Phd

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Aggregation operators based on some extension of fuzzy sets.
(UMT Lahore, 2018-03-22) Raja Noshad Jamil
Bonferroni mean (BM) and heronian mean (HM) operators are useful tools for group decision making problems, when arguments are interrelated to each other. In this thesis, we developed some BM and HM based aggregation operators. We defined some aggregation operators for dual hesitant fuzzy (DHF) sets, for instance, we defined dual hesitant fuzzy geometric bonferroni mean (DHFGBM) and different properties of DHFGBM are discussed. Some special cases are also studied in detail for DHFGBM. In addition, dual hesitant fuzzy weighted geometric bonferroni mean (DHFWGBM) and dual hesitant fuzzy chouqet geometric bonferroni mean (DHFCGBM) proposed. We also model a system of fuzzy soft differential equations (FSDEs) to analyze the behavior over the time of an individual depending on their companion’s actions under any particular situation against some decision by the help of BM. Using the ability of BM to capture the interrelationship of arguments, we defined bonferroni fuzzy soft matrix (BFSM) and weighted bonferroni fuzzy soft matrix (WBFSM) for data representation. WBFSM is a decision matrix and provide optimum fuzzy soft constant (OFSC), which is the key element of FSDEs. By utilizing the OFSC, we developed a system of FSDEs to study a dynamical process with nonlinear uncertain and vague data. We presented a novel efficient technique for analyzing the future attitude of people due to their present decisions. To illustrate the practicality and feasibility of proposed technique, an example is also discussed with the help of phase portrait and line graphs. With respect to multiple attribute group decision making problems, in which the value of the attributes are taken in the form of hesitant 2-tuple (H2T) or intuitionistic 2-tuple (I2T) linguistic information are called CW. H2T linguistic arguments are used to evaluate the group decision making problems which have inter-dependent or interactive attributes. Some operational laws are developed for H2T linguistic elements and based on these operational laws hesitant 2-tuple weighted averaging (H2TWA) operator and generalize hesitant 2-tuple averaging (GH2TA) operator are proposed. Combining choquet integral (CI) with H2T linguistic information, defined hesitant 2-tuple correlated averaging (H2TCA) and generalize hesitant 2-tuple correlated averaging (GH2TCA) operators. In the existing literature review, we observed that during aggregation procedure for H2T, more hesitation produces in the resultant element. We targeted this issue and developed a diminishing hesitant 2-tuple averaging operator (DH2TA) operator for H2T linguistic arguments. DH2TA operator work in the way that it’s aggregate all H2T linguistic elements and during the aggregation process it also controls the hesitation in the translation of the resulting aggregated xv xvi linguistic term. We developed a scalar product for H2T linguistic elements and based on the scalar product, a diminishing weighted hesitant 2-tuple averaging operator (DWH2TA) is introduced. Moreover, combining CI with H2T linguistic information, the diminishing choquet hesitant 2-tuple average operator (DCH2TA) operator is defined. Most of existing operational laws in literature for handling the process for CW are not bounded and hence a logical problem comes. We targeted this issue and developed closed operational laws based on Archimedean t-norm and t-conorm. Some aggregation operators intuitionistic 2-tuple linguistic heronian mean (I2THM) and intuitionistic 2-tuple linguistic chouqet heronian mean (I2TCHM) based on these closed operational laws developed and discussed desired properties of the proposed operators. Linkages between industry and university are the significant parts in the entire advancement of any country. To assess university’s reputation for industry, we proposed a fusion approach by using heronian intuitionistic fuzzy analytic hierarchy process (HIF-AHP), fuzzy geometric bonferroni mean (FGHM) operator and 2-tuple fuzzy linguistic elements. In each chapter, we developed some techniques based on proposed operators and demonstrated the validity and feasibility of these techniques by some examples. Educational note: Sentence case formatting prioritizes readability by limiting capitalization to only the first word of each sentence, proper nouns/adjectives (e.g., "Archimedean," derived from the mathematician Archimedes), and acronyms (e.g., BM, DHF). This consistency avoids visual clutter, which is particularly useful for academic texts like the thesis excerpt provided—readers can quickly parse sentence boundaries without distraction from overcapitalized terms. Acronyms are retained in uppercase to preserve their symbolic meaning and avoid confusion with generic terms (e.g., "BM" remains distinct from the common noun "bm").
Algebraic connectivity and fractional metric dimension of graphs.
