Muhammad Javaid

• Post-Doctorate Mathematics [Graph Theory and Combinatorics]
• Professor (Associate) at University of Management and Technology

A topological index (TI) is a numeric digit that signalizes the whole chemical structure of a molecular network. TIs are helpful in predicting the bioactivity of molecular substances in investigations of quantitative structure-activity relationship (QSAR) and quantitative structure-property relationship (QSPR). TIs correlate various chemical and ph...

In the age of Industry 4.0, the use of cyber-physical production systems (CPPS) is on the rise in the manufacturing industry. These systems combine physical production processes with digital systems to create a more efficient and connected manufacturing environment. Advanced measurement and sensing systems are key components of CPPS, allowing manuf...

Metric dimension is one of the distance-based parameters which are used to find the position of the robot in a network space by utilizing lesser number of notes and minimum consumption of time. It is also used to characterize the chemical compounds. The metric dimension has a wide range of applications in the field of computer science such as integ...

In the studies of the connected networks, metric dimension being a distance-based parameter got much more attention of the researches due to its wide range of applications in different areas of chemistry and computer science. At present its various types such as local metric dimension, mixed metric dimension, solid metric dimension, and dominant me...

Monitoring and controlling complex networks is of great importance to understand different types of technological and physical systems for source localization. Source localization refers to the process of determining the location or position of a signal source in space based on measurements obtained from multiple sensors. Doubly resolving sets, als...

Doubly resolving sets (DRSs) provide a promising approach for source detection. They consist of minimal subsets of nodes with the smallest cardinality, referred to as the double metric dimension (DMD), that can uniquely identify the location of any other node within the network. Utilizing DRSs can improve the accuracy and efficiency of the identifi...

For a signature function $ \Psi:E({H}) \longrightarrow \{\pm 1\} $ with underlying graph $ H $, a signed graph (S.G) $ \hat{H} = (H, \Psi) $ is a graph in which edges are assigned the signs using the signature function $ \Psi $. An S.G $ \hat{H} $ is said to fulfill the symmetric eigenvalue property if for every eigenvalue $ \hat{h}(\hat{H}) $ of $...

Topological indices are the numerical descriptors that correspond to some certain physicochemical properties of a chemical compound such as the boiling point, acentric factor, enthalpy of vaporisation, heat of fusion, etc. Among these topological indices, the Hyper Zagreb index, is the most effectively used topological index to predict the acentric...

Topological index (TI) is a mapping that associates a real number to the under study (molecular) graph which predicts its various physical and chemical properties. The generalized degree distance index is the latest developed TI having compatible significance among the list of distance-based TIs. In this paper, the minimum generalized degree distan...

The area of graph theory (GT) is rapidly expanding and playing a significant role in cheminformatics, mostly in mathematics and chemistry to develop different physicochemical, chemical structure, and their properties. The manipulation and study of chemical graphical details are made feasible by using numerical structure invariant. Investigating the...

The parameter of distance plays an important role in studying the properties symmetric networks such as connectedness, diameter, vertex centrality and complexity. Particularly different metric-based fractional models are used in diverse fields of computer science such as integer programming, pattern recognition, and in robot navigation. In this man...

Let $ G = (V, E) $ be a simple connected graph. A vertex $ x\in V(G) $ resolves the elements $ u, v\in E(G)\cup V(G) $ if $ d_G(x, u)\neq d_G(x, v) $. A subset $ S\subseteq V(G) $ is a mixed metric resolving set for $ G $ if every two elements of $ G $ are resolved by some vertex of $ S $. A set of smallest cardinality of mixed metric generator for...

Metal–organic frameworks (MOFs) are intriguing porous materials that are formed by combining organic materials with metals. MOFs have a vast range of utilizations in distinct medical fields with great efficiency. Recently, Zinc-related MOFs have been investigated and are in demand due to their efficient utilization in medical fields such as biosens...

The variable sum exdeg index, initially introduced by Vukicevic (2011) [20] for predicting the octanol water partition co-efficient of certain chemical compounds, is an invariant for a graph G and defined as SEIa(G)=∑v∈V(G)(dvadv), where dv is the degree of vertex v∈V(G), a is a positive real number different from 1. In this paper, we defined sub-c...

For a connected network $ \Gamma $, the distance between any two vertices is the length of the shortest path between them. A vertex $ c $ in a connected network is said to resolve an edge $ e $ if the distances of $ c $ from its endpoints are unequal. The collection of all the vertices which resolve an edge is called the local resolving neighborhoo...

