Department of Mathematics
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Item Edge Irregularity Strength of Some Path Related Graphs(Library Information Services, COMSATS University, Lahore Campus, 2018) Muhammad Imran,; FA17-RMT-008; r. M. Faisal Nadeem, Assistant Profesor; LHR TP 5646Edge Irregularity Strength of Some Path RelatedGraphs Graph theory is playing a key role in applied and pure mathematics. The scope of graph theory is expanding day by day. In graph theory, there are many categories of graph labeling, such as, graceful label ing, cordial labeling, magic labeling and irregular labeling. Graph labeling is being used in different fields, such as, computer networking, coding theory, circuit design and data management. In this thesis, we will try to find irregular labeling of some path related graphs. Mythesis comprising of six chapters. In the first chapter we will define some funda mental definitions. In the 2nd chapter, we will discuss various types of graph labelings, such as, edge and vertex labeling, total labeling, irregular labeling, edge irregular labeling, vertex irregular labeling and irregularity strength. In the 3rd chapter, we will find vertex irregular labeling and evaluate edge irregularity strength of triangular snake graph, alternate triangular snake graph. In and 4th chapter, we will find vertex irregular labeling and evaluate edge irregularity strength of different variations of comb graph. Results regarding edge irregularity strength and some relevant references are given in the 5th and 6th chapters respectivelyItem 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 5946In 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 chapterItem 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 ProfesorDegree 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.Item Two Population Neural Field Model With Constant External Input(Library Information Services, COMSATS University, Lahore Campus, 2018) Ammara Basharat,; FA16-RMT-032; Dr. Muhammad Yousaf, Assistant Profesor; LHR TP 5195In this work, We have study the two-population neural field Model under the effect of constant external inputs. Earlier, Blomquist et al.[1] investigated the stationary symmetric solution of two population model with no external input.They found that Amari stability analysis and linearized stability analysis produces same results. Later on Yousaf et al. [2] investigated the same model under the influence of spatial dependent external input and proved that the Amari technique and linearized stability technique does not produce same results. They also investigate the number of bumps as a function of amplitude parameter of external input and found at most four bump pairs. In our research work, We have to explore the two population neural field model under the effect of constant external input on bump solutions. We further, have to investigate the stability properties of stationary symmetric solutions by using two approaches (i) Amari approach (ii) standard linearized approach. Numerically, We will investigate the number of bumps as a function of width and amplitude parameter of external inputItem Total Labellings of Some Connected Graphs(Library Information Services, COMSATS University, Lahore Campus, 2019) SP17-RMT-014; Mujahid Shehzad,; Dr. Kashif Ali, Assistant Profesor; LHR TP 5428Ashraf et al. in [3] introduced the concept of vertex H irregular labelling and edge H irregular labelling. Let H be sub graph of given graph G. If each edge of graph G is contained in any subgraph Hi where i ={ 1,2,3,…, k } of G and if each Hi is isomorphic to given subgraph H. then the collection Hi of subgraphs of G is said to be H-covering of G. They also introduced the concept of edge H irregular strength ehs(G) that is the minimum k for which the graph has total H irregular k labelling and vertex H irregular strength vhs(G) that is the minimum k for which the graph has total H irregular k labelling The thesis is devoted to study the total vertex C3 irregular strength wheel graph and triangular ladder graph taking subgraph C3. We also found the total vertex C4 irregular strength ladder graph and grid graph. Furthermore, we find the total vertex C8 irregular strength of a connected graph Qn.Item Analysis of Integral Equations of Ahlfors Map for Multiply Connected Regions(Library Information Services, COMSATS University, Lahore Campus, 2018) Razi Ahmad,; FA16-RMT-025; Dr. Kashif Nazar, Assistant Profesor [; LHR TP 5433This research is to analyze the boundary integral equation of Ahlfors map of multiply connected region also the derivation of a new boundary integral equation related to Ahlfors map. These integral equations are then parameterized and discretized into the system of equations by using the Nystrom method with trapezoidal rule.Item Stokes Flow Through a Permeable Slit With Periodic Reabsorption and Porous Medium(Library Information Services, COMSATS University, Lahore Campus, 2019) Zernab Shafiq,; FA17-RMT-005; Dr. Quart-Ul-Ain Azim, Assistant ProfesorOne of the main and most important organs of the body of an organism is Kidney. Nephron is the functional unit of Kidney which has a tubular part called renal tubule. The walls of renal tubules are porous and fluid reabsorption takes place in these tubules. Due to various diseases, the medium inside the tubules becomes porous and the flow permeability is reduced. There are numerous studies in literature that address uniform reabsorption in renal tubules through porous slits. We have done the study of two di mensional reabsorption flow through porous slits in renal tubules that are diseased and are under the effect of biofouling. We model this physical problem into a mathematical form, which result in a high order differential equation as well as associated bound ary conditions. The problem then solve exactly, and expressions for various physical quantities are computed. At last, the obtained results are analyzed graphicallyItem Cosmological Analysis of Reconstructed Einstein-Aether Models(Library Information Services, COMSATS University, Lahore Campus, 2018) Irfan Ullah Malik; SP 17-RMT-007; Dr. Shamaila RaniIn this thesis, we reconstruct various solutions for the accelerated universe in the Einstein-Aether theory of gravity. For this purpose, we obtain the effective density and pressure for Einstein-Aether theory. We reconstruct the Einstein-Aether model parameter by comparing its energy density with various holographic dark energy models such as Tsallis, R´enyi and Sharma Mittal holographic dark energy models. For this reconstruction, we use two forms of scale factor, power-law and exponential forms. The cosmological analysis of underlying scenario has been done by exploring different cos mological parameters. This includes equation of state parameter, squared speed of sound and evolutionary equation of state parameter via graphical representation. We obtain some favorable results for some model parametric valuesItem Hybrid Projection Algorithm for Generalized Split Equilibrium Problem in Hibert Spaces(Library Information Services, COMSATS University, Lahore Campus, 2019) Hammad Sarwar,; SP17-RMT-008; Dr. Muhammad Aqeel Ahmad Khan, Assistant Profesor [Supervisor]; LHR TP 5430In this thesis we suggest and examine Extra Gradient-Proximal Point algorithm to find the solution of Split Equilibrium Problem and Fixed Point Problem for nonexpansive semigroup in real Hilbert spaces. Our purpose is to address fixed point problems(FPP), variational inequality problems (VIP) and generalized Split Equilibrium Problems(GSEP) in the setting of Hilbert spaces. We propose a hybrid shrinking projection algorithm, which exhibits strong convergence, for the construction of a common element in the set of common fixed points of an infinite family of k-strict pseudo contractions and the set of common solutions of a finite family of Generalized Split Equilibrium Problems(GSEP) in Hilbert spaces. Our results generalize and improve various existing results in the current literature.Item Classifications of Determinantal Varieties(Library Information Services, COMSATS University, Lahore Campus, 2019) Sana Zahoor; , FA17-RMT-054; Dr. Imran Ahmed; LHR TP 5675Let Mns be a space of matrices of order n×s with complex enteries. The determinantal singularities are defined by vanishing of all the minors of a certain size of an n ×s matrix α : Ck → Mns. The class of determinantal singularities is a more general class than the class of complete intersection singularities. The G-finite deteminacy gives a special class of determinantal varieties known as essentially isolated determinantal singularities (EIDS), firstly introduced by Ebeling and Gusein-Zade in 2009. We give a computational method to find linear forms of EIDS by programming in C