(UMT Lahore, 2021-12-03) Mohsin Raza
Let G = (V (G), E(G)) be a graph having V (G) = {vi : 1 ≤ i ≤ n} and E(G) ⊆ V (G) × V (G) as the sets of vertices and edges respectively. A graph Gc is called complement of a graph G with vertex-set V (Gc) = V (G) and edge-set E(Gc) = {uv : u, v ∈ V (G), uv /∈ E(G)}. The number of first neighbors of v ∈ V (G) is defined as degree of the vertex v and it is denoted by d(v) or dG(v). For any two vertices x, y ∈ V (G) the distance (d(x, y)) is the length of shortest path between them. The adjacency matrix (A-matrix) of a graph G of order n is defined as A(G) = [ai,j ]n×n such that ai,j = 1 if vi is adjacent to vj and ai,j = 0 otherwise. The degree matrix (D-matrix) of G is defined by D(G) = [ai,j ]n×n such that ai,i = d(vi) and zero otherwise. The Laplacian matrix (L-matrix) of the graph G is denoted as L(G) and defined as L(G) = D(G) − A(G) where, D(G) and A(G) are degree and adjacency matrices of graph G respectively. For 1 ≤ i ≤ n, the eigenvalues μi = μi(G) and eigenvectors Zi = Zi(G) of L-matrix (L(G)) are the L-eigenvalues and the L-eigenvectors of G respectively. The second smallest eigenvalue of the Laplacian matrix of a graph is known as an algebraic connectivity. It is used as a parameter to measure the connectivity of a graph i.e. how well a graph is connected. Furthermore, any two vertices x, y ∈ V (G) of a simple connected graph G are said to be resolved or distinguished by a vertex z ∈ V (G) if d(x, z) ≠ d(y, z). A set S ⊆ V (G) is called a resolving set of G if each pair of vertices of G is resolved by some vertex in S. A minimum resolving set is known as metric basis and its cardinality is called as metric dimension of the graph G that is denoted by dim(G). For a pair (u, v) of vertices of G, the resolving neighborhood set of G is defined as R(u, v) = {w ∈ V (G) : d(w, u) ≠ d(w, v)}. A resolving function is a real valued function g : V (G) → [0, 1] such that g(R(u, v)) ≥ 1 for each resolving neighborhood of distinct pair of vertices of G, where g(R(u, v)) = ∑ x∈R(u,v) g(x). A resolving function g is called minimal, if there exists a function f : V (G) → [0, 1] such that f ≤ g and f (v) ≠ g(v) for at least one v ∈ V is not a resolving function of G.
Analysis of convective viscous and rate type nanofluids in a channel
(UMT Lahore, 2022-10-26) Kashif Sadiq
This work highlights the convective and unsteady nanofluids flow through a channel under different boundary conditions. The flow is affected by thermal radiation, chemical reaction, heat absorption, magnetohydrodynamics, and the Soret effect. Water, ethylene glycol-water (50%-50%), sodium alginate, and glycerin are considered conventional base fluids, comprising the nanoparticles of copper, silver, titanium oxide, aluminum oxide, and carbon nanotubes. The definitions of time fractional derivatives (Caputo and Caputo-Fabrizio) are applied to the governing equations to have fractional models. Analytical solutions of proposed models are obtained with the help of integral transforms (Laplace and finite sine-Fourier) and semi-analytical results are obtained by the Laplace inversion numerical algorithm of Stehfest’s. The comparison of base nanofluids suspended with different nanoparticles is made, the effect of nanoparticles, as well as the physical parameters like Prandtl number, mass Grashof number, Schmidt number, thermal Grashof number, magnetic parameter, heat absorption, Soret effect, radiation, and fractional parameters on velocity, temperature, and concentration field, are shown and discussed using the mathematical software MathCAD. The skin friction, Sherwood, and Nusselt numbers are also determined by proposed physical models.
Analysis of topological indices and coindices graphs for resultant graphs
(UMT Lahore, 2022-09-28) Muhammad Ibraheem
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. In these fields, the operations of graphs always played an important role for the study of the various complex networks and to develop the different new classes of graphs. For a connected graph Γ, the graphs S(Γ) (subdivided), R(Γ) (triangle parallel), Q(Γ) (line superposition) and T (Γ) (total) are called the derived graphs of Γ, which are obtained by applying the subdivision related operations S (subdivision), R (vertex semi total), Q (edge semi total) and T (total) on the graph Γ, respectively. Let Γ1 and Γ2 be two connected graphs then their F -sum graph (Γ1+F Γ2) is obtained by the cartesian product of the graphs F (Γ1) and Γ2, where F (Γ1) is a derived graph for F ∈ {S, R, Q, T }. The aim of this thesis is to investigate several degree based topological coindices of the different F -sum graphs like S-sum, R-sum, Q-sum, and T -sum. In particular, bounds or exact values for first general zagreb coindex, genernal randi ́c coindex, forgotten coindex, hyper zagreb coindex, sum-connectivity coindex and first multiplicative zagreb coindex are computed. Moreover, a comparison is conducted between the obtained results on the aforesaid topological coindices and existing topological indices for these F -sum graphs. Finally, we numerically analyses that topological coindices for F -sum graphs are better then their topological indices.
Analytical solutions for different motions of differential and rate type fluids with fractional derivatives.
