Department of Mathematics

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    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.
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    Structural Properties Of Certain Molecular Graphs
    (COMSATS University Islamabad, Lahore Campus Library Information Services, CUI Lahore, 2018) Shahid Imran,; FA14-BSM-017; Dr. Muhammad Hussain, Assistant Profeso; LHR TP 5261
    Abstract Let G=(V,E)beasimple connected graph. The sets of ver tices and edges of GaredenotedbyV =V(G)andE =E(G) respectively. In such a simple molecular graph, vertices rep resent atoms and edges represent bonds. In chemical graph theory, we have many invariant polynomials and topological indices for molecular graph. In this thesis we focus on the structure of molecular graph of Bakelite network. Morever the vertices having degrees (2,2),(2,3) and (3,3) gives the generalized form of count of edges in terms of m and n. The m in chemical structure represent the number of rows and n represent the number of column in Bakelite structure. In this thesis we compute the different indices such as Atom bond conecctivity ABC,Geometric arithmatic G(A), Zagreb, Hyper Zagreb, Augmented Zagreb, Randic index and Har monic indices of Bakelite network. Topological indices are real numerical parameter of a chemical structural networks and structural networks. These networks are characterize its topology and graph invariant. The topological indices like Randic, ABC, G(A), are used to diagnose the bioactiv ity of chemical compounds. Actually these topological in dices is designed by transforming a chemical structure into a numeric number. i
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    Chirped And Dipole Soliton In Nonlinear Negative
    (Library Information Services, COMSATS University, Lahore Campus, 2016) Adnan Khalil,; FA14-BSM-012; Dr. Syed Tahir Raza Rizvi, Assistant Profesor
    Chirped and Dipole Soliton in Nonlinear Negative Index Materials A wave which propagates without changing its shape and speed is known as soliton. Whenever two of these waves collide they do not change their shape and speed. Soli tons have a lot of applications in other fields like, fluid, biology and engineering. This dissertation find out the chirped and dipole soliton for nonlinear negative-index ma terials with quadratic-cubic nonlinearity. We will use sub-ODE method [26] to find chirped soliton and ansatz method of Choudhuri and Porsezian [16] to get dipole soli ton
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    Topological Descriptors of OTIS Interconnection, Block Shift and Hierarchical Hypercube Networks
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2020) Iftikhar Ahmed; CIIT/FA14-PMATH-003/LHR; Dr. Sarfraz Ahmad; LHR TP 7453
    Topological Indices(TIs) are the numbers associated with the graphs of chemical compounds/networks that help us understand their properties. The aim of the present work is to compute TIs of OTIs swapped networks, Hierarical hypercube networks and block shift networks. Some of the TIs are computed directly and some are computed indirectly. For indirect computations the M-polynomial has been used. The M-polynomial contains key facts about degree dependent TIs, that is why it is so important. We compute Hosoya polynomials, Harary polynomials, Wiener index, modified Wiener index, Hyper Wiener index, Harary index, generalized Harary index, multiplicative Wiener index, M-polynomials, first Zagreb index, second Zagreb index, modified second Zagreb index, generalized Randi\'c index, inverse Randi\'c index, symmetric division index, Harmonic index, inverse sum index, augmented Zagreb index, Redefined first second and third Zagreb index and Zagreb type polynomials of networks mentioned above. We also computed irregularities, multiplicative version of different TIs . Our results can help the discovery of new medicines and in development of new software.
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    Chromatic Equivalence and Uniqueness of Subdivided K4 Graphs
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2016) Areeb; FA14-MSMATH-009; LHR TP 6935; Dr. Sana Javed
    In this thesis, various chromatically unique and correspondent pairs of several families of graphs with same girths have been attained. Subdivided complete graphs are related to these families, indeed. Our major goal in this paper is to carry on the study of chromaticity of the subdivided complete graphs. We are dealing particularly with four vertices which is called 𝑘4-homeomorph. Distinctiveness of two particular classes of 𝑘4-homeomorph graphs with girth 11 has been explained in detail. The cycles of different lengths from graph has been removed during the discussion. In order to make them chromatically unique, vital and satisfactory conditions have been established. All well known definitions, results and terms are given in Chapter 1. Chapter 2 is covering a brief literature review about chromatic uniqueness and correspondence. While the chapter 3 is focused on the chromatic uniqueness along with chromatic equivalence of two 𝐾4-homeomorph graphs with same girth. At the end of the thesis, few open problems have been recommended as well.