Computers and discovery in algebraic graph theory

Linear Algebra and Its Applications

Volumn 356, Issue 1-3, 2002, Pages 211-230

  Hansen, Pierre ^a   Mélot, Hadrien ^b  

^a HEC MONTRÉAL   (Canada)

^b UNIVERSITY OF MONS   (Belgium)

Author keywords

Computer; Conjecture; Discovery; Graph; Proof; Refutation

Indexed keywords

ALGEBRA; COMPUTERS;

CONJECTURE; DISCOVERY; GRAPH; PROOF; REFUTATION;

GRAPH THEORY;

EID: 84969174978     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0024-3795(02)00386-5     Document Type: Article

References (130)

1. M. Aouchiche, G. Caporossi, P. Hansen, Variable Neighborhood Search for Extremal Graphs 8, Variations on Graffiti 105, Congressus Numer. 148 (2001) 129-144
2. K Appel, and W Haken Every planar map is four colorable. Part I. Discharging Illinois J. Math. 21 1977 429 490
3. K Appel, and W Haken Every planar map is four colorable. Part II. Reducibility Illinois J. Math. 21 1977 491 567
4. K Appel, and W Haken Every planar map is four colorable Contemp. Math. 98 1989 1 741
5. Archimedes The method of mechanical theorem proving cited by J Gray in "Sale of the Century?" Math. Intelligencer 21 1999 12 15
6. K.T Balińska, D Cvetković, M Lepović, and S.K Simić There are exactly 150 connected integral graphs up to 10 vertices Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. 10 1999 95 105
7. K.T. Balińska, M. Kupczyk, S.K. Simić, K.T. Zwierzyński, On generating all integral graphs on 11 vertices, Tech. Rep. 469, Computer Science Center, Techn. Univ. of Poznań, 1999/2000
8. K.T. Balińska, M. Kupczyk, K.T. Zwierzyński, Methods of generating integral graphs, Technical Report 457, Computer Science Center, Techn. Univ. of Poznań, 1997
9. K.T. Balińska, M. Lepović, S.K. Simić, The nonregular, bipartite, integral graphs with maximum degree four. Part II: computer search, in preparation
10. K.T Balińska, and S.K Simić The nonregular, bipartite, integral graphs with maximum degree four. Part I: basic properties Discrete Math. 236 2001 13 24
11. C.A Barefoot, R.C Entringer, and A.P Mullhaupt Computer based proofs by induction in graph theory - A house of cards? Math. Comput. Modelling 17 11 1993 17 23
12. N.S Barghouti, J.M Mocenigo, and L Wenke Grappa: A GRAPh PAckage in Java Lecture Notes in Comput. Sci. 1353 1997 336 343
13. M Behzad, G Chartrand, and L Lesniak-Foster Graphs and Digraphs 1979 Prindle, Weber & Schmidt Boston
14. D Berque, R Cecchini, M Goldberg, and R Rivenburgh The SetPlayer system for symbolic computation on power sets J. Symbolic Comput. 14 1992 645 662
15. J.W Berry Improving discrete mathematics and algorithms curricula with LINK SIGCSE Bulletin (ACM Special Interest Group on Computer Science Education) 29 1997 14 20
16. J.W Berry LINK: A software system for experimentation with graphs and hypergraphs SIAM Discrete Math. Newsletter 2 7 1997
17. J.W Berry, N Dean, M.K Goldberg, G.E Shannon, and S Skiena Graph drawing and manipulation with LINK Lecture Notes in Compt. Sci. 1353 1997 425 437
18. J.W Berry, N Dean, M.K Goldberg, G.E Shannon, and S Skiena LINK: A system for graph computation Software - Practice and Experience 1999
19. T.L Brewster, M.J Dinneen, and V Faber A computational attack on the conjectures of Graffiti: New counterexamples and proofs Discrete Math. 147 1994 35 55
20. R.C Brigham, and R.D Dutton Ingrid: A software tool for extremal graph theory research Congr. Numera. 39 1983 337 352
21. R.C Brigham, and R.D Dutton A compilation of relations between graph invariants Networks 15 1985 73 107
22. R.C Brigham, and R.D Dutton A compilation of relations between graph invariants Networks 21 Suppl. 1 1991 421 455
