PhD

Permanent URI for this collectionhttps://repository.cuilahore.edu.pk/handle/123456789/50

This collection archives the complete set of theses produced by students of the COMSATS University Islamabad, Lahore Campus.

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H-Magic Behaviors of Some Graphs
(Library Information Services, COMSATS University, Lahore Campus, 2015) Imran Zulfiqar Cheema; , FA11-PMATH-001; Dr. Muhammad Hussain,; LHR TP 5297
A graph ( , ) G V E has an H-covering if every edge in E belongs to a subgraph of G isomorphic to H . SupposeG admits an H-covering. AnH-magic labeling is a mapping  from ( ) ( ) E G VG  onto the integers {1,2,...,| ( ) E G VG  ( )|} with the property that, for every subgraph Aof G isomorphic toH , there is a positive integer c such that  A v ( )        v V A ( ) ( ) . e c   e E A ( ) A graph which possess such type of labeling is known as H-magic graph. Further if in a graph vertices are labeled first with smallest positive numbers, then the graph is called H-supermagic. Moreover a graph is said to be H-( , ) ad-anti magic if the magic constant for an arithmetic progression with initial value a and a common difference . d Numerous results on labeling of many families of graphs have been published. In this thesis, research work focuses on to formulate cycle 3 C- ( , ) ad anti-supermagic labeling for the Multi Wheels graph, supermagic labeling for isomorphic copies with its disjoint union of Multi-Wheels graph and cycle ( , ) ad-anti-supermagic labeling for Web graph. Also cycle anti-supermagic labeling for isomorphic copies with its disjoint union of ladder and triangular ladder graphs have been formulated. In addition, investigation of fan, friendship, ladder and wheel line graphs and study of the supermagic and anti-supermagic vertex-edge-face labeling of such graphs and their isomorphic copies have been carried in this thesis. An anti-supermagic labeling of the extension of cycle graphs is also formulated. Lastly the face supermagic labeling of (1,1,1) type of subdivided triangular ladder graph, subdivided mC 4-snake graph and subdivided kmC 4-triangular snake graph with its (1,1,...,1) and (2,2,...,2) string are also the part of this thesis.
Dynamical Properties of Specific Black Holes
(Library Information Services, COMSATS University, Lahore Campus, 2015) Muhammad Umair Shahzad,; FA13-PMATH-005; Dr. Abdul Jawad, Assistant Profesor; LHR TP 5606
Dynamical Properties of Specific Black Holes The present thesis comprises the study of three dynamical phenomenon such as thermal fluctuations, accretion and tidal forces of black holes/regular black holes. We consider the logarithmic corrected entropy in order to analyze the thermal fluctuations. We examine the effects of thermal fluctuations on a regular black hole of the non-minimal Einstein-Yang-Mill theory with gauge field of magnetic Wu-Yang type and a cosmological constant. We investigate the first law of thermodynamics in the presence of logarithmic corrected entropy and non-minimal regular black hole. Furthermore, we discuss the thermal fluctuation problem by utilizing the higher order corrected entropy. We examine the thermodynamical behavior of two well-known black holes such as Reissner-Nordström Anti de Sitter black hole with global monopole and f(R) black hole in the presence of higher order corrected entropy. We also discuss the accretion problem in two phases. In first phase, we analyze the accretion onto static spherically symmetric regular black holes for specific choices of the equation of state parameter. The underlying regular black holes are charged regular black holes using the Fermi-Dirac distribution, logistic distribution, non-linear electrodynamics, respectively, and Kehagias-Sftesos asymptotically flat regular black holes. In second phase, we develop the Hamiltonian dynamical system to tackle the accretion problem. We investigate the accretion of test fluids onto regular black holes such as Kehagias-Sftesos black hole and regular black holes with Dagum distribution function. We analyze the accretion process when different test fluids are falling onto these regular black holes. The behavior of fluid flow and the existence of sonic points is being checked for these regular black holes. Finally, we investigate the tidal forces occurring in a Kiselev black hole surrounded by radiation and dust fluids. We also solve the geodesic deviation equation for radially free-falling bodies toward Kiselev black hole. We explain the geodesic deviation vector graphically and point out the location of the event and Cauchy horizons for specific values of the radiation and dust parameters.
