Graph theory and theoretical physics:
Gespeichert in:
| Weitere Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
London [u.a.]
Academic Press
1967
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| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XV, 358 Seiten graph. Darst. |
Internformat
MARC
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| 245 | 1 | 0 | |a Graph theory and theoretical physics |c edited by Frank Harary, University of Michigan, Ann Arbor, U.S.A. |
| 264 | 1 | |a London [u.a.] |b Academic Press |c 1967 | |
| 300 | |a XV, 358 Seiten |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 650 | 4 | |a Graphes, Théorie des | |
| 650 | 7 | |a Natuurkunde |2 gtt | |
| 650 | 4 | |a Physique mathématique | |
| 650 | 4 | |a Mathematische Physik | |
| 650 | 4 | |a Physik | |
| 650 | 4 | |a Graph theory | |
| 650 | 4 | |a Mathematical physics | |
| 650 | 0 | 7 | |a Graphentheorie |0 (DE-588)4113782-6 |2 gnd |9 rswk-swf |
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| 700 | 1 | |a Harary, Frank |d 1921-2005 |0 (DE-588)117711330 |4 edt | |
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Datensatz im Suchindex
| _version_ | 1804116376423497728 |
|---|---|
| adam_text | Contents
DEDICATION v
CONTRIBUTORS vii
PREFACE ix
1. Graphical Enumeration Problems
FRANK HARARY
I. On the terminology of graph theory 1
II. P61ya s enumeration theorem • 5
III. Operations on permutation groups 8
IV. Configuration counting by permutation groups • • • • 13
V. Unsolved graphical enumeration problems III • ¦ 16
A. Unsolved problems II 16
B. Unsolved problems III 29
VI. How many animals are there? 33
References 38
2. Graph Theory and Crystal Physics
P. W. KASTELEYN
I. The graph theoretical nature of some problems in crystal physics • 44
II. Basic concepts ••¦¦¦•••¦¦ 50
A. Graphs and figures 50
B. Lattice graphs ......... 54
C. Associated graphs 55
D. Generating functions •••¦•••• 56
HI. The random walk problem 59
A. Introduction 59
B. The symbolic method 59
C. The recursion method 61
D. The matrix method 63
E. Walks, circuits and elementary circuits 68
F. Stochastic walks 73
IV. The self avoiding walk problem 74
A. Introduction 74
B. A relation between Hamilton circuits and Euler circuits ¦ • 75
C. A relation between Euler circuits and rooted trees 76
D. The enumeration of rooted trees 78
E. The enumeration of Hamilton circuits 79
F. Application to lattice graphs 81
G. Unsolved problems • • • • 83
V. The dimer problem and the Ising problem 84
A. Introduction 84
xi
Xii CONTENTS
B. Dimer coverings and Pfaffians ...... g4
C. Pfaffians as generating functions of dimer coverings • ¦ 89
D. The construction of an admissible orientation 92
E. Application to lattice graphs ••••••• 95
F. The dimer problem for non planar graphs 97
G. The Ising problem 100
VI. Concluding remarks ........ 106
References 108
3. Graph Theory Applied to Electrical Networks
PETER R. BRYANT
I. Introduction • ¦ ¦ ¦ ¦ • • • • 111
II. The node branch incidence matrix Aa ¦ ¦ • • ¦ 112
III. Loops and the loop matrix ¦ • ¦ • • • 113
IV. Trees and cotrees 114
V. Fundamental sets of loops ¦ • ¦ • • • ¦ 117
VI. Loop space and independent loops • • • ¦ • 118
VII. Cuts and the cut matrix 119
VIII. Fundamental sets of cuts • • ¦ • ¦ ¦ • 121
IX. Cut space and independent cuts • • ¦ ¦ ¦ • 122
X. Cut sets 122
XI. Orthogonality of cut space and loop space • • • • ¦ 123
XII. Electric networks 124
XIII. Algebraic statement of Kirchhoffs laws ..... 126
XIV. Algebraic statement of the element laws • • ¦ ¦ 127
XV. Loop currents and nodal or cut potentials • • • • 128
XVI. Networks with generators ¦ ¦ • • • ¦ ¦ 129
XVII. Loop analysis and nodal analysis • • • • • 130
XVIII. Kirchhoffs and Maxwell s rules 130
XIX. The forms of det P(p) and det Q(p) 133
XX. The multiport synthesis problem • • ¦ • • 134
References ¦ • • • ¦ • • • • • ¦ 136
4. The Decomposition of Complete Graphs into Planar
Subgraphs
LOWELL W. BEINEKE
I. Introduction ..¦••••¦•• 139
II. Lower bounds 140
III. A special matrix ......... 141
IV. The main construction ........ 143
V. Adding three vertices ¦ • • • ¦ • • 147
VI. The thickness of Ki5 150
VII. The thickness of K28 150
VIII. The thickness theorem 153
References ........... 153
CONTENTS xiii
5. Some Classes of Perfect Graphs
CLAUDE BERGE
I. Introduction ¦ • ¦ • • ¦ • ¦ • 155
II. Basic lemmas • • • • ¦ • • • • 156
III. Comparability graphs • • • • ¦ • ¦ 157
IV. Triangulated graphs • ¦ • • • • • 159
V. Unimodular graphs • • • • • • ¦ • 162
References ........... 165
6. Graphs and Matrices
A. L. DULMAGE AND N. S. MENDELSOHN
I. Directed graphs and matrices • ¦ • • • ¦ 167
A. Definitions, notation and background • • ¦ • 167
B. Imprimitive matrices • ¦ • ¦ ¦ ¦ ¦ 172
C. The structure of powers of a non negative reducible matrix • 182
II. The canonical decomposition of a bipartite graph • • ¦ 190
A. The canonical decomposition ...... 190
B. An algorithm for the inversion of sparse matrices • • • 195
C. Dual solutions of the optimal assignment problem • • 198
III. Exponents of primitive matrices ...... 203
