Browsing by Author "Dr. Muhammad Hussain, Assistant Profesor [Supervisor]"

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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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    Metric Dimension Of Different Chemical Structures
    (COMSATS University Islamabad, Lahore Campus Library Information Services, CUI Lahore, 2017) Aqsa Farooq,; FA17-RMT-029; Dr. Muhammad Hussain, Assistant Profesor [Supervisor]; LHR TP 5774
    Let G = (V,E) be a connected graph and distance between any two vertices m and n in G is (m−n) geodesic and is denoted by d(m,n). A set of vertices S resolves a graph G if each vertex is uniquely determined by its vector of distances to the vertices in S. A metric dimension of G is the minimum cardinality of a resolving set of G. In this thesis line graph of honeycomb network , azteic diamond,polycyclic aromatic hy drocarbon,triangualr benzenoid,honeycomb cup network has been derived and than calculate the metric dimension on line graph of Honeycomb network, azteic diamond ,polycyclic aromatic hydrocarbon, triangualr benzenoid and honeycomb cup network .