Graph Theory(I): استاذ الماده: أ.م.د. أكرم برزان عطار
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1 استاذ الماده: أ.م.د. أكرم برزان عطار Graph Theory(I): 1. An introduction to Graphs: The Definition of Graphs - Graphs as Models - Matrix Degree - Subgraphs- Paths and Cycles - the Matrix Representation of Graphs-Fusion. 2. Trees and Connectivity: Definitions and Simple Properties - Bridges-Spanning Trees - Connector Problems-Shortest Path Problems - Cut Vertices and Connectivity. 3. Euler Tours and Hamiltonian Cycles: Euler tours-the Chinese Postman Problem-Hamiltonian Graphs-The Traveling Salesman Problem. 4. Matchings: Matching and Augmenting-The Marriage Problem-The personal Assignment Problem-The Optimal Assignment Problem. 5. Planar Graphs: Plane and Planar Graphs-Euler's Formula-The platonic Bodies-Kuratowski's Theorem-Non Hamiltonian Planar Graphs -the Dual of a Planar Graph. 6. Colouring: Vertex Colouring - Vertex Colouring Algorithms-Critical Graphs-Cliques- Edge Colouring -Map Colouring. 7. Directed Graphs: Definitions-Indegree and Outdegree -Tournaments -Traffic Flow.
2 Graph Theory(II): 1. Traversability: Eulerian Graphs Hamiltonian Graphs 2. Line Graphs: Some properties of line graphs Characterization of line graphs Special line graphs Line graphs and traversability Title graphs 3. Factorization: 1-factrization 2-factrization Arboricity 4. Coverings: Coverings and independence Critical points and lines Line core and point core 5. Network: Flows and Cuts-The Ford and Fulkerson Algorithm-Separating Sets. 6. Ramsey Theorem: A party-a Generalization of the party Problem. 7. Reconstruction: The reconstruction conjecture Reconstruction of regular graphs and disconnected graphs Edge reconstruction The infinite.
3 Qualitative Study of OD.E (I): نظرية المعادالت التف اضلية مدرس المادة :ا.مزدزكمال حامد ياسر 1. Linear deferential equations: Linear stability of OD.E. Hyperbolic OD.E. Elliptic OD.E. Phase portrait of OD.E. Exponential stability. Exponential solution. 2. NonLinear deferential equations: Vector Fields 2. R Solutions of nonlinear OD.E on Existence and uniqueness of solutions. Local structure and global structure. Equivalence of phase portrait. Topological equivalence. 3. Grobman-Hartman Theorm: Stability of critical points. Classification of critical point. 4. First Integrals: Existence of first integrals. The main theorm of first integrals. The relation with Hartman theorm. 5. Limit Cycles: Definition of limit cycles. The types of limit cycles. Polar coordinate to find limit cycles. The main theorm. Paincare-Bendixon theorm.
4 Qualitative Study of Deferential Algebric Equations DAE'S (II): 1. Basic types of DAE'S Why DAE'S. Applications. Constrained variational problems. Networking modelings. Model reduction and singular problems. 2. Theory of DAE's: Solvability and the index. Linear constant coefficient DAE's. Linear time varying DAE's. 3. Nonlinear DAE's: Solvability and the index. Structural form. Index reduction and constraint stability. 4. Stability of critical point: Stability definition. Singular systems. Analytical solution of singular system. Regularity of DAE's. Stability of linear and nonlinear DAE 5. Stability Theory DAE's: Linear stability. Stability of nonlinear DAE's. Liapunov functions. Liapunove stability of DAE.
5 )اختياري(:( optional ) Additional Topics استاذ الماده: أ.م.د. أكرم برزان عطار Matroids Hereditary systems and examples. Properties of matroids. The span Function. The dual of a matroid. Matroid minors and planar graph. Matroid intersection. Matroid union. Random Graphs Existance and expectation. Propertis of almost all graphs. Threshold functions. Evolution graph parameters. Connectivity, cliques and coloring. Martingales. Eigenvalues of Graphs The characteristic polynomial. Linear algebraof real symmetric matrices. Eigenvalues and graph parameters. Eigenvalues of regular graphs. Eigenvalues and expanders. Strongly regular graphs.
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