Non-negative matrices and digraphs

Computational Complexity: Theory, Techniques, and Applications

Volumn 9781461418009, Issue , 2012, Pages 2082-2095

  Berman, Abraham ^a   Shaked Monderer, Naomi ^b  

^a TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

^b The Max Stern Emek Yezreel College   (Israel)

Author keywords

[No Author keywords available]

Indexed keywords

MATRIX ALGEBRA;

GRAPHS AND DIGRAPHS; IRREDUCIBLE MATRICES; LAPLACIAN MATRICES; MATRIX GRAPHS; NON-NEGATIVE MATRIX; RANKING PROBLEMS;

DIRECTED GRAPHS;

EID: 84949179867     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1007/978-1-4614-1800-9_132     Document Type: Chapter

References (57)

1. Alon U (2006) An Introduction to Systems Biology: Design Principles of Biological Circuites. CRC Mathematical and Computational Biology Series. Chapman, Boca Raton
2. Bang-Jensen J, Gutin G (2001) Digraphs: Theory Algorithms and Applications. Springer, London
3. Bapat RB, Raghavan TES (1997) Nonnegative Matrices and Applications. Encyclopedia of Mathematics and its Applications 64. Cambridge University Press, Cambridge
4. Beineke LW, Wilson RJ (2004) Topics in Algebraic Graph Theory. Encyclopedia of Mathematics and its Applications 102. Cambridge University Press, Cambridge
5. Berman A (2003) Graphs of matrices and matrices of graphs. Numer Math J Chin Univ (Engl Ser) 12(suppl):12-14
6. Berman A, Neumann M, Stern RJ (1989) Nonnegative Matrices in Dynamical Systems. Wiley-Interscience, New York
7. Berman A, Plemmons R (1994) Nonnegative Matrices in the Mathematical sciences. SIAM, Philadelphia
8. Berman A, Shaked-Monderer N (2003) Completely Positive Matrices. World Scientific, River Edge
9. Birkhoff G (1946) Tres obseraciones sobre el algebra lineal. Univ Nac Tucuman Rev Ser A 5:147-150
10. Bondy JA, Murty USR (1976) Graph Theory with Applications. North-Holland, New York
11. Brin S, Page L (1998) The anatomy of a large-scale hypertextual Web search engine. Comput Netw ISDN Syst 33(1-7):107-117
12. Brualdi RA (1979) Matrices permutation equivalent to irreducible matrices and applications. Linear Multilinear Algebra 7:1-12
13. Brualdi RA, Parter SV, Schneider H (1966) The diagonal equivalence of a nonnegative matrix to a stochastic matrix. J Math Anal Appl 16:31-50
14. Brualdi RA, Ryser HJ (1991) Combinatorial Matrix Theory. Encyclopedia of Mathematics and its Applications 39. Cambridge University Press, Cambridge
15. Broder A, Kumar R, Maghoul F, Raghavan P, Rajagopalan S, Stata R, Tomkins A, Wiener J (2000) Comput Netw 33(1-3): 309-320
16. Chaiken S (1982) A combinatorial proof of the all-minors matrix tree theorem. SIAM J Alg Dis Meth 3:319-329
17. Chebotarev P, Agaev R (2002) Forest matrices around the Laplacian matrix. Linear Algebra Appl 356:253-274
18. Chung FRK (1997) Spectral Graph Theory. CBMS, Regional Conference Series in Mathematics 92. AMS, Providence
19. Chung F (2005) Laplacians and the Cheeger inequality for directed graphs. Ann Comb 9:1-19
20. Chung F (2006) The diameter and Laplacian eigenvalues of a directed graphs. Electron J Combin 13(4):6
21. Cvetković D, Doob M, Sachs H (1980) Spectra of Graphs: Theory and Applications. Academic Press, New York
22. Eshenbach C, Hall F, Hemansinha R, Kirkland S, Li Z, Shader B, Stuart J, Weaver J (2000) Properties of Tournaments among Well-Matched Players. Amer Math Month 107(10):881-892
23. Frobenius G (1912) Über Matrizen aus nicht negativen El-ementen. Sitzungsbericht. Preussische Akademie der Wis-senschaften, Berlin, pp 456-477
24. Gantmacher FR (1959) The Theory of Matrices. (Translated from Russian), Chelsea, New York
25. Godsil C, Royle G (2001) Algebraic Graph Theory. Graduate Texts in Mathematics 207. Springer, New York
