Department of Mathematics
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Item Edge Irregularity Strength of Rooted Product of Certain Families of Graphs(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2018) Muhammad Usman; CIIT/FA16-RMT-011/LHR; Dr. Faisal Nadeem; LHR TP 5183Graph theory is one of the most powerful tool in the field of mathematics. Different mathe- maticians work in this field and consider it as a powerful tool in different directions because in recent times graph theory has confirmed as a paramount mathematical tool in various subjects. Now it has emerged as an important mathematical discipline in its own right. There are several types of labelings in graph theory. In this thesis we intend to find the irregular labeling of double graph, rooted product as well as the some family of the Teoplitz graphs. we have divided this thesis into seven chapters. First chapter is about the introduction as well as basic definitions of different types of graphs. In this chapter we have discussed some graph operations along with their examples. In chapter second we have clarified different types of labeling which was defined by well known scholars. At the end of this chapter we have written some known results of edge irregularity strength for certain families of graph. In chapter three we have discussed results related to edge irregularity strength of double graph of path and cycle. we have also proved these results. we believe that these results are accurate. In chapter four we have developed some results for edge irregularity strength of rooted product of cycle and path graphs. In chapter five we find edge irregular labeling of different families of Teoplitz graphs and calculated the edge irregularity strength of Teoplitz graphs.Item On Tchebyshev Subdivision of Simplicial Complexes(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2018) Iqra Siddique; CIIT/FA16-RMT-035/LHR; Dr. Sarfraz Ahmad; LHR TP 5197In this work, we studied the Tchebyshev subdivision of a simplicial complex Δ as combinatorial and algebraic point of view. In particular, we have studied f-vector and h-vector transformation maps and properties of corresponding transformation matrices. We have also studied the impact of this subdivision on the algebraic properties of the corresponding Stanely Reisner Rings IΔ, including dimension, primary decomposition and Hilbert series. We extended these results to some other subdivisions of a simplicial complex Δ.Item Implementation of Elgamal Encryption in Electronic Voting(Library Information Services, COMSATS University, Lahore Campus, 2018) SHAHID MEHMOOD,; FA16-RMT-008; Dr. Tariq Javed Zia, Assistant Profesor; LHR TP 5264In this dissertation, in general we have studied the implementation of inite groups in cryptography, especially theoretical based being have mathematical background. In particular the Elgamal encryption scheme based on finite groups has been used being more robust and efficient. It operates on a cyclic group for which the decisional Diffi-Hellman (DDH) assumption holds. This group is the subgroup of Quadratic residue Gq < Zp* of prime order q, where p = 2q + 1 is a safe prime. It is obvious from the definition of Elgamal encryption, which is public key encryption scheme, that the keys generated by the multiple parties (of course finite in numbers) can be combined to form a single (common) public key. The corresponding private keys can be regarded as key shares and, are used in collaborative manner. Consequently we have studied the implementation of modulo groups (Elgamal Algorithm) in particular, in electronic voting by presenting the construction of receipt free protocol from a given primary voting scheme. After applying this construction to the voting protocol we obtain an efficient receipt free 1-out-of-L voting protocol based on homomorphic encryption.Item Characterization of Line Total Simplicial Complexes(Library Information Services, COMSATS University, Lahore Campus, 2018) AYESHA ANDALIB KIRAN,; FA16-RMT-033; Dr. Imran Ahmed; LHR TP 5196In 2017, Ahmed and Mahmood introduced the line simplicial complex ∆L(G) associated to a simple graph G. The pth Betti number is the rank of the pth homology group of a topological space and it refers to the number of n-dimensional holes on a topological surface. In Theorem 3.0.15, we give formula for Betti numbers of the line simplicial complex ∆L(Wn+1) associ ated to wheel graph Wn+1 by writing code in MATLAB. We define first the line total simplicial complex ∆T(G) associated to a sim ple graph G. In Theorem 4.0.24, we prove that the line total simplicial complex ∆T(G) is connected iff G is connected. In Lemma 4.0.19, we find f-vector of the line total simplicial complex ∆T(F2n+1) associated to friend ship graph F2n+1. In Theorem 4.0.20, we give formula for Betti numbers of ∆T(F2n+1) by coding in MATLABItem G-Symplectic Partitioned General Linear Methods(Library Information Services, COMSATS University, Lahore Campus, 2018) Muhammad Sarfraz Khan; , FA16-RMT-022; Dr. Yousaf Habib, Assistant Profesor [Supervisor]; LHR TP 5189The main aim of this thesis is to construct structure preserving numerical methods for Hamiltonian systems. General linear methods are a generalization of RK method and LMSM. G symplectic GLM are suitable for numerical integration of Hamiltonian systems in a similar fashion as that of symplectic RK methods. Order 2 G-symplectic general linear methods were constructed by Butcher for numerically preserving Hamiltonian systems having invariants. After that, Order 4 G symplectic general linear methods were constructed. We have constructed order 6 G-symplectic general linear methods that not only preserve energy of Hamiltonian systems but also show zero parasitism.Item Metric Dimension Of Different Chemical Structures(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2018) Hafiz Muhammad Bilal; FA16-RMT-027; Dr. Muhammad Hussain; LHR TP 5191The Metric Dimension