FACE ANTI-MAGIC LABELINGS AND DEFICIENCY OF GRAPHS
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2016) Ali Tabraiz; FA12-PMATH-001; LHR TP 4824; Dr. Muhammad Hussain
Labeling of graph is a beautiful and rich technique of Mathematics. It has extensive usage in applications such as data transmission, x-ray, circuit design, astronomy, communication links and coding theory. A labeling of a graph is a mapping that carries some set of graph elements into (usually positive integers). An ) , ( d a face anti-magic total labeling of a graph, with p vertices, q edges and f faces, is a one-to-one mapping that takes the vertices, edges and faces into }, , , 2 , 1 { f q p such that the sums of the label on the edges, vertices and faces form an arithmetic progression starting at a and having difference . d An edge magic total labeling of a graph G is a one-to-one mapping from ) ( ) ( G E G V onto the integers }, , , 2 , 1 { q p with the property that there is an integer constant k such that k y xy x ) ( ) ( ) ( for any ). (G E y x Furthermore, the edge-magic deficiency of a graph , G ) (G is defined as the minimum non-negative integer n such that 1 K n G is edge-magic.
Valuations and Metric Dimensions on Graphs
(Library Information Services, COMSATS University, Lahore Campus, 2016) Shahid Imran,; LHR TP 5605; FA12-PmAth-003; Dr. Muhammad Hussain, Assistant Profesor
Labeling of graph is a beautiful and rich technique of Mathematics. It has extensive usage in applications such as data transmission, x-ray, circuit design, astronomy, communication links and coding theory. Labeling of graph is defined as a correspondence from the set } , ,3,2,1{ f e v    to the vertices, edges and faces of a plane graph G in such a way that each vertex, edge and face receives exactly one label and each number is used exactly once as a label. The weight of a face under a labeling of type )1,1,1( is the sum of labels carried by that face and all the edges and vertices surrounding it. A labeling of type )1,1,1( of a plane graph G is called d-antimagic if for every positive integer s the set of weights of all s-sided faces is } )1 ( , , 2 , , { d f a d ad aa      for some integer a and 0d , where f is the number of s-sided faces. In particular when 0d then the d antimagic labeling of type )1,1,1( is called magic labeling of type )1,1,1( . Moreover, if the vertices are labeled by the smallest possible number. It is known as super magic and anti magic accordingly. Furthermore, let G be a connected graph and ) , ( y x d be the distance between the vertices x and y in G. A set of vertices W resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in W. A metric dimension of G is the minimum cardinality of a resolving set of G and is denoted by ). ( dimG This dissertation studies about the construction of super face anti-magic labeling of type )1,1,1( for disjoint union of toroidal fullerenes, magic and anti-magic labeling of type )1,1,1( for union of generalized friendship graphs. Metric dimension of Jahangir related graphs, generalized prism related graphs and graphs obtained by rooted product has been also calculated.
Degree and Distance Based Parameters of Certain Graph Structures
(Library Information Services, COMSATS University, Lahore Campus, 2016) Zeshan Saleem Mufti; , FA12-PMATH-006; Dr. M. Fasial Nadeem, Assistant Profesor
Degree and Distance Based Parameters of Certain Graph Structures Topological indices are a rapidly developing area of research in graph theory. There are various topological indices such as degree-based topological indices, spectrum based topological indices and distance related topological indices. These topological indices correlate certain physico-chemical properties such as boiling point, stability of chemical compounds and many more. In this dissertation, we focus on degree based and spectrum based topological indices. Under the degree based topological indices, we study topological indices such Randic index, general Randic index, Zagreb index, general sum connectivity index, first and second multiple Zagreb indices, geometric index, atom bond connectivity index for para-line graphs of some chemical structures such as Phenylene, Anthracene, Benzenoid, Pentacene. Similarly, we study the spectrum based topological indices such as Estrada index for certain chemical graph structures such as Phenylene, Anthracene, Bi-smuth Tri-iodide, benzene ring embedded in P-type-surface in 2D network (BRE). Also we find the energy of above mentioned structures. We also discuss the notion of distance related parameters such as metric dimension and edge metric dimension. We compute the edge metric dimension for the barcycentric Subdivision of Cayley Graphs. Also we have studied the M-polynomial and entropy of generalized Sierpinski graphs.
