J. Korean Math. Soc. 30 (1993), No. 2, pp. 267{274 INDICES OF IRREDUCIBLE BOOLEAN MATRICES Han Hyuk Cho 1.Introduction Let = f0 1g be the Boolean alge

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1 J. Korean Math. Soc. 30 (1993), No. 2, pp. 267{274 INDICES OF IRREDUCIBLE BOOLEAN MATRICES Han Hyuk Cho 1.Introduction Let = f0 1g be the Boolean algebra of order two with operations (+ ) : 1+0= 0+1= 1+1= 1 1 = 1 & 0+0= 0 1 = 1 0 = 0 0=0and an order : 0 < 1. Then under these Boolean operations, the set B n of all n n matrices over (Boolean matrices) forms a multiplicative matrix semigroup. There have been many researches on various semigroup properties of B n. In this paper we study the indices of irreducible Boolean matrices in B n. Definition 1.1. Let A be an n n Boolean matrix in B n. The index of A and the period of A are the least positive integers q and p respectively such that A q = A q+p. For A 2 B n, index(a) and period(a) denote the index and the period of A respectively, and (A) denotes the number of one's of A. A 2 B n is called a J ;matrix if (A) is n 2, and A is primitive ifa q is a J ;matrix for some integer q. For an n n Boolean matrix A = [a ij ], the associated digraph of A, denoted by G(A), is the digraph with vertices f1 2 ::: ng such that there is an arc from i to j if and only if a ij > 0. A path from i to j of length l in G(A) is a sequence of vertices (v 0 = i v 1 ::: v l = j) such that a vk v k+1 =1foreach k 2f0 1 ::: l; 1g. A path is a simple path if v 1 ::: v l are all distinct, and a simple path is a cycle if v 0 = v l. Then we can interpretes many properties of A 2 B n in terms of its digraph G(A). For example, a Boolean matrix A 2 B n is primitive if and only if its associated digraph G(A) is strongly connected (i.e. for any vertices i and j in G(A) there is a path from i to j). Received February 11, Revised January Partially supported by the Korea Science and Engineering Foundation

2 268 Han Hyuk Cho Definition 1.2. Let A 2 B n be an n n Boolean matrix. Then A is irreucible if A is not permutationally similar to a matrix of the form B1, where B 0 B i 's are square matrices. A is reducible if A is not 2 irreducible. A is nearly reducible if deleting any positive entry of A results in a reducible matrix. If A is an irreducible matrix of period p, then A is called a p;irreducible matrix. A 2 B n is called a cyclically p;partite matrix if A is permutationally similar to the following matrix 0 = D 1 B 1 0 D 2 B 2 0 D 3 B p;1 B p 0 D p 1 C A where the block matrices D k 's on the main diagonal of are square zero matrices. For each k and m, welet k (m) =B k B k+1 B k+m;1 and k = k (p), where B k+i represents B j of ifk + i j(mod p). Lemma 1.3. Consider the matrix in the Denition 1.2. Then, (1) period( i ) = period( j ). (2) j index( i ) - index( j ) j 1. (3) If is irreducible, then index() is the smallest integer q such that k (q) is a J ;matrix for any k. Proof. Refer to Cho [3]. 2. Frobenius Numbers and Circum-diameters For the semigroup R n of n n real matrices, we say M 2 R n is power convergent in R n if the powers M M 2 M 3 M q form a convergent sequence in R n. It is well known that the power convergence of M is closely related to the set of eigenvalues of M. For the semigroup B n of Boolean matrices, any p;irreducible matrix A 2 B n has a nite index in B n, and the circumferences of A is closely related to the index of A as follows.

3 Indices of Irreducible Boolean Matrices 269 Definition 2.1. Let C = fc 1 c 2 c g be a nite set of relatively prime positive integers. The Frobenius number '(C) ofc is the smallest integer q such that any integer h( q) can be expressed as a nonnegative linear combination of c i 's (i.e. h = P a i=1 ic i, where a i 's are nonnegative integers). In general, for a nite set of positiveintegers C = fc 1 c 2 c g with the greatest common divisor gcd(c) =p, the Frobenius number '(C) of C is p '(d 1 d 2 d ), where d i = c i p. Let A 2 B n be an n n Boolean matrix, and let G(A) be its associated digraph with vertices f1 2 ::: ng. If there is a c;cycle (cycle of length c), then such integer c is called a circumference of A (and of G(A)). ;A and (A) denote respectively the set of all the circumferences of A and the cardinality of ;A. Now let A be irreducible and C be a subset of ;A. For any vertices s and t in G(A), P (s t) =f j is a path from s to tg and Q(C s t) =f j is a circumpath from s to t w.r.t. Cg (i.e. is a path from s to t such that meets with a p;cycle for each p 2 C). Then the distance d(s t) from s to t is the minimum of flength of j 2 P (s t)g, and the circum-distance (C s t) from s to t w.r.t. C is the minimum of the set flength of j 2 Q(C s t)g. Finally the circum-diameter A (C) of A w.r.t. C is the maximum of f(c s t)js t 2 G(A)g, and ' A (C) denotes '(C) if C is a subset of ;A. Lemma 2.2. Let A 2 B n be a p;irreducible matrix. Then index(a) ' A (C)+ A (C) for any subset C of ;A with gcd(c) =p. Proof. Refer to Cho [3] B C Consider the 4 by 4 Boolean matrix W A. Then the characteristic polynomial of W (as a real matrix) is 4 ; ; 1, and W is power divergent inr 4 since there exists an eigenvalue whose absolute value is greater than 1. But W (as a Boolean matrix) is power convergent inb n, and index(a) =' W (C)+ W (C). Now consider the primitive matrix