of a graph G is the minimum number of basis element in a resolving set. Let G = (V,E) be a connected graph and length of a shortest path between u and v is known as distance, denoted by d(u,v) in G. Let B = {b1,b2,...,bk} be an ordered set of vertices of G. The representation r(u|B) of u with respect to B is the k − tuple {d(u,b1),d(u,b2),d(u,b3),...,d(u,bk)}, where B is called a resolving set or locating set if every vertex of G is uniquely identified by its distances from the vertices of B or equivalently if distinct vertices of G have distinct representations with respect to B. A resolving set of minimum cardinality is called a basis for G and cardinality is the metric dimension of G is denoted by dim(G). In this thesis we investigated metric dimension of subdivided honeycomb network and Aztec diamond networkItem Thermal Fluctuations Of A Regular And Einstein(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2016) Hafiza Zarish Arshad; , FA14-BSM-001; Dr. Abdul Jawad, Assistant Profesor; LHR TP 5252In this thesis, we study the effects of thermal fluctuations on regular black hole solutions with cosmological constant and Einstein-aether black hole with coupling constant respectively. We consider the logarithmic corrected entropy in order to analyzing the thermal fluctuations on regular black hole solutions and Einstein-aether black hole. We also obtain various thermo dynamical quantities such as entropy, pressure, specific heats, Gibb’s free energy and Helmholtz free energy. We also investigate the stability of reg ular black hole solution and Einstein-aether black hole in terms of γ, phase transition, grand canonical ensemble and canonical ensemble. We analyze that regular black hole are stable when we increase the value of cosmologi cal constant and Einstein-aether black hole are stable when we increase the value of coupling constant respectively.Item General Linear Methods With Projection(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2018) Lubna Mustafa; FA16-RMT-031; Dr. Yousaf Habib; LHR TP 5194The present work deals with the structure-preserving numerical methods for the solution of ordinary differential equations representing dynamic systems from the Hamiltonian perspective of classical mechanics. These differential systems involve quantities that should remain constant over the course of enquiry. Whereas the usual numerical methods do not account for the preservation of these invariants, structure-preserving methods such as the symplectic Runge-Kutta methods and the G-symplectic general linear methods faithfully maintain these characteristic properties of a Hamiltonian system during the numerical discretization of the problem. Physically significant invariant properties of a Hamiltonian system include the conservation of total energy and the symplecticity of flow. In Chapter 1, basic theory of ordinary differential equations is introduced. The exposition combines definitions with the construction and implementation of numerical methods for ordinary differential equations and is central to all the subsequent investigations carried out in this work. In Chapter 2, long - term integrators for a Hamiltonian system are treated in detail. The general linear methods subsume the Runge-Kutta methods as well as the linear multi-step methods introduced previously. In Chapter 3, the important notion of projection is also introduced here. The basic idea of the standard projection technique is generalized to develop the heterogeneous class of numerical methods collectively known as the general linear methods. Projection technique is applied to the general linear methods and the similarity of the symplectic Runge-Kutta methods to the G-symplectic general linear methods is discussed.Item Characterizations of Gallai Total Simplicial Complexes(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2018) ADNAN LIAQUAT; FA16-RMT-003; Dr. Imran Ahmed; LHR TP 5178In 2017, Anwar, Kosar and Nazir introduced the Gallai simplicial complex ∆Γ(G) associated to a simple graph G. The homology is a powerful tool in algebraic topology. It formalizes the concept of holes related to connected components of a simplicial complex. The Betti numbers are topological invariants that are used to calculate the number of holes of each dimension in a space. In Theorem 4.0.26, we compute the formula for Betti numbers of the Gallai simplicial complex ∆Γ(F2n+1) associated to friendship graph F2n+1 by coding in MATLAB. We define first the Gallai total simplicial complex ∆Γ(G) associated to a simple graph G. In Theorem 5.0.30, we prove that the Gallai total simpli cial complex ∆Γ(G) is connected iff G is connected. In Lemma 5.0.31, we compute f-vector of the Gallai total simplicial complex ∆Γ(L∗ 2n) associated to triangular ladder graph L2n. In Theorem 5.0.32, we find the formula of Betti numbers for ∆Γ(L∗ 2n) by coding in MATLABItem Dynamics Of Two Population Synaptic Drive Model(COMSATS University Islamabad, Lahore Campus Library Information Services, CUI Lahore, 2016) Iqra Khan,; FA14-MSMAT-018; Dr. Muhammad Yousaf Bhatti, Assistant Profesor [Supervisor]; LHR TP 4627In this work we have investigated the two population synaptic drive model with a relatively complex temporal kernal for steep firing rate function. We have done this justification by deriving four dimensional autonomous dynamical system from Volterra equations using linear chain trick. Both of the local and global wellposedness of the model has been in vestigated. The existence of stationary point has also been investigated and the general formula for stationary point is derived. We get two different models for delay and non delay case. Theorem on the existence and stability has also been formulated. The stability issue has also been addressed for both cases for the steepness parameter q = 0, and q > 0, which leads us to the conclusion that delays in the system with weak kernel does not effect the stability properties of the model