Dynamical Properties of Specific Black Holes
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2017) Muhammad Umair Shahzad; FA13-PMATH-005; LHR TP 5606; Dr. Abdul Jawad
The present thesis comprises the study of three dynamical phenomenon such as thermal fluctuations, accretion and tidal forces of black holes/regular black holes. We consider the logarithmic corrected entropy in order to analyze the thermal fluctuations. We examine the effects of thermal fluctuations on a regular black hole of the non-minimal Einstein-Yang-Mill theory with gauge field of magnetic Wu-Yang type and a cosmological constant. We investigate the first law of thermodynamics in the presence of logarithmic corrected entropy and non-minimal regular black hole. Furthermore, we discuss the thermal fluctuation problem by utilizing the higher order corrected entropy. We examine the thermodynamical behavior of two well-known black holes such as Reissner-Nordström Anti de Sitter black hole with global monopole and f(R) black hole in the presence of higher order corrected entropy. We also discuss the accretion problem in two phases. In first phase, we analyze the accretion onto static spherically symmetric regular black holes for specific choices of the equation of state parameter. The underlying regular black holes are charged regular black holes using the Fermi-Dirac distribution, logistic distribution, non-linear electrodynamics, respectively, and Kehagias-Sftesos asymptotically flat regular black holes. In second phase, we develop the Hamiltonian dynamical system to tackle the accretion problem. We investigate the accretion of test fluids onto regular black holes such as Kehagias-Sftesos black hole and regular black holes with Dagum distribution function. We analyze the accretion process when different test fluids are falling onto these regular black holes. The behavior of fluid flow and the existence of sonic points is being checked for these regular black holes. Finally, we investigate the tidal forces occurring in a Kiselev black hole surrounded by radiation and dust fluids. We also solve the geodesic deviation equation for radially free-falling bodies toward Kiselev black hole. We explain the geodesic deviation vector graphically and point out the location of the event and Cauchy horizons for specific values of the radiation and dust parameters.
Corporate Derivatives as a Tool for Risk Management: A Comparative Analysis of Pakistan and Malaysia
(Library Information Services COMSATS University Islamaabad Lahore Campus, 2017) Atia Alam; FA10-PMS-003; Dr. Talat Afza; LHR TP 5290
The growing usage of derivatives since the last few decades has increased academic concerns regarding the value relevance of derivatives. Hedging theorists state that considering market imperfections, use of derivatives enhances firm value by minimizing risk. The existing literature has mostly examined the direct effect of derivative usage on firm risk and value, but empirical evidence on ‘how’ and ‘when’ use of derivatives enhances firm value is still missing. Present study primarily adds to the existing literature by identifying ‘how’ and ‘when’ derivative usage increases firm value by using moderated mediation techniques through a bootstrap technique of Hayes (2015). In addition, the current study explores the empirical literature by comparatively analyzing the countries where derivative markets are still in development phase like Pakistan and Malaysia. The sample consists of 167 Pakistani and 266 Malaysian firms for the period of 2004-2010. Findings demonstrate that derivatives play a significant risk management role in both Pakistan and Malaysia, though the effectiveness of the moderating role of derivative usage on the relationship between risk and value is limited in Pakistani non-financial firms. Results on the basis of firm’s types of derivative instruments show no moderating mediating role of a firm’s risk on the relationship of derivative usage and Malaysian firm’s value as they are mostly involved in operational hedging. Whereas, Pakistani firms have high value enhancing benefits of foreign currency derivative usage in case of no ER exposure as firms might have exposed due to imports and other commodity prices. In case of interest rate derivative usage, Malaysian firms use fewer interest rate derivatives in times of low IR exposure as they have more access to external financing at lower costs because of developed capital market.
