Department of Mathematics
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Item On Degree Based Banhatti Indices Of Some Graphs(Library Information Services COMSATS University Islamabad Lahore Campus, 2021) Faheem Yaseen; FA19-RMT-043; LHR TP 7444; Dr. Sarfraz AhmadIn the domains of electronic and electrical engineering, graph theory is crucial. It’s significant in signal processing, networking, communication theory, and a variety of other fields. A topological index (TI) is a numerical value that correlates chemical networks with physical and chemical qualities, as well as chemical reactivity. In this thesis, we compute first and second Banhatti Indices, first and second K-hyper Banhatti indices, first and second multiplicative K-Banhatti indices, first and second multiplicative K-hyper Banhatti indices, K-harmonic Banhatti indices, multiplicative K-harmonic Banhatti indices for certain graphs like as Hierachical hypercube network one and two, block shift network one and two, Octagonal network and line graph of wheel graph. My thesis comprising of four chapters. In the first chapter we define some fundamental definitions. In the 2nd chapter, we find Banhatti indices for Hierachical hypercube network one and two, block shift network one and two. In the 3rd chapter, we find Banhatti indices for Octagonal network and line graph of wheel graph. In the 4th chapter, we present the conclusion of our thesis. In the last, we present the references of our thesis.Item Multiple Lump And Rogue Wave Solutions For …(Library Information Services COMSATS University Islamabad Lahore Campus, 2021) Muhammad Aamir Ashraf; FA19-RMT-092; LHR TP 7421; Dr. Sarfraz AhmadA solitary wave is a localised “wave of transcription” that appears from a balance between dispersive and nonlinear effects. A soliton is a specific type of solitary wave, since it maintains its original shape and speed when collides with other solitons. Nonlinear evolution equations (NLEEs) are the main classification of soliton solutions. In this work, our purpose is to attain lump, lump with one and two-kink soliton, periodic– waves, rogue–waves solution for various forms of NLEEs. We’ll also obtain M-shaped rational soliton, M-shaped interactions with kink and periodic waves, periodic-cross kink solutions, homoclinic breather and multi–wave solutions. In this thesis, we’ll study various forms of NLEEs to attained these types of solutions by the combinations of bilinear, hyperbolic, trigonometric, exponential, and hyperbolic functions. We also seen the degenerations of hybrid waves, through some fittingly bilinear forms and also discussed graphical behavior of our system.Item 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 AhmadIn 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 < ...<