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Cryptography serves as the cornerstone for safe communication and data security in today's digital environment. Because they feature substitution boxes, substitution-permutation networks (SPNs) are crucial for cryptographic algorithms such as the popular Advanced Encryption Standard (AES). The structure and properties of S-boxes have a significant impact on the overall security of cryptographic systems. This article aims to improve cryptographic security through unique S-box construction methodologies. The proposed S-boxes improve the security features by employing chaotic maps and Galois fields, which go beyond traditional design approaches. The S-boxes were analyzed and the weaknesses were removed to design strong candidate S-boxes. The efficiency of the proposed S-boxes in increasing cryptographic resilience is thoroughly explored thereby taking nonlinearity, strict avalanche requirements, bit independence constraints, linear approximation, and differential approximation into account. The dynamic S-boxes have average scores of nonlinearity, strict avalanche criteria(SAC), nonlinearity of Bit Independence Criteria (BIC Nonlinearity), SAC of Bit Independence Criteria (BIC SAC), Linear Approximation Probability (LAP) and Differential Approximation Probability (DAP) is 111.1025, 111.1022, 0.5014, 0.5024, 111.1082, 111.0964, 0.5024, 0.5022, 0.0726, 0.0729 and 0.0214, 0.0219, respectively. Furthermore, given the prevalence of images in modern communication and data storage, this work studies the seamless incorporation of advanced S-boxes into image encryption systems. With its thorough research, the paper contributes to the current discussion on cryptographic security by providing theoretical understandings and practical solutions to improve digital communication and data security in an era of rising cyber dangers and ubiquitous connectivity.

Abstract The mixed metric dimension of a graph is an important parameter in characterizing its structural complexity, specifically in nanoscale networks where precision is paramount. In this article, we calculate the mixed metric dimension of hexagonal nano-network, a modal with significant applications in nanotechnology and material science. Through rigorous analysis, we set up that the mixed metric dimension of the hexagonal nano-network is exactly three, highlighting its minimal but enough resolving set that uniquely identifies all vertices. Furthermore, we check out the exchange property within this context, demonstrating the robust adaptability of the hexagonal network’s resolving sets. Our findings display that the alternate assets aren’t the handiest preserved but stronger in these nano-networks, allowing for flexible adjustments in resolving sets without compromising the network’s integrity. This examination offers critical insights into the fundamental properties of hexagonal nano-networks, offering a theoretical foundation for future research in nanomaterial design and optimization. The results underscore the potential of leveraging mixed metric dimensions and exchange properties to achieve efficient and scalable solutions in nano-network applications.

The property of fuzziness is advantageous for any setting where a problem prediction and solution are intended. In comparison to the classical approach (crisp set), fuzzy graphs are beneficial as they provide greater precision and flexibility. Intuitionistic fuzzy graphs (IFGs) are generalizations of fuzzy graphs, in which each vertex is allocated an ordered pair whose first coordinate represents the membership value and whose second coordinate gives the non-membership value. Sombor indices are recently invented indices and are prime indices in describing the topology of a graph. They have demonstrated strong prediction ability in mathematical chemistry. The main aim of this research draft is to define two milestones, the fifth and sixth versions of Sombor indices in an intuitionistic fuzzy environment, IFGs are much more generalized and universal as compared to crisp graph theory, so we converted these two topological indices in this environment and then formulae for some fundamental families of graphs such as path graph, cycle graph, complete graph etc. are being derived. Our manuscript presents a novel application of opening a chain of schools of a popular brand in a certain vicinity, including the cluster center of the chain and applying intuitionistic fuzzy-based topological indices to assess their performance. With the help of this application, we highlight the practical implications and benefits of employing intuitionistic fuzzy-based techniques on the performance of schools through a comparative analysis. We checked the importance of intuitionistic fuzzy Sombor indices by presenting the application and concluded these indices can be applied to decision-making problems. With the help of the application, the efficiency of both of the intuitionistic fuzzy Sombor indices is also characterized, which must be a future efficacious tool in decision-making issues.

Abstract A relatively recent approach in molecular graph theory for analyzing chemical networks and structures is called a modified polynomial. It emphasizes the characteristics of molecules through the use of a polynomial-based procedure and presents numerical descriptors in algebraic form. The Quantitative Structure-Property Relationship study makes use of Modified Polynomials (M-Polynomials) as a mathematical tool. M-Polynomials used to create connections between a material’s various properties and its structural characteristics. In this study, we calculated several modified polynomials and gave a polynomial description of the magnesium iodide structure. Particularly, we computed first, second and modified Zagreb indices based M-polynomials. Randić index, and inverse Randić indices based M-polynomials are also computed in this work.