Topological In Variants Of Graphs
(Library Information Services, COMSATS University, Lahore Campus, 2017) Imran Nadeem,; FA13-PMATH-003; Dr. Hani Shaker, Assistant Profesor [Supervisor]
Chemical graph theory provides useful tools such as molecular descriptors to develop strong intrinsic relationship between the physicochemical features of chemical com pounds and their molecular graphs. The study of molecular descriptors provides a the oretical basis for the fabrication of chemical materials and is helpful in making up for the lack of chemical experiments. There are two prominent types of molecular descrip tors; the topological indices and counting polynomials. Further, topological indices can be categorized in two major classes: one class is based on degree and the other class is based on distance. In this thesis, we present the study of certain topological indices belonging to degree- and distance-based classes and counting polynomials for some well-known nanostructures. We also present a comparative study between dif ferent topological indices belonging to degree- and distance-based classes for general graphs. In addition, we study the para-line transformation of graphs and obtain the gen eral expressions of certain topological indices for this transformation. We achieve the lower and upper bounds of certain distance-based topological indices for the para-line transformation of graphs
New Measures of Intuitionistic Inclusion and Similarity with Applications
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2018) Madiha Qayyum; SP13-PMATH-002; Dr. Samina Mazhar; LHR TP 4873
Similarity and inclusion are two most important conceptual ways for looking at any possible relationships between objects (sets). A similarity measure is used for estimating the degree of resemblance/similarity between two objects while an inclusion measure expresses the degree to which one of the two is covered by the other one. In this dissertation, we have presented two different yet equally important approaches of defining similarity and inclusion measures for intuitionistic fuzzy sets. The first approach can be regarded as a logical approach implanting intuitionistic fuzzy implication, bi-implications and t-norms while the second one is a purely set theoretic approach having cardinalities and sets operations involved in its construction. In the logical approach, the degree of inclusion/ similarity was obtained by composing the intuitionistic logical operators (implications and bi-implications) with fuzzy measures on intuitionistic fuzzy sets. For this purpose we initially defined some new normal fuzzy measures on intuitionistic fuzzy sets along with a class of scalar cardinality measure for intuitionistic fuzzy sets. Later, we introduced different classes of intuitionistic fuzzy bi-implication operators having axiomatic as well as constructive approaches. A study on the properties of these bi-implication operators by utilizing Lukasiewicz intuitionistic fuzzy implicator revealed some remarkable results. The intuitionistic fuzzy bi-implication operators along with new defined fuzzy measures gave rise to multiple classes of intuitionistic fuzzy bi-implicator based similarity measures for intuitionistic fuzzy sets. Also the same normal fuzzy measures were employed to obtain the degree of inclusion between two intuitionistic fuzzy sets when composed with intuitionistic fuzzy implication operators. The new implication based classes of inclusion and similarity measures fulfilled almost all of the universally accepted criteria’s. In the set theoretic approach, we employed one of the members of new introduced scalar cardinality of intuitionistic fuzzy sets to construct a four parametric family of cardinality based similarity and inclusion measures. Both of these parametric families, xi for different combinations of parameters, generated intuitionistic fuzzy versions of some of the famous crisp measures of time. Lastly, the utility of the new measures (similarity, inclusion, cardinality and others) of intuitionistic fuzzy sets introduced in this work is exhibited by utilizing them as a part of solution techniques to the problems arising in real life situations.
