Department of Mathematics

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    An Anisotropic Version of Tolman VII Solution in Rastall Theory of Gravity
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2020) Dawood Falik Sher; FA18-RMT-022; LHR TP 6457; Dr. Muhammad Zubair
    In our research, we worked for self gravitating system to evaluate stellar in- terior’s possibilities. For this we describe an anisotropic matter distribution , which is based on the Rastall’s theory of gravity with the help of minimal geo- metric deformation approach. As the minimal coupling matter principle is bro- ken down by the Rastall’s gravity, hence we have to provide an exhaustive ex- planation. This explanation deals with Israel-Darmois junction conditions and how it works in this pattern. Further, we have obtained the deformed space- time and use the procedure of mimic constraints. For checking the viability of any proposal, the results has been applied to any well known solution. So as we do. We use famous Tolman VII solution, to check the viability. The whole description of thermodynamical effects presented by the additional sources in mentioned. In addition, we have compared the results with their similes in the shadow of pure general relativity, pure Rastall’s gravity and, also in the struc- ture of general relativity which includes "gravitational decoupling". For mathe- matical and graphical analysis we take α i.e., gravitational decoupling constant and λ i.e., parameter of Rastall as free parameter. The compactness factor which describes the general relativity is taken as 0.2. Besides, in order to get more realistic picture, it requires to bound the parameters α and λ both through the use of real observational data to get the limits of the theory under this model. This methodology suggested to study the applications of neutron and quark stars. Our work also deals with the extended minimal geometric deformation. ixIn this section we check how temporal component effects the value of total en- ergy density, total redial and tangential pressure, anisotropy of pressure, energy conditions and stability conditions. For this we again use famous Tolman VII solution for the values of ν and µ and check the results. The metric potential of this solution is in the form of exponent, hence we use the rules function (ln) to get the suitable values of components. This section show very interesting results as we will discuss in chapter 3 in detail.
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    On Anti-Magic Total Labeling
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2020) Aisha Asghar; SP18-RMT-020; LHR TP 6546; Dr. Muhammad Hussain
    Graph labeling plays a very significant function in implementations in graph theory such as network, correspondence naming, X-ray, circuit modeling, crystallography, radar, rocket navigation, astronomy and data base administration. A labeling is bijective mapping, in which vertices or edges are assigned to natural numbers. If the weights of all vertices are identical, then these labels are considered magic labeling. In 2000, Baca et al.[5] presented the idea of total antimagic labelling of (a, d)-vertex. A labeling in which both vertices and edges compose of domain collection is called total labeling. If we assign the smallest number to vertices and then to edges, a labeling is called super (a, d)-vertex total antimagic labelling. Hussain et. Al. [17,18] formulated (a,d)-vertex antimagic total labeling on Harary graghs, however, many cases are still open. This dissertation learning about the configuration of new results of (a,d)-vertex anti-supermagic total (VAST) labeling of Harary graphs. Also we construct same new esults of super (a,d)-vertex magic total(SVT) labeling on Harary graphs.