26. Gulli A, Signorini A (2005) The indexable web is more than 11.5 billion pages. In: Proc. 14th WWW (Posters), pp 902-903
27. Gupta RP (1967) On basis digraphs. J Combin Theory 3: 309-311
28. Hall P (1935) On representatives of subsets. J London Math Soc 10:26-30
29. Hartfiel DJ (1970) A simplified form for nearly reducible and nearly decomposable matrices. Proc Amer Math Soc 24: 388-393
30. Hershkowitz D (1999) The combinatorial structure of generalized eigenspaces-from nonnegative matrices to general matrices. Linear Algebra Appl 302-303:173-191
31. Hogben L (2007) Handbook of Linear Algebra. Discrete Mathematics and Its Applications Series 39. CRC Press, Boca Raton
32. van der Holst H, Lovász L, Schrijver A (1999) The Colin de Verdière graph parameter. In: Graph Theory and Computational Biology. Bolyai Soc Math Stud 7:29-85
33. Horn RA, Johnson CR (1985) Matrix Analysis. Cambridge University Press, Cambridge
34. Kendall MG (1955) Further contributions to the theory of paired comparisons. Biometrics 11:43-62
35. Kirkland S (2004) Digraph-based conditioning for Markov chains. Linear Algebra Appl 385:81-93
36. Kirkland SJ (2004) A combinatorial approach to the conditioning of a single entry in the stationary distribution for a Markov chain. Electron J Linear Algebra 11:168-179
37. Kirkland SJ (2004/5) Girth and subdominant eigenvalues for stochastic matrices. Electron J Linear Algebra 12:25-41
38. Langville AN, Meyer CD (2005) A survey of eigenvector methods of Web information retrieval. SIAM Rev 47(1):135-161
39. Lee D, Seung H (2001) Algorithms for nonnegative matrix factorization. Adv Neural Process 13:556-562
40. Marcus M, Minc H (1963) Disjoint pairs of sets and incidence matrices. Illinois J Math 7:137-147
41. Marcus M, Minc H (1964) A survey of matrix theory and matrix inequalities. Allyn and Bacon, Boston
42. Minc H (1988) Nonnegative Matrices. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York
43. Perron O (1907) Zur Theorie der Matrizen. Math Ann 64: 248-263
44. Richman D, Schneider H (1978) On the singular graph and the Weyr characteristic of an M-matrix. Aequationes Math 17: 208-234
45. Romanovsky V (1936) Recherche sur les chains de Markoff. Acta Math 66:147-251
46. Rothblum UG (1975) Algebraic eigenspaces of nonnegative matrices. Linear Algebra Appl 12:281-292
47. Schneider H (1956) The elementary divisors associated with 0 of a singular M-matrix. Proc Edinburgh Math Soc 10(2):108-122
48. Schneider H (1977) The concepts of irreducibility and full inde-composability of a matrix in the works of Frobenius, König and Markov. Linear Algebra Appl 18:139-162
49. Schneider H (1986) The influence of the marked reduced graph of a nonnegative matrix on the Jordan Form and on related properties: a survey. Linear Algebra Appl 84:161-189
50. Seneta E (1981) Nonnegative Matrices and Markov Chains. Springer Series in Statistics, 2nd edn. Springer, New York
51. Tam BS (2004) The Perron generalized eigenspace and the spectral cone of a cone-preserving map. Linear Algebra Appl 393:375-429
52. Tutte WT (1948) The dissection of equilateral triangles into equilateral triangles. Proc Cambridge Philos Soc 7:463-482
53. Tutte WT (1980) Graph Theory. Encyclopedia of Mathematics and its Applications 21. Cambridge University Press, Cambridge
54. Victory Jr HD (1985) On nonnegative solutions to matrix equations. SIAM J Alg Dis Meth 6:406-412
55. Wasserman S, Faust K (1994) Social network analysis: Methods and applications. Cambridge University Press, Cambridge
56. Wei TH (1952) The Algebraic Foundations of Ranking Theory. Ph D Thesis, Cambridge University
57. Wu CW (2005) On Rayleigh-Ritz rations of a generalized Lapla-cian matrix of directed graphs. Linear Algebra Appl 402:207-227