A. Gaps in the exponent set of primitive matrices ¦ • • 203
B. Incidence matrix of a finite projective plane ¦ • ¦ 218
References ........... 225
7. Estimation Methods for Mayer s Graphical Expansions
J. GROENEVELD
I. Introduction and summary ....... 230
II. The general procedure • • 231
A. The Mayer expansions ¦ • • • • • 231
B. Approximation methods ....... 232
C. First technique: simplification of configurational integrals • 232
D. The conditional convergence of the Mayer expansions • • 233
E. Second techniques: f h expansions ..... 233
F. The two techniques combined ...... 234
G. Inversion of the problem 234
H. Summary •••••¦•••• 235
III. An ideal gas law estimation ....... 235
A. Introduction 235
B. Definitions and notations • • 235
C. Recurrence relations .•••¦¦¦• 237
D. f h Expansion ......... 237
E. Inequalities ......... 237
F. Discussion •¦•¦¦••••• 238
IV. A Cayley tree estimation, Part I: The pressure and distribution
functions • 239
A. Introduction .....•••• 239
B. Definitions and notations ....... 239
xiv CONTENTS
C. Recurrence relations ........ 242
D. Corollaries 243
E. f h Expansion ...••¦¦•• 245
F. Auxiliary functions and inequalities ..... 246
G. Results 248
H. Corollaries 249
V. A Cayley tree estimation, Part II: The pair correlation function • 249
A. Introduction ......... 249
B. Definitions and notations ....... 250
C. Recurrence relations • • ¦ ¦ ¦ • ¦ 251
D. f h Expansion ......... 252
E. Auxiliary functions and inequalities ..... 253
F. Results 254
G. Corollaries 255
H. Discussion ¦•¦•••¦•¦¦ 256
VI. General discussion .••••¦•¦• 256
References 259
8. Enumerating Labelled Trees
JOHN W. MOON
I. Introduction ... ••¦¦•¦ 261
II. The number of trees with a given degree sequence ¦ • • • 262
III. The number of trees containing a given acyclic graph • • • 265
IV. The complexity of a graph ¦¦¦••••¦ 266
References ........... 271
9. Some applications of a Theorem of De Bruijn
RONALD C. READ
I. The problem •¦•••••••• 273
II. The theorem 274
III. Applications .¦¦••¦•••• 275
IV. Two specific applications •¦•¦¦•¦• 276
V. Further applications ¦••••¦•¦• 278
References 279
10. Generating Functionals and Graphs
GEORGE STELL
I. Introduction • ¦ ¦ ¦ • • ¦ ¦ • 281
II. The introduction of graphs and functionals into statistical physics • 282
III. The formalism 283
IV. Some inversion problems and their relation to statistical mechanics • 287
V. Appendix. A summary of related uses of graphs and functionals • 295
References 297
11. Topics in Graph Theory
W. T. TUTTE
I. Perfect squared squares ........ 301
CONTENTS XV
II. The enumeration of planar maps ...... 303
III. Matroids 308
References ........... 312
12. The Heawood May Colouring Conjecture
J. W. T. YOUNGS
I. Introduction • • • • • • • • ¦ 313
II. Preliminaries ¦ • • ¦ • • • • • 318
III. The manifolds M_and M* ¦_ 320
IV. Construction of M and skeleton S 325
V. Quotient manifold and quotient graph ..... 334
VI. Examples 336
VII. The converse proposition •••••••• 338
VIII. Applications 343
IX. Remarks on vortices ••••¦•••• 345
References ........... 353
Author Index ........... 355
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| spelling | Graph theory and theoretical physics edited by Frank Harary, University of Michigan, Ann Arbor, U.S.A. London [u.a.] Academic Press 1967 XV, 358 Seiten graph. Darst. txt rdacontent n rdamedia nc rdacarrier Grafentheorie gtt Graphes, Théorie des Natuurkunde gtt Physique mathématique Mathematische Physik Physik Graph theory Mathematical physics Graphentheorie (DE-588)4113782-6 gnd rswk-swf Theoretische Physik (DE-588)4117202-4 gnd rswk-swf Theoretische Physik (DE-588)4117202-4 s Graphentheorie (DE-588)4113782-6 s DE-604 Harary, Frank 1921-2005 (DE-588)117711330 edt HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001262144&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Graph theory and theoretical physics Grafentheorie gtt Graphes, Théorie des Natuurkunde gtt Physique mathématique Mathematische Physik Physik Graph theory Mathematical physics Graphentheorie (DE-588)4113782-6 gnd Theoretische Physik (DE-588)4117202-4 gnd |
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| title | Graph theory and theoretical physics |
| title_auth | Graph theory and theoretical physics |
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| title_full | Graph theory and theoretical physics edited by Frank Harary, University of Michigan, Ann Arbor, U.S.A. |
| title_fullStr | Graph theory and theoretical physics edited by Frank Harary, University of Michigan, Ann Arbor, U.S.A. |
| title_full_unstemmed | Graph theory and theoretical physics edited by Frank Harary, University of Michigan, Ann Arbor, U.S.A. |
| title_short | Graph theory and theoretical physics |
| title_sort | graph theory and theoretical physics |
| topic | Grafentheorie gtt Graphes, Théorie des Natuurkunde gtt Physique mathématique Mathematische Physik Physik Graph theory Mathematical physics Graphentheorie (DE-588)4113782-6 gnd Theoretische Physik (DE-588)4117202-4 gnd |
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