Words for Maximal Subgroups in Certain Sporadic Simple Groups
(Library Information Services, COMSATS University, Lahore Campus, 2018) Faisal Yasin,; FA13-PMATH-002; Dr. Adeel Farooq,; LHR TP 5946
In this thesis we found generators for the Maximal Subgroups (MSs) of the Sporadic Sim ple Groups (SSGs) as words in the generators of the parent group. The idea of standard generators was presented by Wilson. He began a project known as online form of Atlas, ”http://brauer.maths.qmul.ac.uk/Atlas” in which he provides the representation and words for MSs of the SSGs. However there are some cases, which have to be dealed. We pursue the work initiated by R. A. Wilson of finding words for the MSs of SSG. Thus the only cases on the list of World-wide-web Atlas which need to be solved are Co1, HN, Fi23, Fi24, B and M. There are 14 MSs of Harada-Norton group HN. Generators of 11 MSs are already known. In this thesis we have provided generators of remaining 3 MSs. Similarly, we have provided words for four MSs of Fi23, words for sixteen MSs ofCo1, words for 11 MSsofFi24 and words for MSs of some linear groups. In World-Wide-Web Atlas of group representation there is only one copy of S5 of the MSs of L3(5), but we have found words for two non-conjugate copies of S5 in Table 15 of second chapter
Space Spectral Time Fractional Finite Difference Method along with Stability Analysis for Fractional Order Nonlinear Wave Equations
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2018) Muhammad Sarmad Arshad; SP13-PMATH-004; Dr. Ayesha Sohail; LHR TP 5299
In this work, nonlinear partial differential equations governing the obscure phenomena of shallow water waves are discussed. Time fractional model is considered to understand the upcoming solutions on the basis of all historical states of the solution. A semi-analytic technique, Homotopy Perturbation Transform Method (HPTM) is used in conjunction with a numerical technique to validate the approximate solutions. With the aid of graphical interpretation, the favorable wave parameters, to avoid wave breaking are estimated. Afterwards, dynamical analysis of fractional order Schr dinger equation governing the optical wave propagation is reported in detail. The validity criteria for the application of the semi-analytic asymptotic methods are exploited. Comparison between the solutions obtained by the two asymptotic techniques, that is, the Fractional Homotopy Analysis Transform Method and the Optimal Homotopy Analysis Method is performed to select the most accurate technique for the stated problem. Space spectral analysis with integrating factor technique and time fraction finite difference method have been implemented to study the pressure waves propagating in bubbly fluids as well as nonlinear phenomena of plasma waves. Dynamical analysis of acoustic/pressure waves propagating in bubbly fluids is of great significance. Such flows arise in many engineering problems including sonochemistry, sonochemical reactors, cavitation around hydrofoils and ultrasonic propagation in medicine and biology. Fractional approach for modeling the propagation of the pressure waves in liquids containing a large number of tiny gas bubbles is proposed. Moreover, numerical solution of the fractional order Modified Korteweg-de Vries equation governing the dynamics is approximated using a novel space spectral time fractional finite difference tool. A spectral technique for space and a multi-step finite difference scheme for time are designed and implemented. The spatial spectral discretization error and the stability bounds are discussed. The nonlinear phenomena of plasma waves are well demonstrated with the aid of graphical analysis. Stability analysis of integer and fractional order KdV equations have been discussed quantitatively with the help of Evans function approximation.
Face Anti-magicLabelingsand Deficiency of Graphs
(COMSATS University Islamabad, Lahore Campus Library Information Services, CUI Lahore, 2018) Ali Tabraiz,; FA14-PMATH-001; Dr. Muhammad Hussain, Assistant Profesor [Supervisor]; LHR TP 5296
Face anti-magic labeling is a significant concept in graph theory that extends the idea of anti-magic labeling from edges and vertices to the faces of planar graphs. A face anti-magic labeling assigns distinct integers to the elements of a graph (vertices, edges, or both) such that the sums associated with each face—typically computed from incident vertices and/or edges—are pairwise distinct. This study explores the existence and construction of face anti-magic labelings for various classes of planar graphs, including cycles, grids, and generalized planar structures. Furthermore, the notion of deficiency is introduced as a measure of how far a graph is from admitting a face anti-magic labeling. Specifically, the deficiency of a graph is defined as the minimum number of isolated vertices or additional elements required to extend the graph so that it satisfies the face anti-magic condition. We investigate bounds, exact values, and constructive techniques for determining the deficiency of selected graph families. The results contribute to a deeper understanding of labeling properties in planar graphs and provide new directions for research in combinatorial optimization and graph labeling theory. Potential applications of these findings include network design, frequency assignment, and coding theory, where distinct labeling constraints are essential.
