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)- 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 , 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.

vii viii PREFAC E

One more feature o f the results on nonnegative matrice s presented i n this boo k consists o f the fact that the consideration o f seemingly pure combinatorial problem s often lead s to interesting results in some special areas o f mathematics. Fo r example, the nonstandard discrete limit distributions obtained in Chapter 4 in the problem on minimal coverings deserves, in our opinion, the attention o f specialists in probability theory. Although thi s boo k i s mainly theoretical , th e author s constantl y kep t i n min d applications o f nonnegative matrices. Amon g these numerous applications the most significant an d promising ones are in the theory o f Markov chains, in linear program- ming fo r constructin g an d analyzin g economi c models , an d i n informatio n theor y for designin g reliabl e information devices . Th e authors hop e that th e methods an d results presente d her e wil l b e usefu l t o specialist s i n thes e area s o f researc h an d engineering. Trying to make the book self-contained, independent , wheneve r possible , o f ad- ditional sources and understandable to a wide range of readers, the authors include d some theorems which already ar e viewed a s classical. Th e book als o contains a sig- nificant numbe r o f results which have not bee n published i n monographs, includin g a serie s o f results obtaine d b y the author s i n the las t fe w years . In presentin g th e material , th e author s usuall y trie d t o mov e fro m simpl e t o more complicate d problems , an d place d mor e specia l result s a t th e end s o f th e sections. Therefore , th e beginning parts o f some sections can be used as a textbook for student s specializin g i n applie d discret e mathematic s an d a s th e materia l fo r special course s an d seminar s i n discrete mathematics . Chapters 4-6 were written b y V. Sachkov, and Chapters 1-3 , b y V. Tarakanov . However th e author s woul d lik e to emphasiz e th e unifie d approac h t o the materia l and t o the idea s o n whic h the boo k i s based . V. Sachkov, V. Tarakanov List o f Notatio n

R i s the se t o f real numbers . R+ i s the se t o f all nonnegative rea l numbers . Rn i s the n-dimensiona l vecto r spac e ove r R . Z i s the se t o f al l integers. Nn i s the se t {1,2,.. . , n). Zn i s the se t {0,1,.. . , n — 1} . |X| i s the cardinalit y o f a se t X; a finite se t X (subset , sample ) i s sometime s called a n |X|-se t (|X|-subset , |X|-sample) . 7£m'n i s the se t o f all nonnegative m x n matrice s (matrice s o f siz e m x n). Sm'n i s the se t o f all (0 , l)-matrices o f siz e m x n . C i s the symbo l o f inclusion fo r sets . fl denote s the operation o f intersection o f sets. U denote s the operatio n o f union o f sets. G i s the symbo l o f inclusion o f an elemen t int o a set . \ i s the symbo l o f the operatio n o f taking th e differenc e o f two sets. 0 i s the empt y set . = denote s isomorphis m o f groups. 6m i s the grou p o f al l permutations o f degree m . {a} i s the cycli c group generate d b y the elemen t a. (a>ij) i s the matri x wit h element s a^ , i = 1 , 2,... , m, j — 1, 2,... , n. AT i s the transpose o f a matrix A. detA o r A (^4) i s the determinant o f a matrix A. per A i s the permanen t o f a matrix A. Jmn o r simpl y J i s the m x n matri x wit h al l elements equa l to one . 0mn o r simpl y 0 i s the m x n matrix wit h al l elements equa l to zero , diag [d\, <^2, • • • ,dn] i s the square matrix o f order n with elements d\, c?2 , • • • , dn on the principa l diagona l an d zero s i n the remainin g positions . In = dia g [1,1,... ,1 ] i s the identit y matri x o f order n. A[I, J], wher e / C 7V m, J C Af n, i s the submatri x o f a n m x n matri x A wit h rows numbers i n I an d colum n number s i n J . (x,y) i s the inne r produc t o f vectors x — (xi,... , xn), y = (yi,... , yn) equa l to xiyi + ... + x nyn. [x\ i s the maxima l intege r whic h doe s no t excee d a numbe r x G R, i n othe r words, the intege r par t o f x. \x] i s the minima l intege r whic h i s not les s than xGR . m (mi, 7722 , • • • 7 r) i s the greates t commo n diviso r o f integers mi, 7712,... , m r. This page intentionally left blank Bibliography

