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Hall exponents of matrices, tournaments and their line digraphs

Richard A. Brualdi, Kathleen P. Kiernan
2011 Czechoslovak Mathematical Journal  
We now show that the upper bound for the Hall exponent in Theorem 4.6 cannot be improved by much, by obtaining a lower bound for the Hall exponent of the line digraph of the special tournament T * n .  ...  In [12] , it is stated that the primitive exponent of a regular or nearly regular tournament of order n is 3 for n 7.  ...  K i e r n a n, Department of Mathematics, University of Wisconsin, Madison, WI 53706 USA, e-mails: brualdi@ math.wisc.edu, kiernan@math.wisc.edu.  ... 
doi:10.1007/s10587-011-0066-2 fatcat:l4ague2v7jfm7n3rpx6fyfnydm

Page 596 of Mathematical Reviews Vol. , Issue 93b [page]

1993 Mathematical Reviews  
In this paper, we first obtain an upper bound for o,(G) and determine the structure of a 2-connected graph G when o,(G) attains the bound.  ...  Let NE, be the set of exponents of primitive, minimally strong digraphs of order n.  ... 

Page 283 of Mathematical Reviews Vol. 41, Issue 2 [page]

1971 Mathematical Reviews  
We say M20 is primitive if M'>0 for some positive integer ¢, the smallest such ¢ being the exponent of M. § 13. The powers of tournament matrices.  ...  If T', (n 2 5) is irreducible then 7’, (i.e., M(7’,)) is primitive, and d<e<d+3, where d is the diameter of 7’, and e the exponent of M(T’,).§ § 14. Scheduling a tournament.  ... 

Generalized exponents of boolean matrices

Bolian Liu
2003 Linear Algebra and its Applications  
Graph Theory 14 (1990) 483) introduced the concept of generalized exponent for primitive Boolean matrices.  ...  In the study of generalized exponents, we focus on problems of the following three types: (1) To determine the maxima (namely, the exact upper bounds) of all kinds of generalized exponents; (2) To characterize  ...  Jai-yu Shao for valuable suggestions.  ... 
doi:10.1016/s0024-3795(02)00714-0 fatcat:j7fzqzadv5cjhizr4igwrrrcje

Page 2607 of Mathematical Reviews Vol. , Issue 96e [page]

1996 Mathematical Reviews  
The extremal exhaustive graphs are determined for n > 7.”  ...  Summary: “Let MD,(n) = {D| D is a primitive ministrong di- graph with » vertices, and the shortest cycle length of D is s}, and b,(n) = max{y(D)| D € MD,(n)}, where y(D) is the primitive exponent of D.  ... 

Page 1721 of Mathematical Reviews Vol. , Issue 84e [page]

1984 Mathematical Reviews  
However, formula (2) in the paper is printed in a misleading way in that the term 2( p— 1)/d should be an exponent of (1 + x??*).  ...  1721 OSC Graph theory vertex set is the collection of all faces of K, and for two the vertices are all the extreme rays of K.  ... 

Page 5763 of Mathematical Reviews Vol. , Issue 95j [page]

1995 Mathematical Reviews  
Pullman, A bound on the exponent of a primitive matrix using Boolean rank (101-116); Karl Gustafson, Matrix trigonometry (117-140); Douglas Hale and George Lady, Qualitative comparative statics and audits  ...  Sidorenko], An analytic approach to extremal problems for graphs and hypergraphs (423-455); W. T.  ... 

MAXIMAL EXPONENTS OF PRIMITIVE GRAPHS WITH MINIMUM DEGREE 3

Byung Chul Song, Byeong Moon Kim
2011 Korean Journal of Mathematics  
In this paper, we find the maximum exponent of primitive simple graphs G under the restriction deg(v) ≥ 3 for all vertex v of G.  ...  Our result is also an answer of a Klee and Quaife type problem on exponent to find minimum number of vertices of graphs which have fixed even exponent and the degree of whose vertices are always at least  ...  Moreover this upper bound is extremal for n ≥ 8, i.e., for each n ≥ 8, there is a primitive graph on n vertices with minimum degree 3 and exponent the smallest value of above inequality.  ... 
doi:10.11568/kjm.2011.19.4.367 fatcat:ygwa2f45bvbo5cmfxd5f3jem44

Generalized exponents of primitive symmetric digraphs

Richard A. Brualdi, Shao Jia-yu
1997 Discrete Applied Mathematics  
For such a digraph there is a minimum integer t, called the exponent of D, such that given any ordered pair of vertices x and y there is a directed walk from x to y of length t.  ...  In this paper we determine the largest value of these generalized exponents over the set of primitive symmetric digraphs whose shortest odd cycle length is a fixed number r.  ...  In [l, 3,4] Brualdi and Liu derived upper bounds for the generalized exponents of primitive digraphs and for primitive digraphs of special type (symmetric digraphs, tournaments, and minimally strong digraphs  ... 
doi:10.1016/s0166-218x(96)00077-7 fatcat:jyb3l4jllrbnln6e2ltbzd2xjq

