholesky factorization to construct a portfolio volatility spillover index. Furthermore, the authors show that spillover risks are persistent and much more common than well-known indicators like the turbulence index and the CBOE VIX index might suggest. Moreover, portfolio volatilities show contributions to and from other portfolio volatilities, which indicate elevated financial market interconnectedness and heightened risk. Foreign exchange contributes strong volatility shocks to other portfolios. More importantly, its contribution has a different magnitude and direction after the Global Financial Crisis (GFC) than before depending on the base currency. To show the causality effects of portfolio volatility spillover the authors apply both econometric models and graph theory, demonstrating that the importance is time varying. This requires monitoring and analysis in decision making regarding portfolio management, portfolio construction, and risk management. [ABSTRACT FROM AUTHOR]

A hypergraph is given by H = (V , E) , where V is a set of vertices and E is a set of nonempty subsets of V , the member of E is named hyperedge. So, a hypergraph is a nature generalization of a graph. A hypergraph has a complex structure, thus some researchers try to transform a hypergraph to a graph. In this paper, we define two graphs, Clique graph and Persian graph. These relations are one to one. We can find the shortest path between two vertices in a hypergraph H , by using the Dijkstra algorithm in graph theory on the graphs corresponding to H. [ABSTRACT FROM AUTHOR]

Recently, in Borzooei et al. [Inverse fuzzy graphs with applications, to appear in New Mathematics and Natural Computation], we defined the concept of inverse fuzzy graph as a generalization of graph, which is able to answer some problems that graph theory and fuzzy graph theory cannot explain. Now, in this paper, we define the notions of dominating set, domination number and different kinds of dominating sets like strong, weak, total, independent and inverse dominating sets with their domination numbers in inverse fuzzy graphs. Then, we investigate the relations among different kinds of dominating numbers. Finally, we give an application in decision making for service buildings. [ABSTRACT FROM AUTHOR]

Over the past few decades, the study of epidemic propagation has caught widespread attention from many areas. The field of graphs contains a wide body of research, yet only a few studies explore epidemic propagation's dynamics in "signed" networks. Motivated by this problem, in this paper we propose a new epidemic propagation model for signed networks, denoted as S-SIS. To explain our analysis, we utilized the mean field theory to demonstrate the theoretical results. When we compare epidemic propagation through negative links to those only having positive links, we find that a higher proportion of infected nodes actually spreads at a relatively small infection rate. It is also found that when the infection rate is higher than a certain value, the overall spreading in a signed network begins showing signs of suppression. Finally, in order to verify our findings, we apply the S-SIS model on Erdös–Rényi random network and scale-free network, and the simulation results is well consist with the theoretical analysis. [ABSTRACT FROM AUTHOR]

We say that a square real matrix M is off-diagonal nonnegative if and only if all entries outside its diagonal are nonnegative real numbers. In this paper, we show that for any off-diagonal nonnegative symmetric matrix M , there exists a nonnegative symmetric matrix M ̂ which is sparse and close in spectrum to M. [ABSTRACT FROM AUTHOR]