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absorption index , 24 0 proper, 17 2 acyclic class , 22 0 strictly proper , 17 2 acyclic matrix , 22 1 covering, 17 1 a-eovering o f a matrix, 9 2 proper, 17 1 a-depth o f a matrix, 90 , 9 1 strictly proper , 17 1 aperiodic class , 22 0 covering number , 91 , 92 aperiodic matrix , 22 1 covering problem , 9 1 aperiodic vertex , 22 0 cycle, 18 6 associated operator , 214 , 23 2 cyclic configuration, 3 6 nonsingular, 24 0 cyclic matrix, 5 , 22 1 cyclic projective plane , 3 7 balanced incomplet e bloc k design , 6 cyclotomic polynomial , 4 0 basis ones , 6 9 basis se t o f positions, 6 9 decomposition o f a digraph int o circuits , 5 0 bipartite graph , 17 3 degree o f a vertex i n a hypergraph , 9 2 Boolean algebra , 118 , 13 9 depth o f a matrix , 9 1 Boolean function , 14 9 determinant o f a matrix, 1 1 Boolean matrix , 11 8 deterministic matrix , 23 5 diagonal, 10 , 19 2 canonical matri x o f a Ryse r class , 4 7 partial, 10 , 19 2 chain, 21 5 positive, 1 0 chainable matrix , 166 , 18 4 row, 19 2 minimal, 18 6 , 16 1 characteristic function , 2 diagonal product , 16 6 circuit, 21 5 diameter, 22 8 elementary, 21 5 diameter o f a component, 21 6 circuit diamete r o f a stron g component , 21 6 diameter o f a strong component o f a digraph, circulant, 5 , 2 7 216 primitive, 2 9 difference set , 3 6 class property, 21 8 distributive lattice , 13 9 collinear elements , 1 0 doubly stochasti c configuration , 4 column-composite class , 6 3 doubly stochasti c matrix , 1 3 communicating vertices , 21 7 doubly stochasti c model , 16 6 complemented lattice , 13 9 component o f a digraph, 21 6 elementary class , 6 3 strong, 21 5 , 20 7 g-universal, 24 4 equivalent configurations , 4 nontransient, 21 5 essential vertex , 21 7 transient, 21 5 essentially nonsingula r matrix , 17 6 composite class , 6 3 strictly, 17 7 configuration, 2 even graph , 5 0 configuration o f subsets, 2 exponent connected graph , 18 5 of a clas s o f matrices, 18 4 consistent syste m o f equations, 15 0 of a graph, 12 0 convergence index , 231 , 244 of a matrix , 119 , 22 4 convergent sequence , 24 4 of a tournament, 22 8 cover rank , 14 , 17 2 extremal matrix , 8 8

267 268 INDEX factorable matrix , 199 , 20 8 nearly decomposabl e matrix , 16 8 forbidden configuration , 8 5 nearly partiall y decomposabl e matrix , 16 8 free Boolea n algebra , 14 1 nonnegative matrix , 2 Frobenius function , 22 1 normal for m o f a matrix , 21 5 Frobenius-Konig Theorem , 1 2 nullity, 159 , 17 1 proper, 17 1 g-circulant, 2 7 g- , 2 7 oscillating sequence , 24 4 graph, 6 packing circui t number , 5 1 graph configuration , 6 partially decomposabl e matrix , 161 , 17 0 hypergraph, 6 , 9 2 path, 18 5 perfectly decomposabl e matrix , 21 2 , 3 period, 231 , 244 formal, 18 1 of a circulant , 2 9 inconsistent syste m o f equations, 15 0 of a clas s o f vertices, 22 0 indecomposable matrix , 215 , 219, 22 1 of a vertex , 21 9 independent generatin g system , 14 1 periodic matrix , 22 1 independent ones , 6 9 periodic sequence , 24 4 index, 21 4 permanent of a matrix , 23 9 decomposition of , 1 1 of a nonnegativ e matrix , 23 0 permanent o f a matrix , 1 1 of an associate d operator , 24 0 , 4 inessential vertex , 21 7 permutation wit h restricte d positions , 12 7 interchange graph , 5 1 cyclic, 12 9 interchange operation , 4 8 permutational automorphism , 7 invariant 0-position , 6 2 permutationally equivalen t matrices , 4 invariant 1-position , 6 2 Ph. Hal l Theorem , 1 5 invariant one , 6 2 positive diagonal , 16 6 invariant se t fo r a Ryse r class , 6 2 positively symmetri c matrix , 23 7 invariant zero , 6 2 prime matrix , 199 , 20 8 irreducible graph , 17 3 primitive graph , 12 0 irreducible matrix , 215 , 22 1 primitive matrix , 119 , 22 4 problem o n forbidde n configurations , 8 5 /c-discordant permutations , 12 9 pseudograph, 6 Konig Theorem , 1 4 quasiconsistent system , 15 0 Konig-Frobenius Theorem , 19 2 Kronecker produc t o f matrices, 2 9 r-indecomposable matrix , 20 0 r-primitivity, 214 , 25 0 Latin configuration , 4 strong, 214 , 25 6 Latin rectangle , 4 r-regular hypergraph , 9 2 Latin square , 5 rank o f a hypergraph , 9 2 lattice, 13 9 reducible graph , 17 4 rook polynomial , 12 9 ra-permutation, 2 4 row-composite class , 6 3 Markov chain , 249 , 26 1 Ryser class , 1 6 homogeneous, 26 1 monotone, 1 7 strongly ergodic , 26 2 weakly ergodic , 26 1 saturation index , 24 1 maximal matrix , 1 7 Schur function , 1 1 minimal a-covering , 9 2 Schur Lemma , 22 5 minimal connecte d graph , 16 8 scope, 20 6 minimal covering , 14 4 semi-irreducible graph , 17 3 monoid wit h uniqu e cycli c decomposition , semi-tot ally indecomposabl e matrix , 17 1 128 Smirnov sequence , 12 4 monomial, 20 8 cyclic, 12 6 multigraph, 6 standard for m o f a matrix, 9 7 multiplicative rank , 20 5 stochastic configuration , 4 multiplier o f a differenc e set , 3 7 strongly connecte d digraph , 21 5 structure, 13 9 structure matrix , 1 7 subconfiguration, 8 5 support o f a matrix, 2 symmetric bloc k design , 3 6