The suppression of harmful information and even its diffusion can be predicted and delayed by precisely finding sources with limited resources. The doubly resolving sets (DRSs) play a crucial role in determining where diffusion occurs in a network. Source detection problems are among the most challenging and exciting problems in complex networks. T...

The study of networks by using topological indices (TIs) have been significantly become a useful attention in the physicochemical properties of compounds, pharmacology and drug delivery in the field of experimental sciences. Thus, TIs help us to study the new networks and they also play an essential role in the study of the quantitative structure p...

In the modern era, mathematical modeling consisting of graph theoretic parameters or invariants applied to solve the problems existing in various disciplines of physical sciences like computer sciences, physics, and chemistry. Topological indices (TIs) are one of the graph invariants which are frequently used to identify the different physicochemic...

The concept of metric dimension is widely applied to solve various problems in the different fields of computer science and chemistry, such as computer networking, integer programming, robot navigation, and the formation of chemical structuring. In this article, the local fractional metric dimension (LFMD) of the cycle-based Sierpinski networks is...

A topological index being a graph theoretic parameter plays a role of function for the assignment of a numerical value to a molecular graph which predicts the several physical and chemical properties of the underlying molecular graph such as heat of evaporation, critical temperature, surface tension, boiling point, octanol-water partition coefficie...

Topological indices (TIs) have been practiced for distinct wide-ranging physicochemical applications, especially used to characterize and model the chemical structures of various molecular compounds such as dendrimers, nanotubes and neural networks with respect to their certain properties such as solubility, chemical stability and low cytotoxicity....

The concept of locating number for a connected network contributes an important role in computer networking, loran and sonar models, integer programming and formation of chemical structures. In particular it is used in robot navigation to control the orientation and position of robot in a network, where the places of navigating agents can be replac...

e connected and acyclic components contained in a network are identified by the computation of its complexity, where complexity of a network refers to the total number of spanning trees present within. e article in hand deals with the enumeration of the complexity of various networks' operations such as sum (), difference (K 2,n ⊖K 2), and the conj...

Locating the sources of information spreading in networks including tracking down the origins of epidemics, rumors in social networks, and online computer viruses, has a wide range of applications. In many situations, identifying where an epidemic started, or which node in a network served as the source, is crucial. But it can be difficult to deter...

1st ReadingAugust 14, 2022 14:58 112-IJFCS 2202002International Journal of Foundations of Computer Science(2022) 1–7© World Scientific Publishing CompanyDOI: 10.1142/S0129054122020026Special Issue:Mathematical Aspects of EvolutionaryComputation and its ApplicationsPrefaceJia-Bao LiuAnhui Jianzhu University, 292 Ziyun RdShushan District, Hefei, Anhu...

For any simple connected graph G of order n, having eigen spectrum μ 1 ≥ μ 2 ≥ ⋯ ≥ μ n with middle eigenvalues μ H and μ L, where H = ⌊(n + 1)/2⌋ and L = ⌈(n + 1)/2⌉, the HOMO–LUMO gap is defined as as ΔG = μ H = μ L. In this article, a simple upper bound for the HOMO–LUMO gap corresponding to a special class of connected bipartite graphs is estima...

Theory of networks serves as a mathematical foundation for the construction and modeling of chemical structures and complicated networks. In particular, chemical networking theory has a wide range of utilizations in the study of chemical structures, where examination and manipulation of chemical structural information are made feasible by utilizing...

Distance based topological indices (TIs) play a vital role in the study of various structural and chemical aspects for the molecular graphs. The first distance-based TI is used to find the boiling point of paraffin. The connection distance (CD) index is a latest developed TI that is defined as the sum of all the products of distances between pair o...

Graph theory (GT) serves as a mathematical foundation that helps us to manipulate, develop, analyze, and comprehend the chemical networks or structures and their characteristics. Molecular graph is a graph made up of vertices (atoms) and edges (chemical bonds between atoms). Chemical GT applied geometrical and combinatorial GT to model the importan...

Metal organic frameworks (MOFs) are distinctive porous chemical materials comprised of metal ions and organic ligands to illustrate marvelous chemical stability, high surface area, distinctive morphology, and large pore volume. MOFs have great significance due to their versatile utilizations, such as purification and separation of various gases, en...

Metric dimension is an effective tool to study different distance-based problems in the field of telecommunication, robotics, computer networking, integer programming, chemistry, and electrical networking. In this paper, we study the latest form of metric dimension called fractional metric dimension of some connected networks such as circular diago...

