Multi-shell Polyhedral Clusters (eBook)

Prof. Dr. Mircea Vasile Diudea is a  Professor of Mathematical Chemistry and Nanoscience at Babes-Bolyai University, Faculty of Chemistry and Chemical Engineering, Cluj, Romania.  Prof. Diudea has edited/authored 11 books:   1. A. R. Ashrafi and M. V. Diudea (Eds), Distance, Symmetry, and Topology in Carbon Nanomaterials, Springer Intl. Pub. Switzerland 2016. 2. M. V. Diudea and C. L. Nagy (Eds), Carbon Materials: Chemistry and Physics. Vol. 6, Diamond and Related Nanostructures. Springer, Dordrecht, 2013. 3. M. V. Diudea (Ed), Nanostructures, Novel Architecture, Nova, New York, 2005. 4. M. V. Diudea, (Ed), QSPR/QSAR Studies by Molecular Descriptors, Nova, New York, 2001. 5. M. V. Diudea, Nanomolecules and Nanostructures - Polynomials and Indices, MCM, No. 10, Univ. Kragujevac, Serbia, 2010. 6. M. V. Diudea and Cs. L. Nagy, Periodic Nanostructures, Springer, 2007. 7. M. V. Diudea, M. S. Florescu, and P. V. Khadikar, Molecular Topology and Its Applications, Eficon, Bucharest, 2006. 8. M.V. Diudea, I. Gutman and L. Jäntschi, Molecular Topology, Nova, New York, 2002. 9. M.V. Diudea and O. Ivanciuc, Molecular Topology, COMPREX, Cluj, 1995 (in Romanian). 10.   M. V. Diudea ; M. Pitea; M. Butan, Fenothiazines and structurally related drugs. DACIA, Cluj, 1992 (in Romanian)., 278p. 11.   M. V. Diudea, S. Todor and A. Igna, Aquatic Toxicology. DACIA, Cluj, 1986 (in Romanian).

Preface 6
Acknowledgements 9
Contents 10
List of Abbreviations 13
Chapter 1: Basic Chemical Graph Theory 14
1.1 Basic Definitions in Graphs 14
1.2 Topological Matrices and Indices 18
1.2.1 Adjacency Matrix 18
1.2.2 Distance Matrix 19
1.2.3 Detour Matrix 20
1.2.4 Combinatorial Matrices 21
1.2.5 Wiener Matrices 21
1.2.6 Cluj Matrices 22
1.2.7 Distance-Extended Matrices 24
1.2.8 Walk Matrices 25
1.2.9 Reciprocal Matrices 27
1.2.10 Layer and Shell Matrices 27
1.2.10.1 Layer Matrices 27
1.2.10.2 Shell Matrices 28
1.2.10.3 Centrality Index 30
1.3 Topological Symmetry 31
References 32
Chapter 2: Operations on Maps 35
2.1 Dual d 35
2.2 Medial m 37
2.3 Truncation t 38
2.4 Polygonal Mapping pn 38
2.5 Snub s 39
2.6 Leapfrog l 39
2.7 Quadrupling q 40
2.8 Septupling sn 41
Chapter 2 Atlas: Single Shell Clusters 43
References 46
Chapter 3: Definitions in Polytopes 48
3.1 Polyhedra 48
3.2 n-Dimensional Structures 51
3.3 Abstract Structures 54
3.3.1 Posets 55
3.3.2 Vertex Figure 56
3.3.3 Abstract Polytope 57
3.4 Polytope Realization 58
3.4.1 P-Centered Clusters 58
3.4.2 Cell-in-Cell Clusters 60
3.4.3 24-Cell and Its Derivatives 62
References 64
Chapter 4: Symmetry and Complexity 66
4.1 Euler Characteristic 67
4.2 Topological Symmetry 68
4.3 Centrality Index 69
4.4 Ring Signature Index 70
4.4.1 Ring Signature in a Translational Network 72
4.4.2 Ring Signature in Spongy Structures of Higher Rank 74
4.4.3 Ring Signature in Spongy Hypercubes 77
4.4.4 Truncation Operation 77
4.5 Pairs of Map Operation 80
References 85
Chapter 5: Small Icosahedral Clusters 87
5.1 Small Cages: Source of Complex Clusters 87
5.2 Truncated MP Icosahedral Clusters 88
5.3 Clusters by Medial Operation 89
5.4 Clusters of Higher Rank 90
Chapter 5 Atlas: Small Icosahedral Clusters 96
References 134
Chapter 6: Large Icosahedral Clusters 135
6.1 Small Complex Clusters 135
6.2 Icosahedral Clusters Derived from the C45 Seed 137
6.3 Clusters of Dodecahedral Topology 142
6.4 Clusters of Icosahedral Topology 143
6.5 Rhomb Decorated Clusters 145
Chapter 6 Atlas: Large Icosahedral Clusters 147
References 195
Chapter 7: Clusters of Octahedral Symmetry 197
7.1 Small Clusters as Seeds for Complex Structures 197
7.2 Clusters Decorated by Octahedra 198
7.3 Clusters Decorated by Dodecahedra 199
7.4 Rhomb Decorated Octahedral Clusters 200
7.5 Cubic Net Transforming 203
Chapter 7 Atlas: Octahedral Clusters 205
References 255
Chapter 8: Tetrahedral Clusters 256
8.1 Small Tetrahedral Clusters 256
8.2 Tetrahedral Clusters of Higher Rank 257
8.3 Tetrahedral Clusters Derived From Ada20 257
8.4 Tetrahedral Hyper-structures Decorated with Only Dodecahedra 260
Chapter 8 Atlas: Tetrahedral Clusters 262
References 288
Chapter 9: C60 Related Clusters 290
9.1 Structures Derived from the Cluster P32@dC60.33 290
9.2 Stellated Clusters 291
9.3 C750 Related Structures 293
9.3.1 Duals of C750 and Related Structures 296
9.3.2 Medials of C750 and Related Clusters 296
9.3.3 Truncated C750 and Related Clusters 299
Chapter 9 Atlas: C60 Related Structures 300
References 343
Chapter 10: Chiral Multi-tori 344
10.1 Design of Chiral Multi-tori 344
10.2 Dodecahedron Related Structures 347
10.3 Cube Related Structures 349
10.4 Tetrahedron Related Structures 350
10.5 C60 Related Structures 351
Chapter 10 Atlas: Chiral Multi-tori 356
Dodecahedron Related Structures 356
References 371
Chapter 11: Spongy Hypercubes 372
11.1 Simple Toroidal Hypercubes 372
11.2 Complex Toroidal Hypercubes 374
11.3 Tubular Hypercubes 375
11.4 Spongy Hypercubes 376
11.5 Truncation of Hypercube 379
11.6 Counting Polynomials in Hypercubes 381
11.6.1 Omega Polynomial 381
11.6.1.1 Omega Polynomial in Hypercubes 382
11.6.1.2 Omega Polynomial in Tubular Hypercubes 382
11.6.1.3 Omega Polynomial in Spongy Hypercubes 384
11.6.2 Cluj Polynomials 386
11.6.2.1 Cluj Polynomials in Hypercubes 387
11.6.2.2 Cluj Polynomials in Toroidal Hypercubes 387
11.6.2.3 Cluj Polynomials in SpongyHypercubes 388
References 392
Chapter 12: Energetics of Multi-shell Clusters 394
12.1 Introduction 394
12.2 C20 Aggregation 395
12.3 Hyper-graphenes by D5 Substructures 398
12.4 Hyper-graphenes by C60 Units 398
12.5 C60 Aggregates with Tetrahedral and Icosahedral Symmetry 402
12.6 C60 Network by [2+2] Cycloaddition 405
12.7 Computational Methods 409
Chapter 12 Atlas: Energetics of Multi-shell Clusters 410
References 445
Index 448