Department of Mathematics
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Item Topological Indices Based on Line Graph of Different Chemical and Non-Chemical Structures(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Alizeh Muhammad; CIIT/SP20-RMT-017/LHR; Dr. Muhammad Hussain; LHR TP 9572Chemical compounds have variety of physicochemical, toxicological and pharmaco- logical properties which predict their nature and how they will react. But the study of these properties requires Herculean and laborious experimentation so its alternative, topological indices were invented about 150 years ago which is the greatest break- through in the field of mathematical chemistry. It provides the substitute to scrutinize the behavior of chemical graphs and an easy approach for substantiating their existing numerous properties and thus helps in designing novel drugs and compounds. This was achieved due to the birth of chemical graph theory in which modeling of chemical compounds can be done. In this research, miscellaneous mathematical and computational techniques are applied for the calculation of several degree-based and spectrum-based topological invariants for boric acid (H3BO3) layer structure’s derivatives (subdivision, line graph and caged chain). The molecular descriptors being computed are the harmonic index; two ver- sions of Randic´, sum-connectivity, atom-bond connectivity, geometric-arithmetic and Zagreb indices. Besides these, the list also includes Estrada index, energy and the distance energy of above-named graphs. Pre-invented formulae are used for their cal- culation and for spectrum-based invariants, polynomials are generated and vindicated based on results.Item Radio Labeling of Different Graphs(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) AMNA RIAZ; CIIT/FA19-RMT-057/LHR; Dr. Muhammad Hussain; LHR TP 7406This thesis is about Radio Labeling. For a graph G, any two vertices, v1 and v2 in G, Let d(v1, v2) represent the distance between v1 and v2. Let diameter of G is diam(G). A Multi-level distance labeling(radio labeling) for G is a function f that assign to each vertex so that they can satisfy this condition d(v1, v2) + |f(v1) − f(v2)| ≥ diam(G) + 1 The main purpose for this research is to find the radio number of Web graph rn(Wn), embedded graph rn(Emn) and radio number of double wheel graph rn(DWn). we use basic concept to find their radio numbers.Item Valuations and Metric Dimensions on Graphs(Library Information Services, COMSATS University, Lahore Campus, 2016) Shahid Imran,; LHR TP 5605; FA12-PmAth-003; Dr. Muhammad Hussain, Assistant ProfesorLabeling 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 } , ,3,2,1{ f e v 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 } )1 ( , , 2 , , { d f a d ad aa for some integer a and 0d , where f is the number of s-sided faces. In particular when 0d 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 ) , ( y x d 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 ). ( dimG 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.Item 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 5284Labeling 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.Item Topological Indices Of Chemical Graphs(COMSATS University Islamabad, Lahore Campus Library Information Services, CUI Lahore, 2018) Sahar Dilber,; SP14-BSM-002; Dr. Muhammad Hussain, Assistant Profesor; LHR TP 5262Topological indices are real numerical parameters of a molecular structural net works and structural networks. These networks are characterize its topology and graph invariant. The molecular networks plays a vital role in QSAR \QSPR study, phisco-chemical properties and topological indices like Randic, atom-bond connec tivity(ABC) and geometric-arithmetic(GA) are used to diagnose the bioactivity of chemical compounds. Actually these topological indices is designed by transforming a chemical structure into a numeric number. In the area of research graph theory has been found to be very useful. In this thesis, we will compute the degree based topological indices the so-called general Randic index Rγ(G) for γ = {1, 12−1,−12 },atom bond connectivity(ABC), geometric-arithmetic(GA), first Zagreb, second Zagreb, second modified Zagreb, third zagreb, hyper Zagreb and augmented zagreb topological indices for subdivision of dif ferent kinds of networks like sierpinksi s(n,k) networks as well as molecular structural networks. This study will give us a very interesting and different results by using new topological index that is first Zagreb, second Zagreb, second modified Zagreb, third zagreb, hyper Zagreb and augmented zagreb topological indicesItem 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 5296Face 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.Item 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 5431Item 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 Topological Indices On Chemical Graphs(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2019) Tayyaba Razzaq; SP16-RMT-002; Dr. Muhammad Hussain; LHR TP 5345Topological 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 graphsItem Irregularities, Energies and Pattern of Bridge Chain Graph Network(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Faisal Javed; SP19-RMT-036; LHR TP 7348; Dr. Muhammad HussainThe graph ? consists of some vertex ?(?) are connected by ?(?) where the position of vertices does not matter. The degree of a graph vertex is basically a number of incident edges ?(?) and vertex ? can be denoted by du. The upper and lower bound of graph energy with minimal and maximal energy is abroad field of Spectral Graph theory. The energy of Graph ?(?) in term of vertices and number of edges in graph ? is involved in absolute value of eigenvalue adjacency matrix of G where λ is the spectral radius. The Graph has energy equal to ?(?) = ∑ |? ? ?=1 |. An irregular graph G is totally defined as ??𝑡(?) = |𝑢 − 𝑣|. Finally, we determine the energy, pattern and irregularities of chain like cyclic graph. Our exploring outputs will rely upon Size and Order of Graph, Degree of Graph, Irregular Graph, Multi Graph, Cyclic Graph, Wheel Graph, Bridge Graph, Prism Graph, Connected and Planarity Graph, and Concept of Distance and Degree in Graph.