(UMT Lahore, 2018-10-24) Muhammad Bilal Riaz
In this dissertation, we present the analytical studies of some fluid flow models. We analyze the fractional models for the flow of non-Newtonian fluids via classical computational techniques to obtain analytical solutions. This study includes the investigation of the unsteady natural convection flow of Maxwell fluid with fractional derivative over an exponentially accelerated infinite vertical plate. Slip condition, chemical reaction, transverse magnetic field and Newtonian heating effects are also considered using a modern definition of fractional derivative. Moreover, the unsteady flow of Maxwell fluid with non-integer order derivatives through a circular cylinder of infinite length in a rotating frame is studied. The motion of Maxwell fluid is generated by a time dependent torsion applied to the surface of the cylinder. As novelty, the dimensionless governing equation related to the non-trivial shear stress is used and the first exact solution analogous to a ramped shear stress on the surface is obtained. The rotational flow of an Oldroyd-B fluid with fractional derivative induced by an infinite circular cylinder that applies a constant couple stress to the fluid is investigated. It is worth mentioning that the considered problem of Oldroyd-B fluid in the settings of fractional derivatives has not been found in the literature. Some unsteady Couette flows of an Oldroyd-B fluid with non-integer derivative in an annular region of two infinite co-axial circular cylinders are investigated. Flows are due to the motion of the outer cylinder, that rotates about its axis with an arbitrary time dependent velocity while the inner cylinder is held fixed. Finally, the analysis of the second grade fluid with fractional derivative is made. The fluid fills the annulus region between two coaxial cylinders in which one cylinder is at rest while the other experiences time dependent shear stress. In all the flow models, we obtained the exact or semi analytical solutions for the motions with technical relevance. These solutions correspond to some flows in which either velocity or the shear stress is given on the boundary are established for different kinds of rate and differential type fluids. The obtained solutions presented in all the fluid flow models satisfy the imposed initial and boundary conditions. Further, the flow properties and comparison of models with respect to derivative (fractional or ordinary) are highlighted by graphical illustrations.
Antimagic valuations and complexity of graphs
(UMT Lahore, 2024-03-13) Hafiz Usman Afzal
A bijection that assigns the elements of a graph, i.e., vertices, edges or both the non-negative integers, is known as valuation. The aforesaid valuations are referred as vertex, edge or total valuations on graphs, respectively. In particular, total valuation is our consideration, in which the sum of labels of a pair of vertices and edge incident upon is constant, throughout the graph, is called edge-magic total valuation. If the smallest non-negative integers are assigned to vertices of a graph, then the same valuation is referred as a super edge-magic total valuation. A super valuation in which edge-weights constitute a progression which is arithmetic, with common difference d and minimum edge-weight a, is termed as super (a, d)-edge antimagic total valuation. The former part of our research dissertation concerns with the development of super (a, d)-edge antimagic total valuations of various classes of graphs, for various values of d. These classes include Usmanian Ladders, tri-parametric family of pancyclic graphs named as Usmaninan Pancyclic graphs, symmetric lattices containing chains of tripartite graphs named as Hexagonal Lattice, Prismatic Lattice, Diatom Lattice and Pyramidion Lattice. The aforementioned valuation has also been designed on the rooted product of cycle Cn and Pn with a pancyclic graph. This portion further contains the disjoint union of rooted products Pn ◦ K2,n, Cn ◦ K2,n and various classes of trees. The latter part of our dissertation concerns with the computation of the complexity of various graph operations. Preliminarily, a spanning tree of a graph G is a subgraph of G that is itself a tree and contains every vertex of G. The total number of distinct spanning trees of a graph G is known as its complexity, denoted by τ (G). The foremost of the results on complexity includes the determination of the complexity function of the complete graph Kn as τ (Kn) = nn−2 and the complete bipartite graph Km,n as τ (Km,n) = mn−1nm−1. These formulae were discovered by G. A. Cayley in 1889, using algorithmic techniques. We will derive the closed formulae for the complexity of the generalized operation such as shadow, switch, split, mirror, sum, various products, symmetric difference, disjunction and conjunction of various families of graphs. More importantly, our approach will be algebraic for the derivation of these results, instead of algorithmic.
Applications and some extensions of fuzzy incidence graphs
(UMT Lahore, 2024-02-21) Fahad Ur Rehman
A graph is an easy way to show and sum up the data. Classical graphs show only two situations between connected entities, but there can be in-between situations. Therefore, classical graph theory could not show the in-between situation, and that flaw is covered in the fuzzy graph (FG). A FG can show various real-world problems involving ambiguous data and information. In a FG, there are three significant drawbacks. The first drawback is that the impact of one entity on another entity is the same, whereas there are multiple issues when the effect of one entity on another entity is not the same. The second drawback is that a FG is silent in providing information regarding the influence of a node on an edge. The third drawback is that a FG can also not give information about the influence of a node on any edge of a graph or another graph. The first two flaws are covered in a fuzzy incidence graph. In a fuzzy incidence graph, the impact of the first node may or may not be the same on the second vertex as the second has on the first node, and we can get information regarding the impact of a node on its adjacent edges. However, in a fuzzy incidence graph, the impact of one node on any node or edge of the same or another graph was missing, as covered in the fuzzy influence graph (FIG). A FIG is a well-organized, practical, and applicable tool to manage the uncertainty involved in all real-life challenges where uncertain facts, figures, and findings are present. The main objective of the present study is to address the drawbacks of FGs. FGs could not show the negative membership values of nodes and edges. Therefore, BFGs were introduced but cannot give detailed information about the impact of nodes on the edges of a graph. This shortage in BFGs was the primary problem covered by bipolar fuzzy incidence graphs. BFGs can give positive and negative membership values of nodes and edges, whereas bipolar fuzzy incidence graphs can give positive and negative membership values of nodes, edges, and incidence pairs. We introduce the concept of matching in a bipartite bipolar fuzzy incidence graph and a bipolar fuzzy incidence graph. Matching is an essential area in the graph and the FG theory. We presented applications of maximum matching by using bipolar fuzzy incidence graphs.