Abstract Eigenvalues have great importance in the field of mathematics, and their relevance extends beyond this area to include several other disciplines such as economics, chemistry, and numerous fields. According to our study, eigenvalues are utilized in chemistry to express a chemical compound’s numerous physical properties as well as its energy form. It is important to get a comprehensive understanding of the interrelationship underlying mathematics and chemistry. The anti-bonding phase is correlated with positive eigenvalues, whereas the bonding level is connected with negative eigenvalues. Additionally, the non-bonded level corresponds to eigenvalues of zero. This study focuses on the analysis of various structures of anticancer drugs, specifically examining their characteristic polynomials, eigenvalues of the adjacency matrix, matching number and nullity. Consequently, the selected structures of the aforementioned anticancer drugs exhibit stability since they are composed of closed-shell molecules, characterized by a nullity value of zero.

Graph operations enable the modification and transformation of graphs, enhancing data representation as well as effectiveness in fields such as computer science, mathematics, and data analysis. Super line graph operations are a concept of advanced graph theory that is used in advanced mathematics and computer science. They are used to solve specific mathematical problems, especially in graph theory and Combinatorics. Topological numbers are numerical values connected with graph structures that are used to analyze structural characteristics and solve problems in fields such as chemistry and mathematics. They are important for simplifying complex data and enabling quantitative analysis. The general Randić number of some super line graph operations of number two with a three-diameter is investigated in this study. The pendant vertices are inserted without disturbing their diameter. Seven graphs of order five, each with a diameter of three, can be used to create twenty-three generalized super line graphs.

Elliptic curves with the special quality known as supersingularity have gained much popularity in the rapidly developing field of cryptography. The conventional method of employing random search is quite ineffective in finding these curves. This paper analyzes the search of supersingular elliptic curves in the space of curves over Fp2. We show that naive random search is unsuitable to easily find any supersingular elliptic curves when the space size is greater than 1013. We improve the random search using a necessary condition for supersingularity. As our main result, we define for the first time an objective function to measure the supersingularity in ordinary curves, and we apply local search and a genetic algorithm using that function. The study not only finds these supersingular elliptic curves but also investigates possible uses for them. These curves were used to create cycles inside the isogeny graph in one particular application. The research shows how the design of S-boxes may strategically use these supersingular elliptic curves. The key components of replacement, which is a fundamental step in the encryption process that shuffles and encrypts the data inside images, are S-boxes. This work represents a major advancement in effectively identifying these useful elliptic curves, eventually leading to their wider application and influence in the rapidly expanding field of cryptography.

A connected acyclic graph in which the degree of every vertex is at most four is called a molecular tree. A number associated with a molecular tree that can help to approximate the physical or chemical properties of the corresponding molecule is called a topological index. It is of great importance to investigate the relation between the structure and the thermodynamic properties of those molecules. In this paper, we investigated the extreme value of the first reformulated Zagreb index with a given order and degree of a graph. Further, we investigated the molecular trees that attain the maximum and minimum values. As an application, we presented the regression models to predict the acentric factor and entropy of octane isomers. Our extremal graphs give the minimum and the maximum acentric factor and entropy, which satisfied the experimental values.

The notion of partition dimension was initially introduced in the field of graph theory, primarily to examine distances between vertices. The local partition dimension extends this idea by incorporating specific conditions into how vertices are represented. In graph theory, it is customary to represent the partition dimension of a graph as $pd(G)$ . Network localization, on the other hand, is the process of precisely determining the position of nodes within a network concerning a selected subset of nodes, called the locating set. The smallest size of its locating set is used to represent the locating number of a network. The generalized hexagonal cellular network provides an innovative framework for network planning and analysis. In our study, we investigate the partition dimension of a generalized hexagonal cellular network and provide a rigorous proof of its exact partition dimension. Hence, our approach ensures the distinct recognition of each node within a generalized hexagonal cellular network. Additionally, we explore the utilization of the metric dimension in flood relief camping by the National Disaster Management Authority (NDMA) Pakistan during floods in 2022. NDMA established relief camps and relief centers with unique codes to rescue humans and animals.

In the aircraft sector, honeycomb composite materials are frequently employed. Recent research has demonstrated the benefits of honeycomb structures in applications involving nanohole arrays in anodized alumina, micro-porous arrays in polymer thin films, activated carbon honeycombs, and photonic band gap honeycomb structures. The resolvability parameter is the area of graph theory that is most commonly explored. This results in an original network reconfiguration. Occasionally in terms of atoms (metric dimension), and sometimes in terms of bounds (edge metric dimension). In this article, we examined the honeycomb rhombic torus’s metric, edge metric, mixed metric, and partition dimension. The application of edge metric dimension is also discussed in it.