For a connected graph, the concept of metric dimension contributes an important role in computer networking and in the formation of chemical structures. Among the various types of the metric dimensions, the fault-tolerant metric dimension has attained much more attention by the researchers in the last decade. In this study, the mixed fault-tolerant...

Topological index (TI) is a graph-theoretic tool that is used to study different physical and structural properties of the networks in various disciplines of science such as computer science, chemistry, and information technology. In this article, we study transition metal tetra-cyano-benzene organic networks by computing their M-polynomials and va...

Topological indices are graph-theoretic parameters which are widely used in the subject of chemistry and computer science to predict the various chemical and structural properties of the graphs respectively. Let G be a graph; then, by performing subdivision-related operations S , Q , R , and T on G , the four new graphs S G (subdivision graph), Q G...

A topological index (TI) is a function that associates a numeric number to the under-study molecular graph. The first TI based on distance was presented by Wiener to find the boiling point of paraffins. In 1972, to compute the total π-electron energy of a molecule, the degree-based indices are used. In this paper, we obtained first and second Zagre...

The degree distance index (DDI) is a vertex-degree weighted version of a well-known index that is called by Wiener index (WI). In extremal theory of graphs, improving the bounds with best possible values is a worth investigating problem. In this note, the exact formulae of the DDI for the four different types of the sum graphs in the terms of vario...

Topological indices or coindices are mathematical parameters which are widely used to investigate different properties of graphs. The operations on graphs play vital roles in the formation of new molecular graphs from the old ones. Let Γ be a graph we perform four operations which are S , R , Q , and T and obtained subdivisions type graphs such tha...

A structural descriptor is a numeric quantity, invariant under symmetry, extracted from a molecular graph Γ using tools from mathematics. Hex-derived networks H D N 1 n and H D N 2 n being dual of (3,12,12) and (4,8,8) carbon allotrope (nanocones, nanotubes) are quite fascinating structures in mathematical chemistry. In this paper, we compute exact...

The Gourava indices and hyper-Gourava indices are graph invariants, related to the degree of vertices of a graph G . Let T n , b denote the collection of all chemical trees with n vertices where b denotes the number of branching vertices, 1 ≤ b < n − 2 / 2 . In the current paper, maximum value for the abovementioned topological indices for differen...

Dendrimers are artificially synthesized polymeric macromolecules composed of frequently branching chains called monomers. Topological indices (TIs) are the molecular descriptors which characterize the topology and help to correlate the distinct psychochemical properties such as stability, boiling point, and strain energy of molecular compounds. TIs...

Topological indices (TIs) have been utilized widely to characterize and model the chemical structures of various molecular compounds such as dendrimers, neural networks, and nanotubes. Dendrimers are extraordinarily comprehensible, globular, artificially synthesized polymers with a structure of frequently branched units. A mathematical approach to...

The Gourava indices and hyper-Gourava indices are introduced by Kulli in 2017. These graph invariants are related to the degree of vertices of a graph G. Let Tn,r be the class of all n−vertex chemical trees with r segments. In this paper, we characterize the graphs with the maximum value of the above indices in the class of chemical trees. In addit...

In theoretical chemistry, topological indices (TIs) have important role to predict various physical and structural properties of the study under molecular graphs. Among all topological indices, Zagreb-type indices have been used more effectively in the chemical literature. In this paper, we have computed first Zagreb, second Zagreb, forgotten, and...

In this article, we address the super edge-antimagic total labeling of the hexagonal lattice $HTT_{m,n}$ and two non-isomorphic forms of prismatic lattice $PTT_{m,n}$ . The aforementioned classes are symmetric lattices involving the finite chain of tripartite networks. Our article further closes with the summary, $3D$ - graphical illustration...

A subset T of the vertex set of a network G is called a resolving set for G if each pair of vertices of G have distinct representations with respect to T . A resolving set B ʹ among all the resolving sets of a network G is called a fault-tolerant resolving set if B ʹ\{ t } is as well a resolving set for each vertex t ϵ B ʹ. A fault-tolerant resolvi...

For having an in-depth study and analysis of various network’s structural properties such as interconnection, extensibility, availability, centralization, vulnerability and reliability, we require distance based graph theoretic parameters. Numerous parameters like distance based dimensions help in designing queuing models in restaurants, public hea...

Topological indices (TIs) are functional tools which correlate with a computational value through a undirected, finite and simple networks. Many physicochemical properties and chemical reactions are studied with the help of these TIs. Recently, they are commonly used in the behavior of quantitative structures activity as well as property relationsh...