Automated Detection and Classification of Brain Tumor from MRI Images using Machine Learning Methods
(COMSATS University Islamabad, Lahore Campus Library Information Services, CUI Lahore, 2018) Ghulam Gilanie; , FA14-PCS-007; Dr. Usama Ijaz Bajwa, Assistant Profesor [Supervisor]; LHR TP 5805
Automated Detection and Classification of Brain Tumor from MRI Images using Machine Learning Methods The focus of this thesis is to report an automated, efficient, and robust method of brain tumor detection and classification from Magnetic Resonance Images (MRI) images. Clinically, it is a challenging issue faced by the researchers working in this domain. In routine health care units, Magnetic Resonance (MR) scanners are being used to generate a massive number of brain slices, underlying the anatomical details. Pathological assessment from this medical data is being carried out manually by the radiologists or neuro oncologists. Due to huge volume of brain anatomical data produced by MRI scanners, it is almost impossible to manually analyze every slice. Conclusively, if automated protocols are executed for auto-interpretation; not only the radiologist will be assisted but also a better pathological assessment process would be expected. Several methods have been suggested to address this problem, but still, accuracy, robustness and optimization is still an open issue to address. The development of such automated procedures is difficult due to complex organization of brain cells, several types of tumor, difference in medical traits of a specific ethnicity and many more factors. To achieve the target, research has been started from reviewing the most popular and prominent state-of-the-art methods. Based upon the reviewed literature, automated brain tumor detection and classification techniques have been reported with high computational cost, low classification rates, detection and classification of only one or a few of brain tumor types, lack of robustness, etc. Therefore, step wise research and experiments based upon empirical scientific methodology have been performed in order to achieve the objectives of brain tumor classification.
T-Norms Based Fuzzy Ordering And Their Application
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2018) Atiq-Ur-Rehman; SP12-PMATH-001; Dr Samina Mazhar; LHR TP 4874
This research work focuses on T-transitive fuzzy orderings, their mathematical structures and representation results on these orderings. We have shown that the consistent nature of a fuzzy preference relation has direct relationship with its being more or less T-transitive. After handling this task, some work is done on applications of T-transitivity in the area of preference modeling. Here we have presented some new algorithms to complete an incomplete fuzzy preference relation. These algorithms are based on T-transitivity of a given fuzzy relation. This work has its roots in multi criteria and multi agent decision making. The target is to achieve a ranking among alternatives while incomplete information is provided by the judges about criteria of selection and pair wise preferences amongst alternatives. We have further extended these results to their interval valued versions where the consistent and consensus degrees are also accommodated. The theory and algorithms provided so far are then used to solve real life problems of industry i.e., the problem of supplier selection in the supply chain management where the successful implementation of the results is demonstrated.
Koszul Loops Algebraic And Topological Description
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2019) Muhammad Nadeem; SP13-PMATH-003; LHR TP 6346; Dr. Sarfraz Ahmad
In this thesis we explore interactions between three well established fields namely Algebra, Combinatorics and Chemistry. Main idea is to solve complicated problems in one field by using comparatively easy approaches of the second. We work it out in three different ways. In the first part, we explore chemical properties of some compounds using combinatorial techniques. In recent days atomic properties of chemical structures are intensively explored using results of graph theory depending upon degrees and distances of the associated graphs. In particular, we work on one of the interesting classes of degree based indices, namely, the Zagreb indices. These indices are very useful in QSPR and QSAR studies. We compute Zagreb index, its co index and multiple index for the topological descriptors of 2-dimensional silicon carbons 𝑆𝑖2𝐶3 − 𝐼𝐼𝐼, � �𝑖𝐶3 − 𝐼𝐼𝐼. On the other hand it is of great importance to explore the irregularity of a graph in Chemistry, Pharmacy and Bio technology. We compute irregularity, total irregularity, variance and irregularity index of some networks including (𝐻𝐻𝐶 − 1), (𝐻𝐻𝐶 −2),(BSN−1),(BSN−2) and nanocone 𝑁𝐶𝑘[𝑛]. In the second part we focus on connections between Algebra and Combinatorics. We construct a family of Wilson loops of different orders with the help of additive and multiplicative groups. We associate these newly obtained algebraic structures to the graphs through link labeling. As a main result of this session we prove that the Latin square of the Wilson loop is connected with a bipartite graph. Similarly its normal sub loop of the Wilson loop is associated with the class of star graphs. Let 𝑅 = 𝑘[𝑥1,…,𝑥𝑛] be a polynomial ring in 𝑛 variables. There is one to one correspondence between set of all square free monomial ideals in 𝑅 and set of all simplicial complexes over 𝑛 vertices. The 𝑓- and ℎ-vectors of the simplicial complex ∆ play an important role to describe algebraic properties of the associated square free monomial ideal 𝐼∆. Let ∆𝑠𝑢𝑏 be subdivided simplicial complex obtained from ∆. In the last part of this thesis, we discuss the transformation maps sending the ℎ-vectors of the simplicial complex ∆ to the ℎ-vectors of the subdivided simplical complex ∆𝑠𝑢𝑏. In the end we give algebraic applications describing the primary decomposition of an ideal 𝐼𝑃 associated to the partially ordered set = 𝐶𝑖 × 𝐶𝑗 , where 𝐶𝑖 is the chain 1 < ...<