The exploration of edge metric dimension and its applications has been an ongoing discussion, particularly in the context of nanosheet graphs formed from the octagonal grid. Edge metric dimension is a concept that involves uniquely identifying the entire edge set of a structure with a selected subset from the vertex set, known as the edge resolving set. Let's consider two distinct edge resolving sets, denoted as Re1 and Re2, where Re1≠Re2. In such instances, it indicates that the graph G possesses a double-edge resolving set. This implies the existence of two different subsets of the vertex set, each capable of uniquely identifying the entire edge set of the graph. In this article, we delve into the edge metric dimension of nanosheet graphs derived from the octagonal grid. Additionally, we initiate a discussion on the exchange property associated with the edge resolving set. The exchange property holds significance in the study of resolving sets, playing a crucial role in comprehending the structure and properties of the underlying graph.

Give us a linked graph, $ G = (V, E). $ A vertex $ w\in V $ distinguishes between two components (vertices and edges) $ x, y\in E\cup V $ if $ d_G(w, x)\neq d_G (w, y). $ Let $ W_{1} $ and $ W_{2} $ be two resolving sets and $ W_{1} $ $ \neq $ $ W_{2} $. Then, we can say that the graph $ G $ has double resolving set. A nanotube derived from an quadrilateral-octagonal grid belongs to essential and extensively studied compounds in materials science. Nano-structures are very important due to their thickness. In this article, we have discussed the metric dimension of the graphs of nanotubes derived from the quadrilateral-octagonal grid. We proved that the generalized nanotube derived from quadrilateral-octagonal grid have three metric dimension. We also check that the exchange property is also held for this structure.

[This corrects the article DOI: 10.1371/journal.pone.0288931.].

Data security and privacy are considered to be the biggest problems faced by service providers who have worked with public data for a long time. A key element of modern encryption that is utilized to increase textual confusion is the Substitution box (S-box) and the algebraic strength of the S-box has a significant impact on how secure the encryption method is. In this article, we present a unique method that uses a linear fractional transformation on a finite field to produce cryptographically robust S-boxes. Firstly, we choose a specific irreducible polynomial of degree 8 in Z2[x] to construct GF(28). Later, we used the action of PGL(2,GF(28)) on GF(28) to generate a robust S-box. The effectiveness of the built-in S-box was evaluated using several criteria including non-linearity, differential uniformity, strict avalanche criteria, linear approximation probability, and bit independence criterion. The proposed S-box's characteristics are compared to those of most recent S-boxes to confirm the higher performance. Additionally, the S box was used to encrypt images to show its usefulness for multimedia security applications. We performed several tests, including contrast, correlation, homogeneity, entropy, and energy, to evaluate the success of the encryption technique. The proposed method for ciphering an image is very effective, as proven by its comparison with several S boxes.

Chemical graph theory represents an interdisciplinary field at the intersection of chemistry and mathematics, focusing on the analysis of various chemical structures through graph-theoretical frameworks. This area facilitates the examination of complex and extensive chemical configurations by leveraging graph theory principles. In chemical structure graphs, atoms are depicted as vertices, with edges representing the bonds between them. By employing chemical graph theory, the exploration of diverse drug compounds and intricate molecular structures becomes more manageable. A key concept in this field is the resolvability parameter, a metric derived from graph theory. It stipulates that each vertex in a structure must possess a distinct representation via elected vertices, known as a metric basis, resolving set, or locating set in different scientific contexts. The metric dimension, defined as the minimum number of vertices in the resolving set, plays a crucial role in characterizing the structural properties of chemical graphs. In this study, we discussed the edge metric dimension of structures of benzodiazepines, alprazolam, chlordiazepoxide, clobazam, clonazepam, diazepam, nitrazepam, lolendazam, alferez, which are types of drug that used for the treatment of depression. We also put to use the edge metric dimension in anti-depression chemical structures, which are different types of Benzodiazepine. For application of this topic, we select the hospitals and health centers in any emergency situation. In any chronic disease situation, these chosen health centers provide all medical facilities to other hospitals efficiently only because we assigned each hospital and health center a unique code.

L-valine is a crucial amino acid that has rising market demand and numerous uses. It can be used to make specific nutrients, animal feed additives, cosmetic ingredients, and other things in the medical and agricultural fields. N-salicylidene-L-valine (NsLv) is attracting a lot of attention due to its unusual structure and enhanced catalytic and cytotoxic activities. Topological index is a numerical value which is associated with the molecular structure. It is very helpful to predict physio-chemical properties and Quantitative structure-activity relationship and Quantitative structure-property relationship modeling. We study the supramolecular chain (Sc) in the dialkyl tin of complexes 2, 3 and 4 of NsLv to better understand this structure and its topological index-related characteristics. Additionally, we compare topological indices and analyze how these structures relate to one another using concrete examples.