BACKGROUND Topological indices (TIs) are mathematical formulas that are applied in mathematical chemistry to predict the physical and chemical properties of various chemical structures. In this study, three different types of polycyclic aromatic hydrocarbon structures (PAHs) (i.e., Hexa-peri-hexabenzocoronene, Dodeca-benzo-circumcoronene, and Hexa-...

The physical and structural properties of molecular structure or graph such as boiling point, melting point, surface tension, or solubility are studied using topological index (TI). Topological index is a mathematical formula that can be applied to any graph which models some molecular structures. The various operations play an important role in gr...

The field of graph theory is broadly growing and playing a remarkable role in cheminformatics, mainly in chemistry and mathematics in developing different chemical structures and their physicochemical properties. Mathematical chemistry provides a platform to study these physicochemical properties with the help of topological indices (TIs). A topolo...

Let N=VN,EN be a connected network with vertex VN and edge set EN⊆VN,EN. For any two vertices a and b, the distance da,b is the length of the shortest path between them. The local resolving neighbourhood (LRN) set for any edge e=ab of N is a set of all those vertices whose distance varies from the end vertices a and b of the edge e. A real-valued f...

A topological index (TI) is a numerical descriptor of a molecule structure or graph that predicts its different physical, biological, and chemical properties in a theoretical way avoiding the difficult and costly procedures of chemical labs. In this paper, for two connected (molecular) graphs G1 and G2, we define the generalized total-sum graph con...

Metric dimension is one of the distance-based parameter which is frequently used to study the structural and chemical properties of the different networks in the various fields of computer science and chemistry such as image processing, pattern recognition, navigation, integer programming, optimal transportation models, and drugs discovery. In part...

A connected graph in which the number of edges is one more than its number of vertices is called a bicyclic graph. A perfect matching of a graph is a matching in which every vertex of the graph is incident to exactly one edge of the matching set such that the number of vertices is two times its matching number. In this paper, we investigated maximu...

The distance centric parameter in the theory of networks called by metric dimension 1 plays a vital role in encountering the distance-related problems for the monitoring of the large-scale networks in the various fields of chemistry and computer science such as navigation, image processing, pattern recognition, integer programming, optimal transpor...

Topological index (TI) is an unaltered numeric number that associates with the graph isomerism. These unaltered numbers mostly appear in the kind of degree based TIs. The first degree based TI was introduced by Gutman and Trinajstić in 1972. Such degree based TIs are further classified in two different ways; degree and connection number based TIs....

The parameter of distance in the theory of networks plays a key role to study the different structural properties of the understudy networks or graphs such as symmetry, assortative, connectivity, and clustering. For the purpose, with the help of the parameter of distance, various types of metric dimensions have been defined to find the locations of...

Topological index (TI) is a function from the set of graphs to the set of real numbers that associates a unique real number to each graph, and two graphs necessarily have the same value of the TI if these are structurally isomorphic. In this note, we compute the HZ−index of the four generalized sum graphs in the form of the various Zagreb indices o...

Chemical structural formula can be represented by chemical graphs in which atoms are considered as vertices and bonds between them are considered as edges. A topological index is a real value that is numerically obtained from a chemical graph to predict its various physical and chemical properties. Thorn graphs are obtained by attaching pendant ver...

The combination of mathematical sciences, physical chemistry, and information sciences leads to a modern field known as cheminformatics. It shows a mathematical relationship between a property and structural attributes of different types of chemicals called quantitative-structures’ activity and qualitative-structures’ property relationships that ar...

The topic of computing the topological indices (TIs) being a graph-theoretic modeling of the networks or discrete structures has become an important area of research nowadays because of its immense applications in various branches of the applied sciences. TIs have played a vital role in mathematical chemistry since the pioneering work of famous che...

Metal-organic networks (MONs) are among the unique complex and porous chemical compounds. So, these chemical compounds consist of metal ions (vertices) and organic ligands (edges between vertices). These networks represent large pore volume, extreme surface area, morphology, excellent chemical stability, highly porous and crystalline materials, and...

Let G=VE,EG be a connected graph with vertex set VG and edge set EG. For a graph G, the graphs S(G), R(G), Q(G), and T(G) are obtained by applying the four subdivisions related operations S, R, Q, and T, respectively. Further, for two connected graphs G1 and G2, G1+FG2 are F-sum graphs which are constructed with the help of Cartesian product of FG1...

A topological index (TI) is a molecular descriptor that is applied on a chemical structure to compute the associated numerical value which measures volume, density, boiling point, melting point, surface tension, or solubility of this structure. It is an efficient mathematical method in avoiding laboratory experiments and time-consuming. The forgott...

In order to identify the basic structural properties of a network such as connectedness, centrality, modularity, accessibility, clustering, vulnerability, and robustness, we need distance-based parameters. A number of tools like these help computer and chemical scientists to resolve the issues of informational and chemical structures. In this way,...

The idea of super -edge-antimagic labeling of graphs had been introduced by Enomoto et al. in the late nineties. This article addresses super -edge-antimagic labeling of a biparametric family of pancyclic graphs. We also present the aforesaid labeling on the disjoint union of graphs comprising upon copies of and different trees. Several problems sh...

The use of numerical numbers to represent molecular networks plays a crucial role in the study of physicochemical and structural properties of the chemical compounds. For some integer k and a network G, the networks SkG and RkG are its derived networks called as generalized subdivided and generalized semitotal point networks, where Sk and Rk are ge...

Polymers, drugs, and almost all chemical or biochemical compounds are frequently modeled as diverse ω-cyclic, acyclic, bipartite, and polygonal shapes and regular graphs. Molecular descriptors (topological indices) are the numerical quantities and computed from the molecular graph Γ (2D lattice). ese descriptors are highly significant in quantitati...

A bijective map ρ from V(Ω) →{1,2, … |V(Ω)|} is called sum divisor cordial labeling for graph Ω so that for every uυ ∈ E(Ω) edge is fixed the label 1 if 2 divides ρ (u) + ρ(υ) and 0 otherwise, then the difference between number of edges labeled with 1 and the number of edges labeled with 0 by at most 1. A graph is called sum divisor cordial graph i...

Topological indices or coindices are one of the graph-theoretic tools which are widely used to study the different structural and chemical properties of the under study networks or graphs in the subject of computer science and chemistry, respectively. For these investigations, the operations of graphs always played an important role for the study o...

The number of spanning trees in a network determines the totality of acyclic and connected components present within. This number is termed as complexity of the network. In this article, we address the closed formulae of the complexity of networks’ operations such as duplication (split, shadow, and vortex networks of Sn), sum (Sn+W3, Sn+K2, and Cn∘...

It is considered that there is a fascinating issue in theoretical chemistry to predict the physicochemical and structural properties of the chemical compounds in the molecular graphs. These properties of chemical compounds (boiling points, melting points, molar refraction, acentric factor, octanol-water partition coefficient, and motor octane numbe...

Distance-based dimensions provide the foreground for the identification of chemical compounds that are chemically and structurally different but show similarity in different reactions. The reason behind this similarity is the occurrence of a set S of atoms and their same relative distances to some ordered set T of atoms in both compounds. In this a...

Distance-based numeric parameters play a pivotal role in studying the structural aspects of networks which include connectivity, accessibility, centrality, clustering modularity, complexity, vulnerability, and robustness. Several tools like these also help to resolve the issues faced by the different branches of computer science and chemistry, name...

Topological index (TI) is a numerical invariant that helps to understand the natural relationship of the physicochemical properties of a compound in its primary structure. George Polya introduced the idea of counting polynomials in chemical graph theory and Winer made the use of TI in chemical compounds working on the paraffin's boiling point. The...

Metric dimension of networks is a distance based parameter that is used to rectify the distance related problems in robotics, navigation and chemical strata. The fractional metric dimension is the latest developed weighted version of metric dimension and a generalization of the concept of local fractional metric dimension. Computing the fractional...

Metric dimension is a distance-based tool that is used in the different fields of computer science and chemistry such as navigation, combinatorial optimization, pattern recognition, image processing, integer programming and formation of chemical compounds. In this paper, we study the latest type of metric dimension called as local fractional metric...

Metal-organic frameworks (MOF(n)) are organic-inorganic hybrid crystalline porous materials that consist of a regular array of positively charged metal ions surrounded by organic ‘linker’ molecules. The metal ions form nodes that bind the arms of the linkers together to form a repeating, cage-like structure. Moreover, in a chemical structure or mol...

e different distance-based parameters are used to study the problems in various fields of computer science and chemistry such as pattern recognition, image processing, integer programming, navigation, drug discovery, and formation of different chemical compounds. In particular, distance among the nodes (vertices) of the networks plays a supreme rol...

In theoretical chemistry, several distance-based, degree-based, and counting polynomial-related topological indices (TIs) are used to investigate the different chemical and structural properties of the molecular graphs. Furtula and Gutman redefined the -index as the sum of cubes of degrees of the vertices of the molecular graphs to study the differ...