Degree and Distance Based Parameters of Certain Graph Structures Zeshan Saleem Mufti
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2019) Zeshan Saleem Mufti; FA12-PMATH-006; LHR TP 5950; Dr. M. Faisal Nadeem
Topological indices are a rapidly developing area of research in graph theory. There are various topological indices such as degree-based topological indices, spectrum based topological indices and distance related topological indices. These topological indices correlate certain physico-chemical properties such as boiling point, stability of chemical compounds and many more. In this dissertation, we focus on degree based and spectrum based topological indices. Under the degree based topological indices, we study topological indices such Randic index, general Randic index, Zagreb index, general sum connectivity index, first and second multiple Zagreb indices, geometric index, atom bond connectivity index for para-line graphs of some chemical structures such as Phenylene, Anthracene, Benzenoid, Pentacene. Similarly, we study the spectrum based topological indices such as Estrada index for certain chemical graph structures such as Phenylene, Anthracene, Bi-smuth Tri-iodide, benzene ring embedded in P-type-surface in 2D network (BRE). Also we find the energy of above mentioned structures. We also discuss the notion of distance related parameters such as metric dimension and edge metric dimension. We compute the edge metric dimension for the barcycentric Subdivision of Cayley Graphs. Also we have studied the M-polynomial and entropy of generalized Sierpinski graphs.
Particle Creation with Adiabatic Transformation Through Thermodynamic Perspective in Modified Theories
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2019) Mubashira Rasheed; FA17-RMT -014; Dr. Muhammad Zubair; LHR TP 5651
Modified gravitational theories play significant role to discuss many con stituent of the accelerated phase of our universe for last few decades. A large numberofdifferent techniques in modified theories are proposed to sort out the cosmological problem. Weanalyzedopenirreversiblethermodynamicsinthescenarioofparticle cre ation for modified theory of gravity known as f(G,T) gravity theory, in which the generic function defines the coupling between GB invariant G and trace of the EMT with non-zero divergence. We discussed the generalized conser vation equation which involves additional contribution from matter geome try coupling for irreversible matter creation process with the help of general ized "second law of thermodynamics" in the light of the “Friedmann-Lematre Robertson-Walker" FLRW space time. Particle creation rate, number of particle creation, creation pressure, entropy production rate, and temperature for this theory have been formulated by choosing some particular model of f(G,T), which help us to explore the particle creation in a significant way. We also discussed non-minimal F(R,T,Q) theory of gravity where (R, T, Q are ricci scalar, trace of EMT and interaction of Ricci tensor and non-minimal coupling of Ricci scalar respectively, and Q = RαβTαβ) is the interaction of Ricci tensor and EMT. Due to the non-minimal coupling of gravity and geometry an extra force present which is perpendicular to four velocity. We also formulated some explicit results of created particle from gravitational field to the created ix matter such as particle creation rate, pressure of created particle and entropy with respect to time variation through irreversible thermodynamics interrelated with adiabatic transformation in the context of FLRW space time. We consid ered the dust universe i.e., P = 0 and the scale factor by power law a(t) = tm to precise our results. It is observed that in this procedure a large amount of co-moving entropy has been generated
Some Studies of Two-Population Neural Field Model
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2019) Zeeshan Afzal; CIIT/FA13-PMATH-009/LHR; Dr. Muhammad Yousaf Bhatti; LHR TP 5949
We investigate the appearance and disappearance of localized activity states, so called bumps in Wilson-Cowan type two population neural ﬁeld model under the inﬂuence of transient spatio-temporal external input with smooth α-type and piecewise-type temporal functions. This two-population model is composed of two coupled nonlinear differential equations derived for the dynamics of spatially localized populations of both excitatory and inhibitory model neurons. The model with no external input corresponds to atmost two bump pair solutions. Such a system can be interpreted as a minimal cortical model for short term working memory, that is the ability of the brain to actively hold stimulus-related information for some seconds in short term memory and discards once it becomes irrele- vant Initially, if there is no activity in the system, persistent activity state can be evoked by switching on a suitable transient excitatory external input. This activity remains stable even though external input is switched off. The effect of external input on appearance and disappearance of bumps for different spatial, smooth α-type and piecewise-type temporal functions of external input is investigated and found that certain parameters play a key role in the generation of persistent activity states in the network e.g. relative inhibition time constant, total duration and the amplitude of external input. It is found that the minimum values of the amplitude and active time to evoke the activity in the network is smaller than those observed in showing that the present choice of temporal functions in the external in- put is more effective and more close to natural behavior.