Chemical study regarding total $\pi$-electron energy with respect to conjugated molecules has focused on the second Zagreb index of graphs. Moreover, in the last half-century, it has gotten a lot of attention. The relationship between the Roman domination number and the second Zagreb index is investigated in this study. We characterize the trees with the maximum second Zagreb index among those with the given Roman domination number.

Silicate minerals make up the majority of the earth's crust and account for almost 92 percent of the total. Silicate sheets, often known as silicate networks, are characterised as definite connectivity parallel designs. A key idea in studying different generalised classes of graphs in terms of planarity is the face of the graph. It plays a significant role in the embedding of graphs as well. Face index is a recently created parameter that is based on the data from a graph's faces. The current draft is utilizing a newly established face index, to study different silicate networks. It consists of a generalized chain of silicate, silicate sheet, silicate network, carbon sheet, polyhedron generalized sheet, and also triangular honeycomb network. This study will help to understand the structural properties of chemical networks because the face index is more generalized than vertex degree based topological descriptors.

Humanity has always benefited from an intercapillary study in the quantification of natural occurrences in mathematics and other pure scientific fields. Graph theory was extremely helpful to other studies, particularly in the applied sciences. Specifically, in chemistry, graph theory made a significant contribution. For this, a transformation is required to create a graph representing a chemical network or structure, where the vertices of the graph represent the atoms in the chemical compound and the edges represent the bonds between the atoms. The quantity of edges that are incident to a vertex determines its valency (or degree) in a graph. The degree of uncertainty in a system is measured by the entropy of a probability. This idea is heavily grounded in statistical reasoning. It is primarily utilized for graphs that correspond to chemical structures. The development of some novel edge-weighted based entropies that correspond to valency-based topological indices is made possible by this research. Then these compositions are applied to clay mineral tetrahedral sheets. Since they have been in use for so long, corresponding indices are thought to be the most effective methods for quantifying chemical graphs. This article develops multiple edge degree-based entropies that correlate to the indices and determines how to modify them to assess the significance of each type.

The p-type networks are designed with the help of CVNET at topo group Cluj and also given support by nano studio. Such networks develop new p-type surfaces and also represent the decorations of the surfaces. This patched network is designed by two repeated units. The first one is triphenylene having a Z-pen formula and the second one is triphenylene with A-phe. Furthermore, these decorations are acquired as the result of map operations represented in the CVNET software, while its assembling is conducted with the help of the nano studio program. In the literature, its topology is discussed by Omega polynomials which is an applied graph theory topic. Another most applied topic of graph theory is known as the resolvability parameter. So this article studied the resolvability parameters of patched networks, such as metric dimension, and edge metric dimension. These parameters are defined as a resolving set is a subset of vertices of a graph with a condition that each vertex of that graph has a unique code or representation with respect to the chosen subset. Its minimum cardinality is known as metric dimension, while the edge metric dimension is defined by the minimum count of members in the edge resolving set and this set is defined as according to a chosen subset each edge of a graph has unique representations, then this set is known as edge resolving set. A resolving set is a subset of vertices of a graph with a condition that each edge of that graph has a unique code or representation with respect to the chosen subset. It is minimum cardinality is known as the edge metric dimension.

Modern cryptographic systems use substitution boxes (S-boxes) throughout the encryption process to enhance the security of the plaintext. The integrity of the communication process is ensured by these S-boxes, which are vital in converting the ciphertext back into the original plaintext during the decryption phase. The cryptographic strength of a certain S-box has a substantial impact on the overall security of a given cipher. As a result, many researchers have used innovative construction techniques to create robust S-boxes. The method used in this paper is a novel combination of a specific map on the direct product of cyclic groups of order $16 \times 16$ and an inversion map of a Galois field with order 256. This strategy aims to produce dynamic S-boxes. The proposed method can produce a large number of strong S-boxes by making little changes to the map’s parameters. Four S-boxes were created and their performance was analyzed using industry standards such as bijectivity, the strict avalanche criterion (SAC), nonlinearity (NL), the bit independence criterion (BIC), linear probability (LAP), and differential probability (DAP). The performance of the recommended S-boxes was compared to that of state-of-the-art S-boxes to show their efficacy. The proposed S-box also exhibits considerable potential as a candidate for modern cryptosystems aiming at securing multimedia information, as shown by a suggested method for protecting the privacy of digital images using it. The effectiveness of the encryption method was then assessed using several tests including contrast, correlation, homogeneity, entropy, energy, Number of Pixel change rate (NPCR), and Unified Average changing intensity (UACI). We observed the efficacy of the suggested method for image encryption by comparing our results with different methods.