Toward automating EA configuration: the parent selection stage

Ekaterina Smorodkina, Daniel Tauritz
2007 2007 IEEE Congress on Evolutionary Computation  
One of the obstacles to Evolutionary Algorithms (EAs) fulfilling their promise as easy to use general-purpose problem solvers, is the difficulty of correctly configuring them for specific problems such  ...  Having a mechanism for automatically configuring parameters and operators of every stage of the evolutionary life-cycle would give EAs a more widely spread popularity in the non-expert community.  ...  Primitive name Description Tournament Selection standard deterministic tournament selection with tournament size of 5 Roulette Selection standard roulette selection Random Selection a routine that uniform  ... 
doi:10.1109/cec.2007.4424455 dblp:conf/cec/SmorodkinaT07 fatcat:zbfm7txexrdhfoxeuvqwqgsuxy

Maximal exponents of polyhedral cones (III)

Raphael Loewy, Micha A. Perles, Bit-Shun Tam
2013 Transactions of the American Mathematical Society  
It is proved that for any positive integers m, n, 3 ≤ n ≤ m, the maximum value of γ(K), as K runs through all n-dimensional polyhedral cones with m extreme rays, equals (n − 1)(m − 1) + 1 2 1 + (−1) (n  ...  For a polyhedral cone K, the maximum value of γ(A), taken over all K-primitive matrices A, is denoted by γ(K).  ...  Acknowledgements The first and third authors would like to thank Hans Schneider and Sergeǐ Sergeev for drawing their attention to the work of Grinberg [11] .  ... 
doi:10.1090/s0002-9947-2013-05879-5 fatcat:frtl2y33avgynfhyh24ozrqhxa

The scrambling index of primitive digraphs

Bolian Liu, Yufei Huang
2010 Computers and Mathematics with Applications  
In this paper, we obtain some new upper bounds for the scrambling index of primitive digraphs.  ...  Moreover, the maximum index problem, the extremal matrix problem and the index set problem for the scrambling index of various classes of primitive digraphs (e.g. primitive digraphs with d loops, minimally  ...  Acknowledgements The authors are grateful to the referees for their helpful comments and useful suggestions on the earlier versions of this paper.  ... 
doi:10.1016/j.camwa.2010.05.018 fatcat:kppiho5i7jhxdliqxllruk6ybi

Skew Hadamard designs and their codes

Jon-Lark Kim, Patrick Solé
2008 Designs, Codes and Cryptography  
Quadratic Residues codes are obtained in the case of the Paley matrix. Results on the p−rank of skew Hadamard designs are rederived in that way. Codes from skew Hadamard designs are classified.  ...  Six new inequivalent [56, 28, 16] self-dual codes over GF (7) are obtained from skew Hadamard matrices of order 56, improving the only known quadratic double circulant code of length 56 over GF (7).  ...  The first author acknowledges partial support by a Project Completion Grant from the University of Louisville.  ... 
doi:10.1007/s10623-008-9173-y fatcat:a2a5afytg5cxfdkcsc7guhr2da

Preserving and Exploiting Genetic Diversity in Evolutionary Programming Algorithms

Gang Chen, Chor Ping Low, Zhonghua Yang
2009 IEEE Transactions on Evolutionary Computation  
Evolution programming (EP) is an important category of evolutionary algorithms. It relies primarily on mutation operators to search for solutions of function optimization problems (FOPs).  ...  Recently a series of new mutation operators have been proposed in order to improve the performance of EP.  ...  ., the number of distinct positions in the population): μ = 100; 2) the number of rounds of tournaments: q = √ μ = 10; 3) the initial standard deviation: η = 0.5; 4) the exponent parameter of the power-law  ... 
doi:10.1109/tevc.2008.2011742 fatcat:utzr4ptt25ex7praq6gkeu76jm

On a conjecture about the kth lower multiexponent

Yufei Huang, Bolian Liu
2010 Computers and Mathematics with Applications  
Liu is proved to be true for the following cases: (1) k = n − i, where i = 2, 3, 4, 5; (2) small n, where n ≤ 8; (3) the class of primitive micro-symmetric digraphs of order n.  ...  Acknowledgements The authors would like to thank the referees for the valuable comments, and for the suggestions to improve the readability of this article.  ...  Then f (D n , k) = 1 + (2n − k − 2) n − 1 k − n − 1 k 2 · k. By Lemma 2.1, D n is an extremal digraph reaching the bound given by f (n, k).  ... 
doi:10.1016/j.camwa.2010.04.027 fatcat:x735rr4hwjfkzergcp2ta5us24
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