^-covered element , 14 6 ^-minimal covering , 14 5 t-quasiconsistent system , 15 0 term rank , 1 0 maximal, 2 6 totally indecomposabl e matrix , 161 , 17 0 tournament, 213 , 22 7 irreducible, 22 7 primitive, 22 8 reducible, 22 7 transitive, 213 , 230 transformation o f a matrix, 18 8 transversal, 1 0 transversal number , 9 2 transversal numbe r fo r a hypergraph, 9 2 undirected graph , 6 unimodal sequence , 6 1 vertex connectivity , 5 4 weak ergodicit y theorem , 25 2 weakly totall y indecomposabl e matrix , 17 1 weight function , 2 width o f a matrix , 9 1 Titles i n Thi s Serie s

213 V . N . Sachko v an d V . E . Tarakanov , Combinatoric s o f nonnegative matrices , 200 2 212 A . V . Mel'nikov , S . N . volkov , an d M . L . Nechaev , Mathematic s o f financial obligations, 200 2 211 Take o Ohsawa , Analysi s o f severa l comple x variables , 200 2 210 Toshitak e Kohno , Conforma l field theor y an d topology , 200 2 209 Yasumas a Nishiura , Far-from-equilibriu m dynamics , 200 2 208 Yuki o Matsumoto , A n introductio n t o Mors e theory , 200 2 207 Ken'ich i Ohshika , Discret e groups , 200 2 206 Yuj i Shimiz u an d Kenj i Ueno , Advance s i n modul i theory , 200 2 205 Seik i Nishikawa , Variationa l problem s i n geometry , 200 1 204 A.M . Vinogradov , Cohomologica l analysi s o f partial differentia l equation s an d Secondary Calculus , 200 1 203 T e Su n Ha n an d King o Kobayashi , Mathematic s o f information an d coding , 200 2 202 V . P . Maslo v an d G . A . Omel'yanov , Geometri c asymptotic s fo r nonlinea r PDE . I , 2001 201 Shigeyuk i Morita , Geometr y o f differentia l forms , 200 1 200 V . V . Prasolo v an d V . M . Tikhomirov , Geometry , 200 1 199 Shigeyuk i Morita , Geometr y o f characteristic classes , 200 1 198 V . A . Smirnov , Simplicia l an d opera d method s i n algebrai c topology , 200 1 197 Kenj i Ueno , Algebrai c geometr y 2 : Sheaves an d cohomology , 200 1 196 Yu . N . Lin'kov , Asymptoti c statistica l method s fo r stochasti c processes , 200 1 195 u Wakimoto , Infinite-dimensiona l Li e algebras, 200 1 194 Valer y B . Nevzorov , Records : Mathematical theory , 200 1 193 Toshi o Nishino , Functio n theor y i n severa l comple x variables , 200 1 192 Yu . P . Solovyo v an d E . V . Troitsky , C*-algebra s an d ellipti c operator s i n differentia l topology, 200 1 191 Shun-ich i Amar i an d Hirosh i Nagaoka , Method s o f informatio n geometry , 200 0 190 Alexande r N . Starkov , Dynamica l system s o n homogeneou s spaces , 200 0 189 Mitsur u Ikawa , Hyperboli c partia l differentia l equation s an d wav e phenomena , 200 0 188 V . V . Buldygi n an d Yu . V . Kozachenko , Metri c characterizatio n o f random variable s and rando m processes , 200 0 187 A . V . Pursikov , Optima l contro l o f distributed systems . Theor y an d applications , 200 0 186 Kazuy a Kato , Nobushig e Kurokawa , an d Takesh i Saito , Numbe r theor y 1 : Fermat's dream , 200 0 185 Kenj i Ueno , Algebrai c Geometr y 1 : From algebrai c varietie s t o schemes , 199 9 184 A . V . Mel'nikov , Financia l markets , 199 9 183 Hajim e Sato , Algebrai c topology : a n intuitiv e approach , 199 9 182 I . S . Krasil'shchi k an d A . M . Vinogradov , Editors , Symmetrie s an d conservatio n laws fo r differentia l equation s o f mathematical physics , 199 9 181 Ya . G . Berkovic h an d E . M . Zhmud' , Character s o f finite groups . Par t 2 , 199 9 180 A . A . Milyuti n an d N . P . Osmolovskii , Calculu s o f variations an d optima l control , 1998 179 V . E . VoskresenskiT , Algebrai c group s an d thei r birationa l invariants , 199 8 178 Mitsu o Morimoto , Analyti c functional s o n the sphere , 199 8

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