Euclidean geometry is utilized to establish the Sombor graph parameter and its invariants. It is sum of all adjacent vertices in graph theory dϒ2+dΓ2 where dϒ is the degree of the vertex ϒ. Geometrical interpretation is used to describe the new Sombor indices types. We examined, recently developed Sombor indices for various nanotube Y-junctions in this article. In specifically, the first area-based Sombor index was introduced by Euclidean geometry. Angular orientation concept to construct the second, fourth, and sixth Sombor graph parameters, while third and fifth Sombor graph parameters are constructed by perimeter.

Manufacturing relatively inexpensive items in every area of engineering and science is the major focus of exploration resultant the world's contemporary economic setback. Making small-sized items that are inexpensive and lightweight while providing high quality is critical in today's and tomorrow's worlds. Nanotechnology has a significant role to play in this situation. Nano-objects or, in general, nanomaterials are especially preferred; nanotubes, especially those comprised of carbon, are one of the most popular types of nanostructures, and they are applied in a variety of chemical, biological and technical applications. This notion prompted us to investigate their many physical and chemical characteristics. We utilized topological descriptors to evaluate diverse nanotube structures such as armchair carbon and semi-capped nanotubes by using vertex-edge based indices to characterize distinct chemical structures via numerical quantitative analysis. Furthermore, we examined uncapped and semi-capped armchair carbon nanotubes and achieved adequate comparative findings.

In November 2019, there was the first case of COVID-19 (Coronavirus) recorded, and up to 3rd of April 2020, 1,116,643 confirmed positive cases, and around 59,158 dying were recorded. Novel antiviral structures of the SARS-COV-2 virus is discussed in terms of the metric basis of their molecular graph. These structures are named arbidol, chloroquine, hydroxy-chloroquine, thalidomide, and theaflavin. Partition dimension or partition metric basis is a concept in which the whole vertex set of a structure is uniquely identified by developing proper subsets of the entire vertex set and named as partition resolving set. By this concept of vertex-metric resolvability of COVID-19 antiviral drug structures are uniquely identified and helps to study the structural properties of structure.

Elliptic curve cryptography has gained attention due to its strong resilience against current cryptanalysis methods. Inspired by the increasing demand for reliable and secure cryptographic methods, our research investigates the relationship between complex mathematical structures and image encryption. A substitution box (S-box) is the single non-linear component of several well-known security systems. Mordell elliptic curves are used because of their special characteristics and the immense computational capacity of Galois fields. These S-boxes are dynamic, which adds a layer of complexity that raises the encryption process’s security considerably. We suggest an effective technique for creating S-boxes based on a class of elliptic curves over GF(2n),n≥8. We demonstrate our approach’s robustness against a range of cryptographic threats through thorough examination, highlighting its practical applicability. The assessment of resistance of the newly generated S-box to common attack methods including linear, differential, and algebraic attacks involves a thorough analysis. This analysis is conducted by quantifying various metrics such as non-linearity, linear approximation, strict avalanche, bit independence, and differential approximation to gauge the S-box’s robustness against these attacks. A recommended method for image encryption involves the use of built-in S-boxes to quickly perform pixel replacement and shuffling. To evaluate the efficiency of the proposed strategy, we employed various tests. The research holds relevance as it can provide alternative guidelines for image encryption, which could have wider consequences for the area of cryptography as a whole. We believe that our findings will contribute to the development of secure communication and data protection, as digital security is becoming increasingly important.

Recent research on nanostructures proved their importance and implementation in various areas. In drug delivery systems, medicine and pharmaceutical, biological sensors, photodetectors, transistors, optical and electronic devices, in all these topics nanostructures are implemented directly or indirectly. Future advanced electronic devices are a motivation for the discovery of carbon nanotubes made Y-shaped junctions. Electronic switches, amplifiers, and three-terminal transistors are of great interest due to their interaction with Y-junctions. The concept of entropy determines the uncertainty of a system or network. Concepts of entropy are also commonly used in biology, chemistry, and applied mathematics. Entropy in the form of a graph can be classified into several types based on the requirements. Graph-based entropy was introduced in 1955. Edge-weighted-based entropy is one of the types of entropy. In this study, we look closely at the abstract form of Y-shaped junctions. For the generic view of Y-shaped junctions, we developed some edge-weight-based entropy formulas. On Y-shaped junctions, we discussed some edge-weighted and topological index-based concepts.

Humanity has always benefited from an intercapillary study in the quantification of natural occurrences in mathematics and other pure scientific fields. Graph theory was extremely helpful to other studies, particularly in the applied sciences. Specifically, in chemistry, graph theory made a significant contribution. For this, a transformation is required to create a graph representing a chemical network or structure, where the vertices of the graph represent the atoms in the chemical compound and the edges represent the bonds between the atoms. The quantity of edges that are incident to a vertex determines its valency (or degree) in a graph. The degree of uncertainty in a system is measured by the entropy of a probability. This idea is heavily grounded in statistical reasoning. It is primarily utilized for graphs that correspond to chemical structures. The development of some novel edge-weighted based entropies that correspond to valency-based topological indices is made possible by this research. Then these compositions are applied to clay mineral tetrahedral sheets. Since they have been in use for so long, corresponding indices are thought to be the most effective methods for quantifying chemical graphs. This article develops multiple edge degree-based entropies that correlate to the indices and determines how to modify them in order to assess the significance of each type.

Sombor index and their types are introduced pertaining to Euclidean geometry. In graph-theoretical terminology, it is the sum of all pairs of adjacent vertices di2+dj2 given di is the degree of ith vertex. Using geometrical interpretation, new types of Sombor indices are introduced. In this article, we discussed newly developed Sombor indices for the supramolecular chain of different complexes. Particularly, the first Sombor index was introduced in terms of area in Euclidean geometry. The second, fourth, and sixth versions of Sombor indices are defined in the form of angular orientation, while the third and fifth Sombor indices are formulated via perimeter.

In a graph G, the distance between two vertices is the length of the shortest path between them. The maximum distance between a vertex to any other vertex is considered as the eccentricity of the vertex. In this paper, we introduce the first general Zagreb eccentricity index and found upper and lower bounds on this index in terms of order, size and diameter. Moreover, we characterize the extremal graphs in the class of trees, trees with pendant vertices and bipartite graphs. Results on some famous topological indices can be presented as the corollaries of our main results.

Abstract: Neutrosophic set (NS) is a framework used when the imprecision and uncertainty of an event are described based on three possible aspects, i.e., the membership degree, neutral membership degree and non-membership degree. On the other hand, neutrosophic graphs (NG) are applicable to deal with bulk information events. Furthermore, the incidence graph concept in neutrosophic sets contains a handful of problems like decision-making as well as, social and communication networks. This paper aims to propose the interval-valued neutrosophic sets to incidence graph and represent a new concept, namely interval-valued neutrosophic incidence graph (IVNIG). An IVNIG is a generalization of the concept of single-valued neutrosophic incidence graph (SVNIG). Moreover, some properties related to IVNIG, such as strong edge, strong pair, strong cut pair and neutrosophic incidence cut pair, are also discussed using suitable examples. The defined new concept of IVNIG is applied and investigated on a practical problem of safe root travelling.

Graph operations play an important role in constructing complex network structures from simple graphs. Computation of topological indices of these complex structures via graph products is an important task. In this paper, we generalized the concept of subdivision double-corona product of graphs and investigated the exact expressions of the first and second Zagreb indices, first reformulated Zagreb index, and forgotten topological index (F-index) of this graph operation.

The Wiener index WI is one of the connectivity parameters used to know the biochemical and physicochemical properties of compounds depending upon their molecular structures. Intuitionistic fuzzy graphs IFGs are a convenient tool to represent the objects and relations between them with two types of information using truth membership degree and falsity membership degree. This research work presents the concept of WI under the structure IFGs, IF trees, and IF cycles. Some bounds on WI are investigated. The relationship between WI and connectivity index CI is also studied. In the end of this study, an application of the WI in transport network flow is proposed.

Graph operations are utilized for developing complicated graph structures from basic graphs, and these basic graphs can help to understand the properties of complex networks. While on the other side, the topological descriptor is known as a numeric value that is associated with the graph of a network. It has enormous practical applications in chemistry and other fields of science. This particular work in this draft is the extended work and investigated the first, second, first multiplicative, first reformulated Zagreb indices, and the forgotten index of subdivision double corona and subdivision double neighborhood corona products.

The concept of M-polynomials determines the algebraic form of a system or network. It creates a structure into an algebraic equation and makes work easy to do on such a structure. This has diverse uses in different applied mathematics and as well as in engineering fields. In this study, we look closely at the abstract form of Y-shaped junctions. For the generic view of Y-shaped junctions, we developed some vertex-degree-based M-polynomials formulas. On Y-shaped junctions, we discussed some topological index-based concepts as well and verified the results available in the literature.

