Combinatorics o f Nonnegative Matrice s This page intentionally left blank 10.1090/mmono/213 Translations o f MATHEMATICAL MONOGRAPHS Volume 21 3 Combinatorics o f Nonnegative Matrice s V. N. Sachko v V. E. Tarakano v Translated b y Valentin F . Kolchl n | yj | America n Mathematica l Societ y Providence, Rhod e Islan d EDITORIAL COMMITTE E AMS Subcommitte e Robert D . MacPherso n Grigorii A . Marguli s James D . Stashef f (Chair ) ASL Subcommitte e Steffe n Lemp p (Chair ) IMS Subcommitte e Mar k I . Freidli n (Chair ) B. H . Ca^KOB , B . E . TapaKaHO B KOMBMHATOPMKA HEOTPMUATEJIBHbl X MATPM U Hay^Hoe 143/iaTeJibCTB O TBE[ , MocKBa , 200 0 Translated fro m th e Russia n b y Dr . Valenti n F . Kolchi n 2000 Mathematics Subject Classification. Primar y 05-02 ; Secondary 05C50 , 15-02 , 15A48 , 93-02. Library o f Congress Cataloging-in-Publicatio n Dat a Sachkov, Vladimi r Nikolaevich . [Kombinatorika neotritsatel'nyk h matrits . English ] Combinatorics o f nonnegativ e matrice s / V . N . Sachkov , V . E . Tarakano v ; translate d b y Valentin F . Kolchin . p. cm . — (Translations o f mathematical monographs , ISS N 0065-928 2 ; v. 213) Includes bibliographica l reference s an d index. ISBN 0-8218-2788- X (acid-fre e paper ) 1. Non-negative matrices. 2 . Combinatorial analysis . I . Tarakanov, V . E. (Valerii Evgen'evich ) II. Title . III . Series. QA188.S1913 200 2 512.9'434—dc21 200207439 2 Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them, ar e permitted t o make fai r us e of the material, suc h a s to copy a chapter fo r use in teachin g o r research . Permissio n i s granted t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customary acknowledgmen t o f the source i s given. Republication, systemati c copying , or multiple reproductio n o f any material in this publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission shoul d b e addressed to the Acquisitions Department , America n Mathematica l Society , 201 Charle s Street , Providence , Rhod e Islan d 02904-2294 , USA . Requests ca n also b e made b y e-mail t o reprint-permiss ion0ams.org. © 200 2 b y the American Mathematica l Society . Al l rights reserved . The America n Mathematica l Societ y retain s al l rights except thos e grante d t o the United State s Government . Printed i n the United State s o f America . @ Th e paper use d i n this boo k i s acid-free an d falls withi n the guidelines established t o ensure permanenc e an d durability. Visit th e AMS hom e pag e a t http: //www. ams. org/ 10 9 8 7 6 5 4 3 2 1 0 7 06 05 04 03 0 2 Contents Preface vi i List o f Notation i x Chapter 1 . Matrice s an d Configuration s 1 Introduction 1 1.1. Definition s an d example s 2 1.2. Ter m rank . Arrangemen t o f positive element s 9 1.3. Combinatoria l theor y o f cyclic matrices 2 7 Chapter 2 . Ryse r Classe s 4 5 Introduction 4 5 2.1. A constructive descriptio n o f Ryser classe s 4 6 2.2. Invarian t set s 5 8 2.3. Estimate s o f the term ran k 6 9 Chapter 3 . Nonnegativ e Matrice s an d Extrema l Combinatoria l Problem s 8 3 Introduction 8 3 3.1. Forbidde n configuration s 8 4 3.2. Coverin g proble m 9 0 3.3. Th e va n de r Waerden-Egorychev-Falikman Theore m 10 6 Chapter 4 . Asymptoti c Method s i n the Stud y o f Nonnegative Matrice s 11 7 Introduction 11 7 4.1. Nonnegativ e matrice s an d graph s 11 8 4.2. Asymptotic s o f the numbe r o f primitive (0 , l)-matrices 13 1 4.3. Asymptotic s o f the permanen t o f a random (0 , l)-matrix 13 5 4.4. Rando m lattice s an d Boolea n algebra s 13 8 4.5. Covering s o f sets an d (0 , l)-matrices 14 3 4.6. Rando m covering s o f sets 15 1 Chapter 5 . Totall y Indecomposable , Chainable , an d Prim e Matrice s 15 9 Introduction 15 9 5.1. Totall y indecomposabl e an d chainabl e matrice s 16 1 