Generalizations of Cyclic Refinements of Jensen’s Inequalities
(Library Information Services COMSATS University Islamabad Lahore Campus, 2020) Nasir Mehmood; FA14-PMATH-005; LHR TP 7451; Dr. Saad Ihsan Butt
In recent years, the concept of convex functions has been generalized extensively. Applications of convex functions are widely seen in many areas of modern analysis. Convex functions also have significant relation with the theory of inequalities and many useful inequalities are the result of the applications of convex functions. The Jensen's inequality has tremendous implications in many fields of modern analysis. It helps computing useful upper bounds for several entropic measures used in information theory. We consider discrete and continuous cyclic refinements of Jensen's inequality and extend them from convex to higher order convex function by using new Green functions introduced by us and employing different interpolating polynomials and identities. We formulate monotonicity of the linear functionals for nconvex functions at a point. We calculate some new Grüss and Ostrowski type bounds. As an application of our obtained results we give new bounds for Shannon, Relative and Zipf-Mandelbrot entropies.
Algebraic Characteristics of Graphs
(Library Information Services COMSATS University Islamabad Lahore Campus, 2020) Muhammad Asim Razzaq; FA13-PMATH-004; LHR TP 7454; Dr. Kashif Ali
Graph theory is a vital and special branch of discrete mathematics. It deals with geometric computational study of various objects. The main component and object of the theory is graph and its generalization approach. Therefore, it has vast application to other mathematical and provides computational assistance to non-mathematical sciences. The most prominent use of graph theory as a significant tool in chemistry, this branch is named as chemical graph theory. It provides helpful assistance for obtaining molecular descriptor to develop essential relationship between molecular graphs and chemical compounds. Graph theory also serves its specialization in algebraic graph theory specifically labellings. In this dissertation, we formulate and discuss all these types of indices. We develop a new approach to calculate the eccentricities of vertices of any graph by using computer aid softwares. This approach help us to find the exact expression of the ECI and ABC5 for the butterfly, bene and torodial grid graphs. We also express a new type of counting polynomial associated with the previous counting polynomials. Moreover, we work some topological indices relative to degree dependent on generalized subdivision of line graphs. We also determine the general results about subdivision of line graphs for neighborhood valency-based indices. We also discuss the stability of line graph of chemical molecules.”Line0graphs furnish0a tool for0studying the0topological properties of0alpha-systems.”In addition to this, we study four important counting polynomials called omega, theta, PI and Sadhana for V-Phenylenic. We also introduced a new polynomial Upsilon on the basis of ”co” relation and find new results relative to new counting polynomial.”We also determine that inertia indices for V-phenylenic nanotube and these indices are not0equal for the line0graph of V-phenylenic0nanotube. We also find that the0nullity for this0nanotube and of line0graph of these0nanotube.”Lastly, we also discuss H-group magic labellings of some graphs. We mainly study about the H-groupmagic0total graph0labelings G(v; e) over finite0abelian0group Zt Zv, where H =K2; Ct and e = (t 􀀀1)v: