Browsing by Author "Dr. Muhammad Hussain"

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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.
Distance and Eccentricity Based Invariants of Certain Networks, Nanotubes, Nanotori and Fullerene
(Library Information Services, CUI Lahore, 2021) Haseeb Ahmad; FA17-PMT-004/; Dr. Muhammad Hussain
Topological indices are numerical parameters to associate with graphs of networks, nanotube, nanotri etc and help us to understand the properties of concerned structures. These numbers remain invariant upto graph isomorphism. Like topological indices, algebraic polynomials are also important invariants to understand the topology of underlined structures and help us to recover many topological indices directly. In this thesis, we computed Hosoya, Harary and eccentricity connectivity polynomials of triangular oxide, regular triangular oxide, silicate, regular silicate networks, TUC nanotori and fullerenes. Moreover, Wiener, modified Wiener, hyper-Wiener, multiplicative Wiener, Harary and eccentric indices of above mentioned structures are also computed in this thesis. Our results may help to develop a better understanding about the nanomaterials and can be used to enhance the ability of existing nanostructures/networks.
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.
Fractional Metric Dimension Of Different Chemical
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2020) Ahmad Kaleem; FA18-RMT-052; LHR TP 6458; Dr. Muhammad Hussain
Graph Energies of Generalized Chain on Wheel Like Graphs
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2022) Muhammad Shahzad Khalid; FA20-RMT-016; LHR TP 7920; Dr. Muhammad Hussain
The energy of a graph G is the absolute sum of all eigen-values of its adjacency matrix A(G). Eigen-values are used to calculate the limit of how much data may be sent via a communication channel such as a phone line or the air. Moreover energy which are the absolute sum of all eigen-value are also use in various field's especially in computer networking to transmit data. Here we calculate and generalize graph energy of different families of graph.
Graph Energies on Bridge Chain for Composition of Graphs
(Library Information Services, CUI Lahorez v, 2023) Ambar Munir; FA20-RMT-008; Dr. Muhammad Hussain
The graph𝐺 consists of some vertex 𝑉(𝐺) are connected by 𝐸(𝐺) wherethe position of vertices does not matter. The degree of a graph vertex is basically a number of incident edges 𝐸(𝐺) and vertex 𝑢 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 𝐸(𝐺)in term of vertices and number of edges in graph 𝐺 is involvedin absolute value of eigenvalue adjacency matrix of G where λis thespectral radius. The Graph has energy equal |.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.
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 Hussain
The 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.
Local Fractional Metric Dimension of Different Chemical Structures
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2020) MUHAMMAD ZULQARNAIN; FA18-RMT-078; LHR TP 7012; Dr. Muhammad Hussain
In this thesis we find out different chemical structures and then convert them into planar network graphs. Consequently, to calculate the local fractional metric dimension of these network graphs. We are also interested to extend the proposal to the rooted product of different families of graphs. We also find local metric dimension and local fractional metric dimension on the line graph of subdivided honeycomb network.
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
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
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.
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 7406
This 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.
Tiplogical industries on line graphic of cellulose network
(Library Information Services, CUI Lahore, 2021) Muhamma Awais Iqbal; UR43/TOV/234; Dr. Muhammad Hussain
Topological index is a numeric parameter which describe some silent features (boil- ing point, stability and melting point, heat of formation and freezing point etc.) 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 molecular descriptor. These molecular descriptors are widely used in the study of Quantitative structure-activity relationships(QSARs) that provide insight into animate effects based on chemical structures.We’re going to investigate degree based topological indices such as Randic index, Sum connectivity index, Harmonic index, First Zagreb, second zagreb and third zagreb in- dices, Augmented zagreb index, Hyper zagreb index, Atom bond connectivity index
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 9572
Chemical 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.
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
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.
Zagreb Connection Numbers of Different Chemical Graphs
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2020) ASAD ZUBAIR; FA18-RMT-051; LHR TP 6029; Dr. Muhammad Hussain
Topological indices are real numerical parameters of a molecular structural networks 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 connectivity(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 connection based topological indices the so-called first Zagreb connection index, second Zagreb connection index and first modified Zagreb connection index of Honeycomb Network, Linegraph of Honeycomb Network, Line graph of subdivision of Honeycomb Network Triangular Benezenoid Network,line graph of sub-division of Triangular Benezenoid Network and Hourglass system. This study will give us a very interesting and different results by using new topological index that is first Zagreb, second Zagreb, first modified Zagreb connection index.