The first general Zagreb index MγG or zeroth-order general Randić index of a graph G is defined as MγG=∑v∈Vdvγ where γ is any nonzero real number, dv is the degree of the vertex v and γ=2 gives the classical first Zagreb index. The researchers investigated some sharp upper and lower bounds on zeroth-order general Randić index (for γ

The act of accessing the exact location, or position, of a node in a network is known as the localization of a network. In this methodology, the precise location of each node within a network can be made in the terms of certain chosen nodes in a subset. This subset is known as the locating set and its minimum cardinality is called the locating number of a network. The generalized hexagonal cellular network is a novel structure for the planning and analysis of a network. In this work, we considered conducting the localization of a generalized hexagonal cellular network. Moreover, we determined and proved the exact locating number for this network. Furthermore, in this technique, each node of a generalized hexagonal cellular network can be accessed uniquely. Lastly, we also discussed the generalized version of the locating set and locating number.

Multiplicative Zagreb indices have been studied due to their extensive applications. They play a substantial role in chemistry, pharmaceutical sciences, materials science and engineering, because we can correlate them with numerous physico-chemical properties of molecules. We use graph theory to characterize these chemical structures. The vertices of graphs represent the atoms of a compound and edges of graphs represent the chemical bonds. We present upper and lower bounds on the general multiplicative Zagreb indices for graphs with given number of vertices and cut-edges called bridges. We give all the extremal graphs, which implies that our bounds are best possible.

For a simple graph G=(V(G),E(G)), this paper deals with the existence of an edge labeling χ:E(G)→{0,1,…,k−1},2≤k≤|E(G)|, which induces a vertex labeling χ∗:V(G)→{0,1,…,k−1}in such a way that for each vertex v, assigns the label χ(e1)⋅χ(e2)⋅⋯⋅χ(en)((modk)), where e1,e2,…,enare the edges incident to the vertex v. The labeling χ is called a k-total edge product cordial labeling of G if |(eχ(i)+vχ∗(i))−(eχ(j)+vχ∗(j))|≤1 for every i,j,0≤i

For a colour cluster C = (C1, C2, C3, …, Cℓ), where Ci is a colour class such that ∣Ci∣ = ri, a positive integer, we investigate two types of simple connected graph structures G1C, G2C which represent graphical embodiments of the colour cluster such that the chromatic numbers χ(G1C) = χ(G2C) = ℓ and $\min\{\varepsilon(G^{C}_1)\}=\min\{\varepsilon(G^{C}_2)\} =\sum\limits_{i=1}^{\ell}r_i-1$, and ɛ(G) is the size of a graph G. In this paper, we also discuss the chromatic Zagreb indices corresponding to G1C, G2C.

Planar graphs play an effective role in many practical applications where the crossing of edges becomes problematic. This paper aims to investigate the complex q-rung orthopair fuzzy (CQROF) planar graphs (CQROFPGs). In a CQROFPG, the nodes and edges are based on complex QROF information that represents the uncertain knowledge in the range of unit circles in terms of complex numbers. The motivation in discussing such a topic is the wide flexibility of QROF information in the expression of uncertain knowledge compared to intuitionistic and Pythagorean fuzzy settings. We discussed the complex QROF graphs (CQROFGs), complex QROF multigraphs (CQROFMGs), and related terms followed by examples. Furthermore, the notion of strength and planarity index (PI) of the CQROFPGs is defined and exemplified followed by a study of strong and weak edges. We further defined the notion of complex QROF face (CQROFF) and complex QROF dual graph (CQROFDG) and exemplified these concepts. A study of isomorphism, coweak and weak isomorphism, is set up, and some results relating to the CQROFPG and isomorphisms are explored using examples. Furthermore, the problem of short circuits that results due to crossing is discussed because of the proposed study where an algorithm based on complex QROF (CQROF) information is presented for reducing the crossing in networks. Some advantages of the projected study over the previous study are observed, and some future study is predicted.

The inspection of the networks and graphs through structural properties is a broad research topic with developing significance. One of the methods in analyzing structural properties is obtaining quantitative measures that encode data of the whole network by a real quantity. A large quantity of graph-associated numerical invariants has been used to examine the whole structure of networks. In this analysis, degree-related topological indices have a significant place in nanotechnology and theoretical chemistry. Thereby, the computation of indices is one of the successful branches of research. The noncomplete extended p-sum NEPS of graphs is a famous general graph product. In this paper, we investigated the exact formulas of general zeroth-order Randić, Randić, and the first multiplicative Zagreb indices for NEPS of graphs.

Let G=VG,EG be a molecular graph, where VG and EG are the sets of vertices (atoms) and edges (bonds). A topological index of a molecular graph is a numerical quantity which helps to predict the chemical/physical properties of the molecules. The Wiener, Wiener polarity, and the terminal Wiener indices are the distance-based topological indices. In this paper, we described a linear time algorithm (LTA) that computes the Wiener index for acyclic graphs and extended this algorithm for unicyclic graphs. The same algorithms are modified to compute the terminal Wiener index and the Wiener polarity index. All these algorithms compute the indices in time On.