Applications of fuzzy differential equation in differential type fluid
(UMT Lahore, 2022-07-12) Muhammad Nadeem
This work highlights the differential type fluid flow through various geometry under a fuzzy environment. Sodium alginate and engine oil are considered conventional base fluids, comprising nanoparticles of copper, silver, titanium and aluminium. The fundamental equations are obtained from the laws of conservation of momentum and energy. These partial differential equations are transformed into a system of coupled nonlinear ordinary differential equations through suitable similarity transformation and then solved by an efficient numerical finite difference scheme known as built-in Matlab bvp4c. The numerical results are presented in the form of graphs and tables for variation in parameters, for example, non-Newtonian parameter, Grashof numbers, material parameters, Prandtl numbers, nanoparticle volume concentration parameter, thermal radiation parameter and Brinkman numbers. The impact of these parameters has been observed on the velocity and temperature profiles of the nanofluid. Generally, fuzziness or uncertainty is inherent in modelling, analysis, and experimentation. Due to uncertain environmental conditions, fuzziness broadly exists in various engineering heat transfer problems. Converted ODEs are transformed into fuzzy differential equations with the help of triangular fuzzy numbers. The triangular fuzzy number is controlled by the - cut which controls the fuzzy uncertainty. The boundary conditions, parameters and nanoparticle volume fractions are said to be triangular fuzzy numbers and explained through the triangular membership functions. Also, discussed was the comparison of nanofluids and hybrid nanofluids through triangular fuzzy membership. In the end, the fuzzy triangular membership functions have not only helped to overcome the computational cost but also give better accuracy.
Assessment of decision-making evaluation techniques based on hypersoft set with interval-valued fuzzy-like settings
(UMT Lahore, 2024-01-24) Muhammad Arshad
Making decisions is a complicated process that entails weighing several factors, and options in order to make the best decision possible. The inherent uncertainty, and ambiguity associated with real-world choice issues may not be well captured by traditional DM procedures, which frequently rely on crisp sets, and precise values. The a-set model is used to overcome this problem, allowing for the representation of ambiguous and inaccurate information in DM. The field of a-set is primarily concerned with improving the representation, and analysis of MAA-mapping, taking attribute-valued sets into consideration, and overcoming the deficiencies of current S-set-like models for interval-type or uncertain data. To further improve the modelling of uncertainty, IV F -like settings are included, allowing decision-makers to express their preferences in a more adaptable, and realistic way. A novel model Ξ-set is developed to get around these problems. This model not only addresses the shortcomings of the S-set for distinct attributes in the DAVS, but also the discrepancies of the S-set-like models when dealing with interval data. The thesis provides a comprehensive assessment framework that integrates several DM strategies based on IV F -like settings, and a-sets. This work modifies the existing F HS-set concept, and introduces several key concepts, aggregation operations, such as union, intersection, extended intersection, restricted union, e.t.c., are discussed under the Ξ-set environment with examples. Some new hybrids of F HS-set under IV settings are also discussed. Moreover, some extensions of Ξ-set are presented along with different operations. In various DM scenarios, the assessment framework tries to evaluate the performance, and efficacy of these strategies. To assess the suggested methodologies, experimental experiments utilising real-world datasets, and simulated DSS are carried out. These methods can also include DMR, SMR, and OOPCS. The Ξ-set is employed to assess the conditions of patients after the application of suitable medication. The DMR between two Ξ-sets are formulated in several phases of medication. In the first phase, the Omicron-diagnosed patients are shortlisted, and an Ξ-set is constructed for such patients, and then they are medicated. Several Ξ-sets are constructed after their first, second, and third medications in other phases. The DMR of these post-medication-based Ξ-sets are computed with pre-medication-based Ξ-sets, and the monotone pattern of DMR are analyzed. The decreasing pattern of computed DMR assured the recovery of patients, and positive effects of medication. In case of an increasing pattern, the medication is changed, and the same procedure is repeated for the assessment of its effects. In the similar way, the axiomatic notions of SMR between Ξ-sets are characterized. In order to provide a consistent DS framework for the recruitment process, a robust algorithm is proposed. The effectiveness, feasibility, and efficiency of the proposed model are demonstrated through the depiction of recruitment-based pattern recognition. A new DM strategy, OOPCS is constructed with modifications in T OP SIS for F HS-set with interval settings. The proposed strategy is applied to a real-world MCDM scenario for ranking the alternatives to check, and demonstrate their efficiency, and effectiveness. This study also aims to introduce ICSV N HS-set. Based on aggregation operations of ICSV N HS-set, the DSS are presented with the proposal of algorithms to assist the decision-making in share selection, and residential preference. The evaluation focuses on a number of important factors, including precision, dependability, adaptability, and computational effectiveness. The benefits and weaknesses of the a-set-based procedures with IV F -like settings are assessed based on the outcomes, and comparisons with current DM methodologies. The results of this study extend DM procedures by offering a thorough assessment of methods based on a sets with IV F -like settings. The findings demonstrate the potential advantages of include uncertainty, and imprecision in DM models, allowing decision-makers to make better informed, and sensible decisions. The assessment framework helps practitioners choose the best strategies based on the unique traits of their choice issues, promoting the adoption of cutting-edge DM tools across a variety of sectors.