5.2. Rectangula r nonnegativ e matrice s 17 0 5.3. Rectangula r nonnegativ e chainabl e matrice s 18 4 5.4. Extensio n o f partial diagonal s 19 2 5.5. Prim e Boolea n matrice s 19 9 5.6. Prim e nonnegativ e matrice s 20 8 vi CONTENT S Chapter 6 . Sequence s o f Nonnegative Matrice s 21 3 Introduction 21 3 6.1. Directe d graph s o f nonnegative matrice s 21 5 6.2. Irreducibl e an d primitiv e matrice s 22 1 6.3. Tournamen t matrice s 22 7 6.4. Associate d operato r 23 2 6.5. Sequence s o f powers o f a nonnegative matri x 24 3 6.6. Ergodicit y o f sequences o f nonnegative matrice s 25 0 Bibliography 26 3 Index 267 Preface The subject o f this book i s nonnegative matrices. Th e variety o f combinatoria l properties o f such matrices is widely discussed i n mathematical literature, and ther e are lots o f papers on this topic. However , there ar e only a fe w monographs devote d specially t o th e combinatoria l propertie s o f nonnegative matrices . Th e author s o f the present boo k have tried to concentrate not o n traditional algebrai c and, i n par- ticular, spectra l propertie s o f nonnegativ e matrices , bu t rathe r o n thei r relation s to various mathematical structure s studie d i n combinatorics. I n additio n t o appli - cations i n grap h theory , Marko v chains , tournaments , an d abstrac t automata , w e consider relation s betwee n nonnegativ e matrice s an d suc h structure s a s covering s and minimal coverings o f sets by families o f subsets. Alon g with the study o f combi- natorial notion s which can be interpreted usin g nonnegative matrices , considerabl e attention i s given to the study o f various properties o f the matrices themselves an d also o f the classe s forme d b y th e matrice s havin g a give n structure . Asymptoti c properties o f nonnegative matrice s a s som e o f the parameter s ten d t o infinit y ar e also investigated . In th e book , bot h enumerativ e an d extrema l combinatoria l problem s ar e pre - sented. I n the study o f enumerative problems, along with combinatorial methods we use th e probabilisti c approach . Amon g th e extrema l problems , w e conside r bot h problems directl y relate d t o matrice s an d problem s wher e nonnegativ e matrice s provide suitable tool s o f investigation . In connectio n wit h ou r combinatoria l approach , a n essentia l rol e i s playe d b y the binar y structur e o f a nonnegative matrix , tha t is , the arrangemen t o f positiv e elements and zeros. I t should be noted that this structure is related to the properties of matrice s which , a t firs t sight , ar e no t o f combinatoria l character . On e suc h example i s th e applicatio n o f nonnegativ e matrice s i n th e stud y o f ergodicit y o f Markov chains i n Chapter 6 . Th e binary characte r o f nonnegative matrice s ca n b e seen i n the fac t tha t man y essentia l properties o f such matrices ar e determined b y the propertie s o f their supports , tha t is , b y the (0 , l)-matrices whic h ar e obtaine d by replacing the positiv e element s o f a matrix b y ones . Therefore , w e shall devot e a lo t o f attention t o (0 , l)-matrices. In analyzing (0 , l)-matrices, we distinguish between local and global properties. Global propertie s characteriz e a matri x a s a whole , wherea s loca l propertie s ar e concerned with relations between parts o f the matrix. Example s o f global properties are th e distributio n o f positiv e an d zer o element s i n row s an d columns , th e ter m rank, an d the cove r rank. Loca l properties include , fo r example , various condition s on the inne r products o f rows and column s (viewe d a s vectors o f the correspondin g space ove r rea l numbers ) an d th e presenc e (o r absence ) o f submatrices o f a give n structure.