23. R.C Brigham, R.D Dutton, and F Gomez INGRID: A graph invariant manipulator J. Symbolic Comput. 7 1989 163 177
24. G Brinkmann Fast generation of cubic graphs J. Graph Theory 23 1996 139 149
25. G Brinkmann Isomorphism rejection in structure generation programs P Hansen, P Fowler, M Zheng, Discrete Mathematical Chemistry DIMACS Series on Discrete Mathematics and Theoretical Computer Science vol. 51 2000 AMS Providence, RI 25 38
26. G Brinkmann, and A.W.M Dress A constructive enumeration of fullerenes J. Algorithms 23 1997 345 358
27. CaGe. The chemical and abstract environment. Available from Homepage: http://www.mathematik.uni-bielefeld.de/∼CaGe/
28. M Capobianco, and J.C Molluzzo Examples and Counterexamples in Graph Theory 1978 North Holland New York
29. G Caporossi, D Cvetkovic, I Gutman, and P Hansen Variable neighborhood search for extremal graphs 2. Finding graphs with extremal energy J. Chem. Inf. Comp. Sci. 39 1999 984 996
30. G Caporossi, A.A Dobrynin, P Hansen, and I Gutman Trees with palindromic Hosoya polynomials Graph Theory Notes NY 37 1999 10 16
31. P.W. Fowler, P. Hansen, G. Caporossi, A. Soncini, Polyenes with maximum HOMO-LUMO gap (Variable Neighbourhood Search for Extremal Graphs 7), Chem. Phys. Lett. 342 (2001) 105-112
32. G Caporossi, I Gutman, and P Hansen Variable neighborhood search for extremal graphs 4. Chemical trees with extremal connectivity index Comput. Chem. 23 1999 469 477
33. G. Caporossi, P. Hansen, Variable Neighborhood Search for Extremal Graphs 5. Three ways to automate finding conjectures, Discrete Math. (to appear)
34. G. Caporossi, P. Hansen, Finding relations in polynomial time, in: Proceedings of the XVIth International Joint Conference on Artificial Intelligence (IJCAI), vol. 2, Stockholm, 1999
35. G Caporossi, and P Hansen Variable neighborhood for extremal graphs 1. The system AutoGraphiX Discrete Math. 212 2000 29 44
36. Y Carbonneaux, J.-M Laborde, and M Madani Cabri-graphs: a tool for research and teaching in graph theory Lecture Notes in Computer Science vol. 1027 1995 Springer Berlin 123 127
37. G Chartrand, H.V Kronk, and C.E Wall The point arboricity of a graph Isr. J. Math. 6 1968 169 175
38. P.Z Chinn Disovery-method teaching in graph theory Ann. Discrete Math. 55 1993 375 384
39. F Chung The average distance is not more than the independence number J. Graph Theory 12 1988 229 235
40. V Chvátal The smallest triangle-free 4-chromatic 4-regular graph J. Combin. Theory 9 1970 93 94
41. M. Conder, Semi-automated theorem proving. The impact of computers on research in pure mathematics, First Asian Technology Conference in Mathematics, Singapore, December 1995
42. M Conder An infinite family of 5-arc-transitive cubic graphs Ars Combin. 25A 1988 95 108
43. M Conder A surprising isomorphism J. Algebra 129 1990 494 501
44. M Conder Hexagon free subgraphs of hypercubes J. Graph Theory 17 1993 477 479
45. M Conder, and P Lorimer Automorphism groups of symmetric graphs of valency 3 J. Combin. Theory Ser. B 47 1989 60 72
46. R.J Cook G Chartrand, Y Alavi, D.L Goldsmith, L Lesniak-Foster, D.R Lick, The Theory and Applications of Graphs, Proceedings of the Fourth International Conference on the Theory and Applications of Graphs, (Western Michigan Univ., Kalamazoo, MI, 6-9 May 1980) 1981 Wiley New York 610
47. D Cvetković Chromatic number and the spectrum of a graph Publ. Inst. Math. (Beograd) 14 28 1972 25 38
48. D Cvetković Discussing graph theory with a computer, II: Theorems suggested by the computer Publ. Inst. Math. (Beograd) 33 47 1983 29 33