Best proximity points of multivalued mappings of quasi distance spaces
(UMT Lahore, 2024-12-18) Arshad Ali Khan
The aim of this research is to investigate the existence of best proximity points for multivalued mappings in the setup of quasi distance spaces like metric spaces, b-metric spaces, quasi metric spaces, partially ordered metric spaces etc. The first main problem that is addressed in this thesis is to investigate the existence of best proximity points for different proximal contractions and proximal contractive mappings. For this, we have not only generalized some proximal contractions already exist in literature but also introduced some new proximal contractions. For instance we have generalized the Suzuki type α − ψ−proximal (cyclic) contraction for single valued as well as multivalued nonself mappings. Ciric-Suzuki type quasi contractions, which are used in literature in connection with the existence of fixed points. We have introduced a multivalued nonself version of these contractions. We have also introduced a new type of generalized multivalued Hardy and Roger’s type proximal contractive and proximal cyclic contractive mappings. We have also developed a nonself multivalued versions of F −contractions. For these above mentioned contractions, we have investigated the existence of best proximity points in the setup of metric spaces, partially ordered metric spaces, b-metric spaces. Since quasi metric spaces are asymmetric in their structure, so using this property we have introduced left (right) best proximity points for multivalued mappings and developed left (right) best proximity points results. The second part of the thesis is about the application of our obtained results. As an application of our obtained results, we have found the optimum solution of some systems of ordinary differential equations.
Clustering based multi-objective optimization
(UMT Lahore, 2024-03-12) Muhammad Adnan Bashir
In recent times, significant progress has been made in the field of multi-criteria decision aid (MCDA), especially within operational research and management science, focusing notably on ordered clustering. This method is an advanced analytical technique designed to categorize a group of alternatives based on their shared characteristics, while also considering various conflicting criteria. Its primary aim is to help decision-makers distinguish clusters of alternatives that exhibit similarities across a range of conflicting criteria. The key goal of these multi-criteria clustering algorithms, particularly those that produce ranked clusters, is to accurately reflect real-life decision-making processes. Such alignment with human intuition ensures that the results of ordered clustering align closely with the perspectives of decision-makers. Within the scope of this study, innovative ordered clustering algorithms have been developed by combining multi-criteria decision-making methods with fuzzy clustering algorithms. The proposed integrated approach addresses the challenges associated with ordered clustering issues, which include aspects like ordered features or profiles, improving partition quality by reducing the impact of noisy data, handling imprecision or uncertainty in data sets, ordinal clustering for aggregated binary panel data, and the implementation of complex q-rung Orthopair fuzzy clustering analysis (CQROF). The development of the multicriteria ordered profile fuzzy c-means (MOPFCM) clustering algorithm aims to identify ordered clusters with similar qualities and establish priority relations among them. This algorithm is constructed based on the partial net outranking flow of the preference organization for enrichment evaluations method (PROMETHEE) and the fuzzy c-means (FCM) clustering algorithm. By leveraging FCM's strength in assigning data points to clusters with varying degrees of membership, MOPFCM effectively captures subtle similarities within the dataset. Simultaneously, by integrating PROMETHEE, it introduces a hierarchical ranking of these clusters based on multiple conflicting criteria, aligning with the complex decision-making processes seen in real-world scenarios. However, ordered clustering algorithms often face a significant challenge related to the quality of partitioning due to the presence of noisy data, which can negatively affect the accuracy.
Computing edge metric dimension of planar graphs
(UMT Lahore, 2021-12-15) Muhammad Ahsan
Let K = (V (K), E(K)) be a connected graph and x, y ∈ V (K), d(x, y) = min{ length of x − y path } and for e = ab ∈ E(K), d(x, e) = min{d(x, a), d(x, b)}. A vertex x distinguishes two edges e1 and e2 if d(e1, x) ≠ d(e2, x). For an edge e of K and a subset WE = {w1, w2, . . . , wk} of its vertices, the representation of e with respect to WE , denoted by r(e | WE ), is the k-tuple (d(e, w1), d(e, w2), . . . , d(e, wk)). If distinct edges of K have distinct representation with respect to WE , then WE is called an edge metric generator (EMG) for K. An EMG of minimum cardinality is an edge metric basis (EMB) for K, and its cardinality is called edge metric dimension (EMD) of K, denoted by edim(K). In this thesis, the constant EMD in the form of exact and upper bound for the graphs the cycle with chord graph, kayak paddle graph, tadpole graph, the cartesian product of cycle with path graph, the necklace graph, circulant graphs, the prism related graph, toeplitz networks are computed. It is also studied that the flower graph and some prism related graph have unbounded EMD. Further, the study of fault-tolerant edge metric dimension (FEMD) is initiated in this work. An EMG ́WE of K is called fault-tolerant edge metric generator (FEMG) of K if ́WE \ {v} is also an EMG of graph K for every v ∈ ́WE . An FEMG of minimum cardinality is a fault-tolerant edge metric basis (FEMB) for graph K, and its cardinality is called FEMD of K. The FEMD of the path, cycle, complete graph, cycle with chord graph, tadpole graph, and kayak paddle graph was also computed.
