EMERGING TOPICS ON DIFFERENTIALGEOMETRY AND GRAPH THEORY -- CONTENTS -- PREFACE -- APPLICATIONS OF GRAPH THEORY IN MECHANISMANALYSIS -- Abstract -- 1. Introduction -- 2 Graph Representation of Mechanisms -- 2.1. Topological Graph Representation of Kcs with Simple Joints -- 2.2. Bicolored Graph Representation of Kcs with Multiple Joints -- 2.3. Tricolored Graph Representation of Glms -- 2.4. Combinatorial Graph Representation of Glkcs -- 3. Detection of Isomorphism Among Kcs and Glkcs -- 3.1. Detection of Isomorphism Among Kcs -- (1) Theory to Detect Isomorphism among Kcs -- (2) Method to Detect Isomorphism among Kcs -- (3) Illustrations -- Example 1. Determination of Isomorphism of Kcs. -- Example 2: Determination of Isomorphism of Graphs -- 3.2. Detection of Isomorphism among Glkcs -- (1) Theory to Detect Isomorphism Among Glkcs -- (2) Method to Detect Isomorphism Among Kcs -- (3) Illustrations -- Example 1: Determination of Isomorphism of the Glkcs, as Shown in Fig. 10(A) and (B) -- Example 2: Determination of isomorphism of the GLKCs, as shown in Fig. 11(a) and (b). -- 4. Topology-Loop Characteristics of Kcs -- 4.1. The Number of Topological Loops of Bicolored Graph -- 4.2. The Number of Topological Loops of Tricolored Graph -- 5. Structural Decomposition of Mechanisms -- 5.1. Principle of Structural Decomposition -- 5.2. Calculation of Transformation Number -- 5.3. Types of Kinematic Units -- 5.4. Criteria of Choosing the Sequential Circuits -- 5.5. Examples of Structural Decomposition -- Example 1: Fig. 15(A) Shows A PLM with one DOF, and Fig. 15(B) the Weighted Graphand Fig. 15(C) the Decomposing Procedure. -- Example 2: Fig. 16(A) Shows a Hydraulic Mechanism with one DOF, and Fig. 16(B) theWeighted Bicolored Graph and Fig. 16(C) the Decomposing Procedure.

Example 3: Fig. 17(A) Shows a GLM with one DOF, and Fig. 17(B) the WeightedTricolored Graph and Fig. 17(C) the Decomposing Procedure. -- Example 4: Fig. 18 Shows a Complex GLM, its Tricolored Graph, and the DecomposingProcedure. -- 6. Conclusion -- References -- A CATEGORICAL PERSPECTIVE ON CONNECTIONSWITH APPLICATION IN THE FORMULATION OFFUNCTORIAL PHYSICAL DYNAMICS -- Abstract -- 1. Introduction -- 2. The Extension/Restriction of Scalars Categorical Adjunction -- 2.1. The Adjoint Pair of Extension/Restriction Functors -- 2.2. The Universal Object of Differential 1-Forms -- 2.3. The Notion of Connection -- 2.4. The Algebraic De Rham Complex and the Notion of Curvature -- 3. The Abstract Equivalent Monadic Notion of Connection -- 3.1. The Extension/Restriction of Scalars Monad-Comonad Pair -- 3.2. Categorical Monadic Reformulation of Connections -- 4. General Theory of Relativity from the ClassicalExtension/Restriction Monad-Comonad Pair -- 5. Functorial Physical Theories of Dynamics -- 5.1. Conceptual Analysis of Basic Principles -- 5.2. Topological Sheaf-Theoretic Field Dynamics and Abstract DifferentialGeometry -- 5.3. Quantum Functorial Dynamics from the Classical to Quantum Extension/Restriction Monad-Comonad Pair -- 6. Conclusion -- References -- THE NOTION OF CR HAMILTONIAN FLOWSAND THE LOCAL EMBEDDING PROBLEMOF CR STRUCTURES -- Abstract -- 1. Introduction -- 2. CR Structures and Its Deformation Theory -- 3. CR Hamiltonian Vector Fields -- 4. CR Hamiltonian Flows -- 5. On the Existence of the Solution -- 6. An Application to the Local Embedding Problem of CRStructures (Setting) -- 7. The Solution -- References -- EQUIVARIANT METHODS IN COMBINATORIALGEOMETRY -- Abstract -- 1. Introduction -- 2. Topological Methods -- 2.1. Equivariant Obstruction Theory -- 2.2. Some Equivariant Bits and Tricks.