49. D Cvetković Discussing graph theory with a computer, IV: Knowledge organisation and examples of theorem proving Proceedings of the Fourth Yugoslav Seminar on Graph Theory, Novi Sad 1983 43 68
50. D Cvetković Discussing graph theory with a computer, VI: Theorems proved with the aid of the computer Cl. Sci. Math. Natur., Sci. Math., T. XCVII 16 1988 51 70
51. D Cvetković, M Doob, I Gutman, and A Torgasev Recent results in the theory of graph spectra Ann. Discrete Math. 36 1988 1 306
52. D Cvetković, M Doob, and H Sachs Spectra of Graphs 1995 Johann Ambrosius Barth Heidelberg
53. D Cvetković, and L Kraus "Graph" an expert system for the classification and extension of the knowledge in the field of graph theory, User's manual 1983 Elektrotehn. Fak Beograd
54. D Cvetković, L Kraus, and S Simić Discussing graph theory with a computer, I: Implementation of graph theoretic algorithms, Ser. Mat. Fiz. No. 716-No. 734 1981 University of Beograd Publ. Elektrotehn. Fak Beograd 100 104
55. D Cvetković, and I Pevac Discussing graph theory with a computer III: Man-machine theorem proving Publ. Inst. Math. (Beograd) 34 48 1983 37 47
56. D Cvetković, and I Pevac Man-machine theorem proving in graph theory Artificial Intelligence 35 1988 1 23
57. D Cvetković, and S Simić Graph theoretical results obtained by the support of the expert system "Graph" Cl. Sci. Math. Natur., Sci. Math., T. CVII 19 1994 19 41
58. D Cvetković, S Simić, G Caporossi, and P Hansen Variable neighbourhood search for extremal graphs 3. On the largest eigenvalue of color-constrained trees Linear and Multilinear Algebra 2 2001 143 160
59. N Dean, and Y Liu Discrete mathematics software for the K-12 education D.S Franzblau, F.S Roberts, J.G Rosenstein, Discrete Mathematics in the Schools: Making an Impact DIMACS Series vol. 36 1997 AMS Providence, RI
60. E. De la Vina, Bibliography on conjectures of Graffitti. Available from http://cms.dt.uh.edu/faculty/delavinae/research/wowref.htm, 2000
61. G di Battista, P Eades, R Tamassia, and I.G Tollis Algorithms for drawing graphs: an annotated bibliography Comput. Geometry: Theory Appl. 4 5 1994 235 282
62. G di Battista, P Eades, R Tamassia, and I.G Tollis Graph Drawing: Algorithms for the Visualization of Graphs 1999 Prentice-Hall Englewood Cliffs, NJ
63. P Eades, and K Sugiyama How to draw a directed graph? J. Inform. Process 13 4 1990 424 434
64. C.S Edwards, and C.H Elphick Lower bounds for the clique and the chromatic numbers of a graph Discrete Appl. Math. 5 1983 51 64
65. S.L Epstein On the discovery of mathematical theorems in: Proceedings of the Tenth International Joint Conference on Articial Intelligence, Milan, Italy 1987 194 197
66. S.L Epstein Learning and discovery: one system's search for mathematical knowledge Comput. Intelligence 4 1988 42 53
67. S.L Epstein, and N.S Sridharan Knowledge representation for mathematical discovery: Three experiments in graph theory J. Appl. Intelligence 1 1991 7 33
68. S. Fajtlowicz, Written on the Wall, A regulary updated file accessible from http:// www.math.uh.edu/clarson/
69. S Fajtlowicz On conjectures of Graffiti - II Congr. Numer. 60 1987 187 197
70. S Fajtlowicz On conjectures of graffiti Discrete Math. 72 1988 113 118
71. S Fajtlowicz On conjectures of Graffiti - III Congr. Numer. 66 1988 23 32
72. S Fajtlowicz On conjectures of Graffiti - IV Congr. Numer. 70 1990 231 240
73. S Fajtlowicz On conjectures of Graffiti - V in: Seventh International Quadrennial Conference on Graph Theory vol. 1 1995 367 376
74. I.A Faradzev Constructive enumeration of combinatorial object Problèmes Combinatoires et Théorie des Graphes Colloque international CNRS 260 1978 131 135