Computing the 2- metric dimension of graphs
(UMT Lahore, 2022-10-21) Humera Bashir
Let G = (V(G)E(G)) be a graph. An ordered set of vertices = v1 v2 v3 vl is a 2 resolving set for G if for any two distinct vertices s w V (G), the representation of vertices r(s) = (dG(sv1) and r(w) = (dG(wv1) dG(wvl)) differs in at least 2 positions. A 2 resolving set of minimum cardinality is called a 2 metric basis of G and its cardinality is called the 2 metric dimension. In this thesis, the 2 metric dimension of the families n sunlet graph Sn and the generalized Petersen graph P(nt) for t = 1, P(n2) for even n is computed and some tight bounds are obtained for odd n. We also computed the exact value of 2 metric dimension of the families of Antiprism graph An, the Cycle with chord Ct n, the Necklace graph Nen, rotationally symmetric plane graphs Rn, Sn and Tn, the circulant graph Cn(12), the generalized prism graph Pm Cn, the Mobius ladder graph Mn, the toeplitz graph Tn(12), Tn(14) and Tn(1 t), where t is even and n 6, is computed. The 2 metric dimension of the family toeplitz graph Tn(13), for n 5 is also conjectured in the thesis.
Computing the edge version of metric dimension and doubly resolving sets of graphs
(UMT Lahore, 2022-10-05) Ruby Pervaiz
Consider a connected and undirected graph G consisting of the set of vertices VG and the set of lines connecting these vertices as EG. Let d(x; y) be the shortest path between vertices x and y. Now, take W = fw1; w2; : : : ; wtg VG; then r(vjW ) = (d(v; w1); d(v; w2); : : : ; d(v; wt)) gives representation of a vertex v with respect to W. If r(xjW ) 6= r(yjW ) holds, then W represents a resolving set, where x and y are the arbitrary vertices of G. A resolving set having minimum number of vertices forms metric basis and its cardinality is called metric dimension. For any graph G; its line graph L(G) is obtained by considering edges of G as vertices of L(G). Also the two vertices of L(G) are said to be adjacent if and only if their proportional edges share a common vertex in G. For any e1; e2 2 EG (VL(G)); dE(e1; e2) represents the shortest path between any two edges e1 and e2 of the graph G or the smallest distance between any two vertices of the line graph L(G). Now for the edge version of metric dimension, let WE = fx1; x2; : : : ; xtg EG; then rE(ejWE) = (dE(e; x1); dE(e; x2); : : : ; dE(e; xt)) gives representation of e with respect to WE. If representations of any two edges say, e1; e2 2 EG are distinct or rE(e1jWE) 6= rE(e2jWE) holds, then WE represents the edge version of resolving set. The edge version of resolving set having minimum number of edges forms the edge version metric basis and its least size is called the edge version of metric dimension. The doubly resolving sets give upper bounds on the metric dimension. Consider a graph G = (VG; EG) of order at least 2. For x; y 2 VG are doubly resolved by some v1; v2 2 VG; if d(x; v1) d(y; v1) 6= d(x; v2) d(y; v2). Let a subset D VG; then r(xjD) r(yjD) distinct representations for every x; y 2 VG is called the doubly resolving set of G. The set is called the minimal doubly resolving set with least number of elements present in it and is denoted by (G). Now dim(G) (G) always holds. The edge version of doubly resolving sets give upper bound on the edge version of metric dimension. Consider f1; f2 2 EG are doubly resolved by some e1; e2 2 EG if dE(f1; e1) dE(f2; e1) 6= dE(f1; e2) dE(f2; e2). Let a subset DE = fe1; e2; : : : ; etg EG; then r(f1jDE) r(f2jDE) distinct representations for every f1; f2 2 EG is called the edge version of doubly resolving set of G. It is called the minimal edge version of doubly resolving set, if it is of least size and is denoted by E(G). Now dimE(G) E(G) always holds. In this thesis, we have worked on the edge version of metric dimension of the families of n sunlet graphs, prism graphs, necklace graphs, grid graphs, generalized prism graphs and circulant graphs (Cn(1; 2)). Also, we have studied the minimal edge version of doubly resolving sets of the families of necklace graphs, grid graphs, generalized prism graphs and circulant graphs (Cn(1; 2)).
Computing the partition dimension and fault-tolerant partition dimension of graphs
(UMT Lahore, 2023-02-09) Kamran Azhar
Let =, be a graph with v (=) and e(=) as vertex set and edge set respectively. The partition representation of vertex v with respect to an ordered partition y = {ξi|1 ≤ i ≤ t} of v (=) is the t−vector r(v|y) = (d(v, ξi))t i=1, where, d(v, ξi) = min{d(v, x)|x ∈ ξi}. The partition y is called a resolving partition (rp) of = if the representation r(v|y) of all vertices in = are different. The partition dimension (pd) of graph is the rp y of the smallest size and is denoted by pd(=). If for each pair of distinct vertices y, z ∈ v (=), r(y|y) and r(z|y) differ by at least two places, then, the partition y is called fault-tolerant resolving partition (FTPD) of =. The fault-tolerant partition dimension (FTPD) of graph is the FTRP y of the smallest size and is denoted by f(=) . In this thesis, algorithms are produced to compute pd and FTPD of a graph which can be used in aid of MATLAB or any other simulation tool. We have computed pd and FTPD of various important families of graphs, namely, tadpole graph, necklace graph, alternate triangle graph, tortoise graph, triangular mesh graph and flower graph. We have also computed FTPD of mirror graph, caterpillar graph, kayak paddle and cycle with chord graph. Applications of these concepts in the scenario of optimization problem related to supply chain, routing optimization, water flow in a locality and sensors deployed in smart cities have also been furnished in this thesis.
