Advances in Applied Combinatorics

Description

Advances in Applied Combinatorics talks about the subject of binomial coefficients, permutations, the combinational proofs, the graph theory, derangements, partitions, linear recurrences, graph algorithms and permutation groups, to give a far-fetched insight on applied combinatorics. This book also discusses about the fractional sums and the differences, harmonic numbers and the cubed binomial coefficients, the recursive algorithms, linear recurrences and the fibonacci numbers. The generating functions and the sequence of numbers and polynomials. Cover Title Page Copyright DECLARATION ABOUT THE EDITOR TABLE OF CONTENTS List of Contributors List of Abbreviations Preface SECTION 1: BINOMIAL COEFFICIENTS, PERMUTATIONS AND COMBINATORIAL PROOFS Chapter 1 Fractional Sums and Differences with Binomial Coefficients Abstract The Fractional Differences And Sums With Binomial Coefficients Conclusion References Chapter 2 The Identical Estimates of Spectral Norms for Circulant Matrices with Binomial Coefficients Combined with Fibonacci Numbers and Lucas Numbers Entries Abstract Introduction Preliminaries The Identities of Estimations For Spectral Norms Numerical Examples Conclusion Acknowledgments References Chapter 3 Harmonic Numbers and Cubed Binomial Coefficients Abstract Introduction Integral Representations And Identities References Chapter 4 A Generalization of a Combinatorial Identity by Chang and Xu Abstract Introduction And Main Results Proofs And Auxiliary Results An Alternative Proof Of Theorem 1 Acknowledgments References SECTION 2: GRAPH THEORY AND PARTIALLY ORDERED SETS Chapter 5 Total Dominator Chromatic Number of Paths, Cycles and Ladder Graphs Abstract Introduction Main Results References Chapter 6 Modular Leech Trees of Order at Most 8 Abstract Introduction Taylor’s Condition For Modular Leech Trees Computational Results References Chapter 7 Recursive Algorithms for Phylogenetic Tree Counting Abstract Background Serial Sampling Constraints Conclusions Appendix 1: Algorithm For Counting Frs Trees Appendix 2: Algorithm For Counting Fully Ranked Resolutions of A Fully Ranked Constraint Tree Acknowledgements Authors’ Contributions References SECTION 3: DERANGEMENTS AND THE EULER’S TOTIENT Chapter 8 A Note on Some Identities of Derangement Polynomials Abstract Introduction Some Identities Of Derangement Polynomials Arising From Umbral Calculus Results And Discussion Conclusion Acknowledgements Authors’ Contributions References SECTION 4: PARTITIONS AND GENERATING FUNCTIONS Chapter 9 A Rademacher Type Formula for Partitions and Overpartitions Abstract Background A Common Generalization A Proof of Theorem 2.1 Acknowledgments References Chapter 10 On The Exponential Generating Function For Non-Backtracking Walks Abstract Introduction Exponential Generating Function For Undirected Graphs. Exponential Generating Function For Directed Graphs. Computing The Centrality Vectors. Block Matrix Interpretations Star Graph Analysis Numerical Tests Summary Acknowledgements References SECTION 5: LINEAR RECURRENCES AND THE FIBONACCI NUMBERS Chapter 11 On Sequences of Numbers and Polynomials Defined by Linear Recurrence Relations of Order 2 Abstract Introduction Main Results And Examples Identities Constructed From Recurrence Relations Solutions Of Algebraic Equations and Differential Equations Acknowledgments References Chapter 12 On The Partial Finite Sums of the Reciprocals of the Fibonacci Numbers Abstract Introduction Reciprocal Sum Of The Fibonacci Numbers Reciprocal Square Sum Of The Fibonacci Numbers Acknowledgements References SECTION 6: GRAPH ALGORITHMS Chapter 13 Shortest Augmenting Paths for Online Matchings on Trees Abstract Introduction Related Work Preliminaries Shortest Paths On Trees Playing Against An Adversary References Chapter 14 Subgraph-augmented Path Embedding for Semantic User Search on Heterogeneous Social Network Abstract Introduction Related Work Problem Formulation S-Path Construction S-Path Embedding End-To-End Training Experiments Conclusion Acknowledgments References Chapter 15 A Hybrid Optimized Weighted Minimum Spanning Tree for the Shortest Intrapath Selection in Wireless Sensor Network Abstract Introduction Problem Formulation Methodology Experimental Set-Up And Results Conclusion References SECTION 7: PERMUTATION GROUPS Chapter 16 The Commuting Graph of the Symmetric Group Sn Abstract Groundwork And Main Result Commuting Elements of Sn The Commuting Graph ΔRn The Commuting Graph ΔRn∪Cnn For Even N An Upper Bound on The Diameter of ΔSn For Composite N and N - 1 The Existence Of Elements At Distance 5 In ΔSn Proof Of Main Res References Chapter 17 Modeling Quantum Behavior in the Framework of Permutation Groups Abstract Introduction Formalism Of Quantum Mechanics Emergence Of Geometry Within Large Hilbert Space Via Entanglement .304 Constructive Modification Of Quantum Formalism Modeling Quantum Evolution Summary References Index Back Cover