75. I.A Faradzev Generation of nonisomorphic graphs with given degree sequence (Russian), Nauka, Moscow Algorithmic studies in Combinatorics 1978 11 19
76. I.A Faradzev, and M.H Klin Computer package for computations with coherent configurations in: Proceedings of the ISSAC-91, Bonn 1991 ACM Press New York 219 223
77. I.A Faradzev, M.H Klin, and M.E Muzychuck Cellular rings and groups of automorphism of graphs in: Investigations in Algebraic Theory of Combinatorial Objects, Dordrecht Mathematics and its Applications vol. 84 1994 Kluwer Academic Publishers Dordrecht 1 152
78. O Favaron, M Mahéo, and J.-F Saclé Some results on conjectures of graffiti. I Ars Combin. 29 1990 90 106
79. O Favaron, M Mahéo, and J.-F Saclé On the residue of a graph J. Graph Theory 15 1991 39 64
80. O Favaron, M Mahéo, and J.-F Saclé Some eigenvalue properties in graphs (conjectures of Graffiti. II) Discr. Math. 111 1993 197 220
81. S Fiorini, and R.J Wilson On the chromatic index of a graph II in: Combinatorics (Proc. British Combinatorial Conf.) (Univ. Coll. Wales, Aberystwyth) London Mathematical Society Lecture Notes Series vol. 13 1973, 1974 Cambridge University Press London 37 51
82. M Frölich, and M Werner Demonstration of the Interactive Graph-Visualization System da Vinci Lectures Notes in Computer Science vol. 894 1994 Springer Berlin 266 269
83. GAP. Groups, Algorithms and Programming. Available from Homepage: http://www. groups.dcs.st-andrews.ac.uk/gap/
84. C Gregory, and B Latka Minimal antichains in well-founded quasi-orders with an application to tournaments J. Combin. Theory Ser. B 80 2000 258 276
85. R Grund, A Kerber, and R Laue MOLGEN: ein Computeralgebra-System für die Konstruktion molekularer Graphen Commun. Math. Chemistrymatch 27 1992 87 131
86. I Gutman Total π-electron energy of benzenoid hydrocarbon Topics Curr. Chem. 162 1992 29 63
87. I Gutman, and S Cyvin Introduction to the Theory of Benzenoid Hydrocarbons 1989 Springer Berlin
88. I Gutman, and Y Hou Bipartite unicyclic graphs with greatest energy Commun. Math. Comp. Chem. 43 2001 17 28
89. P. Hansen, H. Mélot, Variable Neighborhood Search for Extremal Graphs 6. Analysing Bounds for the Connectivity Index, J. Chem. Inform. Comput. Sci. (to appear)
90. P. Hansen, H. Mélot, D. Stevanović, Integral Complete Split Graphs, Les Cahiers du GERAD, G-01-43, submitted
91. P Hansen, and N Mladenović Variable neighborhood search: Principles and applications European J. Oper. Res. 130 2001 449 467
92. P Hansen, and M.L Zheng Sharp bounds on the order, size, and stability number of graphs Networks 23 1993 99 102
93. F Harary, and E.M Palmer Graphical Enumeration 1973 Academic Press New York
94. M Himsolt GraphEd: A Graphical Platform for the Implementation of Graph Algorithms Lectures Notes in Computer Science vol. 894 1994 Springer Berlin 182 193
95. M Himsolt The graphlet sytem S North, Graph Drawing Lectures Notes in Computer Science vol. 1190 1996 Springer Berlin 233 240
96. D Knuth The Stanford Graphbase: A Platform for Combinatorial Computing 1993 Addison-Wesley Reading, MA
97. B Latka Finitely constrained classes of homogeneous directed graphs J. Symbolic Logic 59 1 1994 124 139
98. L.-Q Lee, J.G Siek, and A Lumsdaine The generic graph component library ACM SIGPLAN Notices 34 1999 399 414
99. D.R Lick A class of point partition numbers in: Recent Trends in Graph Theory (Proc. Conf. New York 1970) Lectures Notes in Mathematics vol. 186 1971 Springer Berlin 185 190
100. L Lovász Combinatorial Problems and Exercices 1979 North-Holland New York