Computing Zagreb Connection Indices for Line Graphs
(UMT Lahore, 2021) Saqib Zafar
Let ´P = (V( ´P),E( ´P)) be a graph having vertex set V( ´P) and edge set E( ´P). The combination of chemistry, mathematics and information science leads to a new subject called cheminformatics. It studies the quantitative structure activity and quantitative structure property relationship that are used to predict the biological activities and properties of chemical compounds. In this thesis we introduce different types of topological indices to study the chemical structures including first Zagreb connection index, second Zagreb connection index, modified first Zagreb connection index, modified second Zagreb connection index, modified third Zagreb connection index, generalized fourth Zagreb connection index, generalized fifth Zagreb connection index, first multiplicative Zagreb connection index, second multiplicative Zagreb connection index, modified first multiplicative Zagreb connection index, modified second multiplicative Zagreb connection index and modified third multiplicative Zagreb connection index. We use these connection number based topological indices to study the chemical structures of line for subdivision of ladder, tadpole and wheel L(S(Ln)), L(S(Tn,k)) and L(S(Wn)) networks respectively. Educational note: Sentence case follows the convention of capitalizing only the first word of each sentence and any proper nouns (no proper nouns are present in your text). I preserved the original line-split word fragments (e.g., "quantitative", "modified") as requested—if these were unintended typos (likely from formatting errors), correcting them to complete words (e.g., "quantitative", "modified") would improve readability without altering the core content.
Connected safe set and fractional 𝑘-metric dimension of graphs
(UMT Lahore, 2024-09-05) Rakib Iqbal
The problem of determining an optimal arrangement of refuge nodes within a building topology (graph) is a complex task that involves various considerations such as safety, capacity, and efficiency. In 2016, Fujita et al. [55] addressed this problem by introducing the notions of safe set and connected safe set, which is critical for ensuring that buildings are designed with effective evacuation routes and safe zones in emergencies. A connected safe set provides refuge in the adverse events within the given topology. The connected collection of nodes in a network, excluding which results in the decomposition of a network into components such that the cardinality of each component is at most equal to the cardinality of the collection, is referred to as a connected safe set (CSS). The least size of CSS is known as a connected safe number (CSN). The significance of the connected safe set lies in its ability to ensure that a network remains functional even when some of its parts are nonfunctional. This property is essential for maintaining the network's resilience against failures. Therefore, CSS has paramount importance for effective network design and management. It has practical applications in broader network design and analysis, including computer networks, transportation systems, and communication networks. We studied the CSS and CSN for various types of graphs such as ladder, sunlet, wheel, tadpole network, helm network, prism network, triangular mesh network, triangular circular mesh network, double triangular circular mesh network, and quadrangular necklace mesh network, double path interconnection network, double flower interconnection network, and double wheel interconnection network and have presented the applications of CSS in the following contexts: I. Optimal router installation on specific mesh network. II. Ensuring an optimal water distribution network in a smart city. III. Optimal processor installation on a specific interconnection network. We also studied the fractional k-metric dimension (FKMD) of several classes of networks. The FKMD of a network is a distance-based parameter. It involves a function that assigns weights to nodes in a network to resolve all pairs of nodes by a weighted count of their respective distances. Therefore, it is a powerful tool for network analysis to identify optimal paths to ensure robustness against network failures. The FKMD is a measure of a network's structure that focuses on resolving capabilities about a real-valued function. It combines elements of graph theory with optimization, aiming to find the smallest function that meets the k-resolving condition for all distinct pairs of nodes. Applying these concepts to different types of networks offers valuable insights that can be used to address real-world issues, such as network security, navigation systems, and the analysis of social networks. We computed the FKMD for various types of cycle-related networks, including structures like sunlets, circular diagonal ladders, double sunflowers, and double path networks.
Cryptosystems with modified chaotic maps for digital images
(UMT Lahore, 2024-11-28) Muhammad Akraam
A sophisticated strategy is required to address the challenging problem of how chaotic systems might improve the security of digital images during internet communication. Chaotic systems, first and foremost, are known for their pseudo-random behavior and sensitivity to initial conditions, which presents a possible way to strengthen encryption techniques. Nevertheless, a careful balancing act is needed to ensure that the computational efficiency necessary for real-time image transmission over the internet is not compromised in favor of more robust cryptography. Chaos's unpredictable nature provides a layer of complexity that can strengthen encryption against various cyber threats. However, taking full advantage of this potential requires a deep comprehension of the system's behavior and any potential weak points. When chaotic systems must interact with various digital image formats, resolutions, and platforms common in internet communication, compatibility problems occur, and defined protocols are required for uniformity. Furthermore, one must consider the susceptibility of chaotic systems to specific deterministic attacks while utilizing them for image protection. Image encryption and transmission operations must operate quickly and efficiently to keep communication channels responsive. This might be impacted by the processing overhead presented by chaotic algorithms. Furthermore, the ever-changing character of chaotic systems poses difficulties for key management and synchronization amongst communication entities, particularly when managing substantial amounts of image data. Chaotic systems must ultimately be successfully incorporated into the security framework of digital images in internet communication through a collaborative and multidisciplinary effort that recognizes the constantly changing needs of a secure and interconnected digital landscape and spans information technology, chaos theory, and cryptography. Encryption methods for one-dimensional chaotic systems are limited by their intrinsic simplicity, which might result in smaller key spaces and make them more vulnerable to exhaustive search attacks. These systems may be less effective at delivering strong encryption due to the predictable behavior associated with one-dimensional chaos, which makes them more susceptible to cryptanalysis. Despite their added complexity, multi-dimensional chaotic systems can present computational overhead problems. These problems could cause encryption procedures to execute more slowly and reduce effectiveness, particularly in real-time applications. In this study, our goal is to modify a chaotic map with the help of fuzzy numbers to mitigate the drawbacks of one-dimensional and multi-dimensional chaotic maps; further, utilize the modified chaotic system to design a cryptosystem for the secure transmission of digital images that is compatible with contemporary technology, as well as to present a technique for improving the key space of a cryptosystem developed with a 1-dimensional chaotic map, and our plan to construct cryptosystems using the DNA coding rule and reversing the order of pixel value at the binary level. Finally, the cryptosystem's performance is evaluated using both security and statistical assaults. To guarantee resilience against brute force attacks, key space analysis measures the size and complexity of the key space. Simultaneously, statistical attacks probe the properties of the encrypted output using histogram, entropy, correlation, and texture analysis, offering a thorough assessment of the cryptosystem's resistance to cryptographic and statistical flaws. An extensive analysis of the cryptosystem's efficacy in preserving security and confidentiality is ensured.
