Department of Mathematics

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Author: search.filters.author.Dr. Muhammad HussainEnd date: 2019

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    Valuations and Metric Dimensions on Graphs
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2017) Shahid Imran; CIIT/FA12-PMATH-003/LHR; Dr. Muhammad Hussain; LHR TP 5284
    Labeling of graph is a beautiful and rich technique of Mathematics. It has extensive usage in applications such as data transmission, x-ray, circuit design, astronomy, communication links and coding theory. Labeling of graph is defined as a correspondence from the set {1,2,3,, v  e  f } to the vertices, edges and faces of a plane graph G in such a way that each vertex, edge and face receives exactly one label and each number is used exactly once as a label. The weight of a face under a labeling of type (1,1,1) is the sum of labels carried by that face and all the edges and vertices surrounding it. A labeling of type (1,1,1) of a plane graph G is called d-antimagic if for every positive integer s the set of weights of all s-sided faces is {a, a  d, a  2d,, a  ( f 1) d } for some integer a and d 0 , where f is the number of s-sided faces. In particular when d  0 then the d- antimagic labeling of type (1,1,1) is called magic labeling of type (1,1,1) . Moreover, if the vertices are labeled by the smallest possible number. It is known as super magic and anti- magic accordingly. Furthermore, let G be a connected graph and d (x, y) be the distance between the vertices x and y in G. A set of vertices W resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in W. A metric dimension of G is the minimum cardinality of a resolving set of G and is denoted by dim(G). This dissertation studies about the construction of super face anti-magic labeling of type (1,1,1) for disjoint union of toroidal fullerenes, magic and anti-magic labeling of type (1,1,1) for union of generalized friendship graphs. Metric dimension of Jahangir related graphs, generalized prism related graphs and graphs obtained by rooted product has been also calculated.
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    Metric Dimension Of Different Chemical Structures
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2019) Muhammad Umer Farooq; SP17-RMT-001; Dr. Muhammad Hussain; LHR TP 5431
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    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 5191
    The 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 network
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    Topological Indices On Chemical Graphs
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2019) Tayyaba Razzaq; SP16-RMT-002; Dr. Muhammad Hussain; LHR TP 5345
    Topological index is a numeric parameter which describe some silent features (con nectivity, boiling point, stability and melting point) of a Molecular graph representing some chemical compounds. These indices characterize topology and are usually hold mathematical properties of Graph invariant.The process of transforming chemical information of chemical molecules into useful numerical results is known as molec ular descriptor. These molecular descriptors are extensively used in the study of Quantitative structure-activity relationships (QSARs) which give insight to the bio logical properties based upon chemical structures. We will investigate R˜andic indices Rλ(G), where λ ∈ R. First, second and third Z˜agreb indices, First and second Z˜agreb Polynomials, Multi Z˜agreb indices, Atom bond connectivity ABC index, Geometric arithmetic GA index, fourth version of ABC index, Harmonic index and fifth version of GA index to estimate biological properties of chemical graphs
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    FACE ANTI-MAGIC LABELINGS AND DEFICIENCY OF GRAPHS
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2016) Ali Tabraiz; FA12-PMATH-001; LHR TP 4824; Dr. Muhammad Hussain
    Labeling of graph is a beautiful and rich technique of Mathematics. It has extensive usage in applications such as data transmission, x-ray, circuit design, astronomy, communication links and coding theory. A labeling of a graph is a mapping that carries some set of graph elements into (usually positive integers). An ) , ( d a face anti-magic total labeling of a graph, with p vertices, q edges and f faces, is a one-to-one mapping that takes the vertices, edges and faces into }, , , 2 , 1 { f q p such that the sums of the label on the edges, vertices and faces form an arithmetic progression starting at a and having difference . d An edge magic total labeling of a graph G is a one-to-one mapping from ) ( ) ( G E G V onto the integers }, , , 2 , 1 { q p with the property that there is an integer constant k such that k y xy x ) ( ) ( ) ( for any ). (G E y x Furthermore, the edge-magic deficiency of a graph , G ) (G is defined as the minimum non-negative integer n such that 1 K n G is edge-magic.
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    Anti-Magic Labeling on Harary Graph and Subdivided Caterpiller
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2016) Muhammad Mubashar; FA14-RMT-029; LHR TP 6839; Dr. Muhammad Hussain
    Graphs play an important and vital role in Mathematics. Graph labeling creates new ideas graph theory. Can labeling be helped us to solve problems in Mathematics and other fields? The answer of this question increases the deep value of Graph Labeling. A magic labeling of a graph is described by Kotzig and Rosa [22] in 1970 as an important iniative. A graph labeling (or valuation) is a unique way that gives the labels to graph elements (commonly, these are non negative or positive integers). if we consider the domain set of all vertices and edges; such labelings are called total labelings. Sometime we use labelings for vertex set or the edge set alone and these labelings would be called as vertex-labeling and edge-labeling respectively. Many other domains are also possible. There are some famous labelings like as harmonious, cordial, graceful and antimagic. In 2000, Baˇca et al. [2] discussed a (a,d)-vertex-antimagic total labeling. A (a,d) vertex-antimagic total labeling is called super (a,d)-vertex-antimagic total labeling, when we give smallest labels to the vertices. An (a,d)-edge-antimagic total labeling was defined by Simanjuntak et al. [41]. An (a,d)-edge-antimagic total labeling is called a super (a,d)-edge-antimagic total labeling, when we give the least labels to the vertices. A graph G, which is made up of cycles with chords is known as Harary Graph. A brief discussion exposed important results on super (a,d)-edge-antimagic total labeling Harary graph by M. Baˇca and M. Murugan [1, 6]. This work is designed for the construction of (a,d)-vertex-antimagic total labeling of Harary graph as well as for super (a,d)-vertex antimagic total labeling of Harary graph for the different value of d.