2.3. Topology of the Complement of the Arrangement -- The Point Class -- The Broken Point Class -- 3. The Motivating Problem -- 3.1. The Formulation of the Partition Problem -- 3.2. The Configuration Space / Test Map Paradigm -- 3.3. The Equivariant Problem -- 3.4. Target Extension Scheme -- 4. Computations -- 4.1. The Q4n Simplicial and Cellular Structures on S3 -- 4.2. The Inverse Image of the Singular Set and oQ4n(f)(e) -- 4.3. Calculating Coinvariants H2(W (m−1)n \ [ A -- Z)Q4n -- 4.4. 4-Fan Partitions of Two Measures -- The obstruction cocycle for the general ( an, bn, cn , dn) case -- The Main Theorem for 4-Fans -- 4.5. 3-Fan Partitions of Measures -- In per Suite of the Hyper Arrangement J -- The Obstruction Cocycle -- The Homology H2(Wn\ [A(J , ) -- Z) and Coinvariants H2(Wn\ [A(J , ) -- Z)Q4v -- The Main Theorem for (a, a + b, b) Case of 3-Fans -- References -- SOMETHING NEW ABOUT RECONSTRUCTION -- Abstract -- 1. Introduction -- 2. Graph Reconstruction from Metric Balls -- 2.1. Definitions and Notations -- 2.2. Exact Reconstruction from the Metric Balls of Radius r = 2 -- 2.3. Exact Reconstruction from the Metric Balls of Radius r ≥ 2 -- 2.4. Open Questions and Conjectures -- 2.5. Connection with Reconstruction of Chemical Compounds -- 3. Vertex Reconstruction from Metric Balls -- 3.1. Definitions and Notations -- 3.2. Hamming and Jonson Graphs -- 3.3. Simple, Regular and Cayley Graphs: General Results -- 3.4. Cayley Graphs on the Symmetric Group -- 3.4.1. The Cayley Graphs on Symn Generated by Transpositions -- 3.4.2. The Cayley Graphs on Symn Generated by Reversals -- 3.5. Cayley Graphs on the Group Hyperoctahedral Group -- 3.5.1. The Cayley Graphs on Bn Generated by Transpositions -- 3.5.2. The Cayley Graphs on Bn Generated by Reversals -- 3.6. Open Questions and Conjectures -- 3.7. Connections with Some Other Problems -- 4. Conclusion.