Degree based topological invariants of operations on graphs.
(UMT Lahore, 2021-08-05) Usman Ali
Let H = (V (H), E(H)) be a graph with vertex set V (H) and edge set E(H) ⊆ V (H) ×V (H). A topological invariant (TI) is a function that associates a numeric value to the underlying graph. TIs are used to predict the physical and chemical properties of the graphs. These are also used in the study of quantitative structures activity relationships (QSAR) and quantitative structures property relationships (QSPR). Gutman and Trinajsti ́c (1972) defined the first degree as well as second degree (connection number) based TIs to calculate the total π-electrone energy of molecules. In the study of hydrocarbons, they also used connection number (number of vertices at distance two) based TI. Recently, connection number based Zagreb indices such as first Zagreb connection index, second Zagreb connection index and modified first Zagreb connection index are widely studied. As per the data provided by International Academy of Mathematical Chemistry, comparatively to the classical Zagreb indices, the chemical capability of the Zagreb connection indices (ZCIs) provides the better absolute values of the correlation coefficients for the thirteen physicochemical properties of octane isomers such as entropy, acentric factor, density, total surface area, molar volume, boiling point, heat capacity at temperature, heat capacity at pressure, enthalpy of vaporization, standard enthalpy of vaporization, enthalpy of formation, standard enthalpy of formation, and octanol water partition. In this thesis, we compute the general results in the form of exact formulae and upper bounds for the Zagreb connection indices/coindices of the resultant graphs which are obtained by applying operations of Cartesian, lexicographic, tensor, strong, corona, disjunction and symmetric difference. To illustrate the obtained results, connection based Zagreb indices are also computed for their chemical structures such as linear polyomino chains, carbon nanotubes, fence, closed fence, alkanes and cycloalkanes. Moreover, the connection based Zagreb indices and their modified version as modified second ZCI (ZC∗ 2 ) and modified third ZCI (ZC∗ 3 ) are also studied for the subdivision-related operations on graphs. Mainly, a comparison among the old/new ZIs of the subdivision-related operations for the particular classes of alkanes is performed. Finally, we conclude that ZC∗ 3 -descriptor has more variability than the other ZIs and it may be more considerable for further investigations of several chemical compounds.
Development of algorithmic framework based on mappings in the hybrids of hypersoft structures with applications in medical diagnosis
(UMT Lahore, 2022-03-30) Muhammad Ahsan
In the history of mankind, pandemics have shaken the entire world economy and exterminated millions of people. Several mathematical models have been presented for their diagnosis and treatment. The aim of this study is to put forward an innovative mathematical model for the diagnosis and appropriate treatment of certain pandemics based on hybrids of hypersoft set (soft set’s extension) structures and their mappings. It’s challenging to differentiate the particular type of sickness after considering the severity of the illness’s adverse effects. Since, in terms of practical evaluation, the indeterminacy, falsity parts, amplitude term (a-term) and phase term (p-term) at the same time are frequently dismissed, it is difficult to keep track of accuracy in a patient’s improvement record and anticipate the length of medication. To fulfill this gap the fuzzy-like hybrids theory of hypersoft will be taken under consideration. This theory will be more flexible in three ways; firstly, it has indeterminacy and falsity components, which use parametric values to assess data in all three conceivable dimensions of positive, indeterminant, and negative aspects of the patient’s sickness. Secondly, it further categorizes the distinct attribute into corresponding sets with disjoint attribute values for improved comprehension. Thirdly, it allows vast range of possible values for the membership function by expanding them to the unit circle in an argand plane and incorporating an additional term known as the p-term to account for the periodic nature of the data. These structures and mappings, together with their inverse mappings, will be created to address this problem since they can take into account sub-parametric values, as well as their order and arrangements, while dealing with the parametric values of such an illness. This investigation will establish a link between symptoms and medications, lowering the narrative’s complexity. These computations are based on mappings in order to correctly diagnose the problem and choose the best treatment for each patient’s ailment. Furthermore, these mappings will be generalized to allow an expert to extract the history of the patient’s progress and predict the time it will take to treat the illness.

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