In theoretical medicine, topological indices are defined to test the medicine and pharmacy characteristics, such as melting point, boiling point, toxicity and other biological activities. As basic molecular structures, hexagonal jagged-rectangle and distance-regular structure are widely appeared in medicine, pharmacy and biology engineering. In this paper, we study the chemical properties of hexagonal jagged-rectangle from the mathematical point of view. Several vertex distance-based indices are determined. Furthermore, the Wiener related indices of distance-regular structure are also considered. Keywords: Theoretical medicine, Molecular graph, Distance-based index, Hexagonal jagged-rectangle

Topological index is a number that can be used to characterize the graph of a molecule. Topological indices describe the physical, chemical, and biological properties of a chemical structure. In this paper, we derive the analytical closed formulas of face index of some planar molecular structures such as TUC4, TUC4C8S, TUHC6, TUC4C8R, and armchair TUVC6.

Several graph invariants have been defined and studied, which present applications in nanochemistry, computer networks, and other areas of science. One vastly studied class of the graph invariants is the class of the topological indices, which helps in the studies of chemical, biological, and physical properties of a chemical structure. One recently introduced graph invariant is the face index, which can assist in predicting the energy and the boiling points of the certain chemical structures. In this paper, we drive the analytical closed formulas of face index of silicon carbides Si2C3−Ia,b, Si2C3−IIa,b, Si2C3−IIIa,b, and SiC3−IIIa,b.

Among topological descriptors, connectivity indices are very important and they have a prominent role in chemistry. The atom-bond connectivity index of a connected graph G is defined as ABC(G)= where dv denotes the degree of vertex v of G and the eccentric connectivity index of the molecular graph G is defined as ξ(G)=, where ε(v) is the largest distance between v and any other vertex u of G. Also, the eccentric atom-bond connectivity index of a connected graph G is equal to ABC5(G)=. In this present paper, we compute this new Eccentric and Connectivity index for an infinite family of Linear Polycene Parallelogram Benzenoid.

A massive of early drug tests implies that there exist strong inner relationships between the bio-medical and pharmacology characteristics of drugs and their molecular structures. The forgotten topological index was defined to be used in the analysis of drug molecular structures, which is quite helpful for pharmaceutical and medical scientists to grasp the biological and chemical characteristics of new drugs. Such tricks are popularly employed in developing countries where enough money is lacked to afford the relevant chemical reagents and equipment. In our article, by means of drug molecular structure analysis and edge dividing technology, we present the forgotten topological index of several widely used chemical structures which often appear in drug molecular graphs.

A novel topological index, the face index ( F I ), is proposed in this paper. For a molecular graph G, face index is defined as F I ( G ) = ∑ f ∈ F ( G ) d ( f ) = ∑ v ∼ f , f ∈ F ( G ) d ( v ) , where d ( v ) is the degree of the vertex v. The index is very easy to calculate and improved the previously discussed correlation models for π - e l e c t r o n energy and boiling point of benzenoid hydrocarbons. The study shows that the multiple linear regression involving the novel topological index can predict the π -electron energy and boiling points of the benzenoid hydrocarbons with correlation coefficient r > 0 . 99 . Moreover, the face indices of some planar molecular structures such as 2-dimensional graphene, triangular benzenoid, circumcoronene series of benzenoid are also investigated. The results suggest that the proposed index with good correlation ability and structural selectivity promised to be a useful parameter in QSPR/QSAR.

The first Zagreb index $M_1$ of a graph $G$ is equal to the sum of squares of degrees of the vertices of $G$. Goubko proved that for trees with $n_1$ pendent vertices, $M_1 geq 9,n_1-16$. We show how this result can be extended to hold for any connected graph with cyclomatic number $gamma geq 0$. In addition, graphs with $n$ vertices, $n_1$ pendent vertices, cyclomatic number $gamma$, and minimal $M_1$ are characterized. Explicit expressions for minimal $M_1$ are given for $gamma=0,1,2$, which directly can be extended for $gamma>2$.

For a graph G without isolated vertices, the inverse degree of a graph G is defined as I D ( G ) = ∑ u ∈ V ( G ) d ( u ) − 1 where d ( u ) is the number of vertices adjacent to the vertex u in G. By replacing − 1 by any non-zero real number we obtain zeroth-order general Randić index, i.e., 0 R γ ( G ) = ∑ u ∈ V ( G ) d ( u ) γ , where γ ∈ R − { 0 } . Xu et al. investigated some lower and upper bounds on I D for a connected graph G in terms of connectivity, chromatic number, number of cut edges, and clique number. In this paper, we extend their results and investigate if the same results hold for γ < 0 . The corresponding extremal graphs have also been identified.