101. L Lovász, and J Pelikan On the eigenvalues of trees Period. Math. Hungar. 3 1973 175 182
102. MathWorks, Inc. Matlab: The Language of Technical Computing, The MathWorks, Inc
103. B.D. McKay, Nauty user's guide (version 1.5). Technical Report. TR-CS-90-02, Department of Computer Science, Australian National University, 1990
104. B.D McKay Isomorph-free exhaustive generation J. Algorithms 26 1998 306 324
105. K Mehlhorn, and S Nähger Leda: A platform for combinatorial and geometric computing Commun. ACM 38 1 1995 96 102
106. M. Mevenkamp, N. Dean, C. Monma, NETPAD user's guide and reference guide, Bellcore, 1990
107. P.F Newberry The Design of an Extendible Graph Editor Lectures Notes in Computer Science vol. 704 1993 Springer Berlin
108. e(3,3;5) J. Graph Theory 32 1999 41 49
109. G Polya, R.C Read, and R.C Combinatorial Enumeration of Groups 1987 Springer Berlin
110. C.E Praeger, and L.H Soicher Low rank representations and graphs for sporadic groups Australian Mathematical Society Lecture Series vol. 8 1997 Cambridge University Press Cambridge
111. F. Pujol, Etude de conjectures en théorie des graphes. Internship Report under direction of P. Hansen (GERAD), 1998
112. Z Radosavljević, and S Simić There are just thirteen connected nonregular nonbipartite integral graphs having maximum vertex degree four R Tosić, D Acketa, V Petrović, Graph Theory, Proceedings VI Yugoslav Seminar on Graph Theory, Dubrovnik, April 18-19, 1985 1987 University of Novi Sad, Institute of Mathematics Novi Sad 183 187
113. Z Radosavljević, and S Simić The nonregular nonbipartite integral graphs with the maximum degree four J. Comb. Inf. Syst. Sci. 20 1-4 1995 9 26
114. S.P Radziszowski Small Ramsey numbers. Dynamic survey 1 Electr. J. Combin. 1994 Updated 1998
115. M Randić On characterization of molecular branching J. Am. Chem. Soc. 97 1975 6609 6615
116. R.C Read Every one a winner Ann. Discrete Math. 2 1978 107 120
117. R.C Read Some applications of computers in graph theory L.W Beineke, R.J Wilson, Selected Topics in Graph Theory 1978 Academic Press London, New York 417 444
118. N Robertson, D Sanders, P Seymour, and R Thomas The four-color theorem J. Combin. Theory Ser. B 70 1997 2 44
119. P Rowlinson On the index of certain outerplanar graphs Ars Combin. 29C 1990 271 275
120. G Sander Graph Layout through the VCG Tool Lectures Notes in Computer Science vol. 894 1994 Springer Verlag 194 205
121. G. Shannon, L. Meeden, D. Friedman, SchemeGraph: an object-oriented environment for manipulating graphs, Software and documentation, 1990
122. S Simić On the largest eigenvalue of unicyclic graphs Publ. Inst. Math. (Beograd) 42 56 1987 13 19
123. S Simić Some results on the largest eigenvalue of a graph Ars Combin. 24A 1987 211 219
124. S Skiena Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica 1990 Addison-Wesley Reading, MA
125. L.H Soicher GRAPE: a System for Computing with Graphs and Groups L Finkelstein, W.M Kantor, Groups and Computation DIMACS Series in Discrete Mathematics and Theoretical Computer Science vol. 11 1993 AMS New York 287 291
126. L.H Soicher Three new distance-regular graphs European J. Combin. 14 1993 501 505
127. R.P Stanley A bound on the spectral radius of graphs with e edges Linear Algebra Appl. 87 1987 267 269
128. D. Stevanović, Some compositions of graphs and integral graphs (in Serbian), Ph.D Thesis, University of Niš, 2000
129. Vega. A system for manipulating discrete mathematical structures, Available from Project Homepage: http://vega.ijp.si/
130. Wolfram, Research Inc. Mathematica Language and Software, Wolfram Research Inc