References -- COALESCENCE OF GRAPHS -- Abstract -- 1. Introduction -- 2. Basic Equation -- 2.1. The Master Equation -- 2.2. Generating Functional and the Evolution Equation -- 3. Coalescence of Trees and Coagulation -- 4. Thermodynamic Limit for Random Graphs -- 5. Gelation Catastrophe and Giant Component -- 6. Exact Solution of Evolution Equation -- 7. Bipartite Graphs -- 8. Bipartite Graphs in the Thermodynamic Limit -- 9. Summary -- 10. Conclusion -- Appendix A. Polynomials Pg( ) -- Definition and Exponential Generating Function -- Integral Equation for the Generating Function -- Recurrences -- Recurrences for Derivatives of y(˘, ) -- Calculation of Ag -- Asymptotic Behavior of Pg( ) -- Concluding Comments -- Appendix B. Polynomials Pm,n( ) -- Definition and Exponential Generating Function -- Sum Rules and Recurrences -- Polynomials Fm,n(x) -- Generating Function and Integral Equations -- Asymptotic Behavior of Pm,n( ) -- References -- GRAPH ANALYSIS WITH APPLICATIONTO ECONOMICS -- Abstract -- 1. Introduction -- 2. Graph Theory and Small Size Applications -- 2.1. Graph Building and Embeddings -- 2.1.1. Preliminaries and Notations -- 2.1.2. Application to the CS Sub-model -- 2.1.3. Circular Embeddings and Condensation -- 2.2. Essential Graph Theory -- 2.2.1. All-Pair Shortest Paths Matrix -- 2.2.2. Eccentricity, Central and Peripheral Vertices, Cut-points -- 2.2.3. Rooted Embedding and Graph Traversals -- 2.3. Vertex Typology of a Maximal SCC -- 2.3.1. Vertex Typology -- 2.3.2. Structural Properties of the CS Sub-model -- 3. Further Graph Theory for Economic Models -- 3.1. Matching Theory -- 3.1.1. Introduction -- 3.1.2. Perfect Bipartite Matching -- Definitions -- Number of perfect matchings -- 3.1.3. Maximal Matching Solutions -- 3.1.4. Multiple Matchings -- 3.1.5. Finding All the Perfect Matchings -- 3.2. Connectivity Theory.

3.2.1. Components, Independency, Cuts and Connectivity -- 3.2.2. Menger's Theorem -- 3.2.3. Computational Methods -- 3.2.4. Connectivity in Weighted Graphs -- 3.3. Circuit Theory -- 3.3.1. Definitions -- 3.3.2. Enumeration Problems -- 3.3.3. Tarjan's Algorithm -- 3.3.4. Upper Bounds on Time and Space -- 3.3.5. Set of Circuits and Non-edge Disjoint Circuits -- 3.4. Dynamic Steady-State Graph -- 4. Graph Theory and Large-Size Applications -- 4.1. Graph Characteristics of a Large-Size Model -- 4.2. Connectivity of Large Graphs -- 4.2.1. Connectivity Properties -- 4.2.2. Depth - and Breadth-First Search -- 4.2.3. Thresholds of Connectivity -- 4.2.4. Vertex Typology -- 4.3. Circuit Enumeration of Large Graphs -- 4.3.1. Enumeration of the Circuits -- 4.3.2. Non-edge Disjoint Circuits -- 4.4. Dynamic Large Graphs -- 4.4.1. Short-run Dynamic Graph Structures -- 4.4.2. Long-run Dynamic Graph Structures -- A. Ford-Fulkerson Algorithm for Matching -- B. Matching Problem of the CS Sub-model -- C. Backtracking Procedure -- D. Graph Theory with Mathematica -- D.1. Introduction -- D.2. Klein-Goldberger Model -- D.2.1. Description of the Model -- D.2.2. Graph of the Model -- D.2.3. Block Triangularity -- D.3. Structure Analysis of the KG Model -- D.3.1. Graph Embeddings -- D.3.2. Strongly Connected Components -- D.3.3. DAGs of the Static and Dynamic Versions -- D.3.4. Lexicographically Ordered Circuits -- D.3.5. Edge-disjoint Cycles -- D.4. Vertex Typology of the KG Model -- D.4.1. All-Pairs Shortest Paths Matrix -- D4.1.1. Static KG Model -- D.4.2. D4.1.2. Dynamic KG Model -- D.4.3. Vertex Typology -- D.5. Conclusion -- E. List of Variables of the FAIR Model -- References -- APPLICATION OF HILBERT SPACES TO THESTABILITY STUDY OF FLOWS ON A SPHERE -- Abstract -- 1. Introduction -- 2. Spherical Harmonics -- 3. Geographical Coordinate Maps.

4. Orthogonal Projectors and Fractional Derivatives.