4 270 Han Hyuk Cho A = : B A Then ;A = f4 5 7g ' A (4 5 7) = 7 A (4 5 7) = 7: But index(a) = 12 < 14 = ' A (4 5 7) + A (4 5 7): Thus there exists a Boolean matrix A such that index(a) < ' A (C) + A (C). Note that if p and q are relatively prime integers, then '(p q) =(p ; 1)(q ; 1): 3. Gaps in the Index Set E p n For the given positive integers n and p(p n), let En p be the index set findex(a)ja 2 B n is p;irreducible g and men p be the maximum element of En. p Also let G p n be the set fgjg is a positive integer less than men p and g =2 En p g. Then any integer in Gp n is called a gap of En. p Shao [11] and Min [9] proved that there is no gap less than! n +1in the index set En(n 1 6= 11), where! n = n2 ;2n+2. Moreover 2 Lewin and Vitek [7] specied the gaps greater than! n + 1 in En. 1 In this section we investigate the gaps in the index set En p of n n p;irreducible matrices using their results and the index properties of cyclically p;partite matrices. Definition 3.1. E n (i j) denotes an n n Boolean matrix whose (i j);entry is the only nonzero entry. Each n n permutation matrix P can be expressed as a Boolean sum P n E i=1 n(i (i)), where is an element of the symmetric group S n representing P. Let n = p + with =[ n p ] for some positive integers p( n), and let M n p denote the set fa 2 B n ja is p;irreducible and index(a) >p(! +1)+g. Lemma 3.2. Let C = fc 1 c 2 c g be a nite set of positive integers with 2 c 1 < c 2 < < c n and gcd(c) = p. If c + c ;1 n, then E p n(c) E p n, where Ep n(c) = f'(c) + c ; p '(C)+c ; p +(n ; c ;1 )g.

5 Indices of Irreducible Boolean Matrices 271 Proof. Let u = c ;1, v = c, and M(C) = (v) + P E i=1 v(c i ). Here, + is the Boolean sum, (v) =E v (v 1) + P v;1 E i=1 v(i i +1),and E v (c i ) = P j2s i E v (c i j), where S i = fjjj > 0 and c i ; j +1 2 Cg. For integers s and t with 0 s v ; u and 0 t < u, consider the (v + t) (v + t) matrix where M(C s) =M(C)+ P s E i=1 v(i + u). For n with v n<u+ v M(C s t) Q M n (C s t) = R 0 is an n n matrix such that the i;th row (respectively column) of R(Q) is the rst row (column) of M(C s t) for any i. If u + v > n, then the index of the above M n (C s t) is '(C) +(v ; p) +(v ; u ; s) +t. Now let u + v = n, and consider the n n matrix S = Mn;1 (C 0 u; 1) 0 +E n (u ;1 n)+e n (n u+1). Then the index of S is '(C)+(v ; p)+(n ; u). Therefore we have the lemma. It is well known that any p;irreducible matrix is a cyclically p;parti te matrix. Also note that permutationally similar matrices have the same index. Thus without loss of generality we will assume that the matrix A of Lemma 3.3 and 3.4 is of the form (in Denition 1.2). Lemma 3.3. If A 2 M p n( 3), then ;A = fpg phg with g+h >.

6 272 Han Hyuk Cho Proof. First let p = 1. It is known that for a primitive matrix A 2 B, (A) 3 means index(a)! +1[7]. If (A) is 1, then the girth of A is 1 and index(a)! +1. Also if ;A = fg hg with g + h, then index(a) (g ; 1)( ; g ; 1) + 2 ; g ; 1 = ;g 2 + g( ; 1) + g! +1. Now let p 2. If index( 1 ) is the minimum of the set findex( i )g and if there are many i 's such that index( i ) > index( 1 ), then index(a) p(index( 1 )) +. Thus if (A) = 1, then the girth of A is p and index(a) p + p(! +1)+. Next if (A) 3, then index(a) p(! +1)+ since there are at most many i 's whose order is greater than. Now let ;A = fpg phg with g < h, and let be the minimum of forder of i g and the order of k be. Note that if g + h, then index( k ) (g ;1)(;g ;1)+2 ;g ;1! +1. Since at least p; many i 's are of order less than +1, index(a) p(! +1)+. Thus if index(a) >p(! +1)+, then ;A = fpg phg with g + h>. Lemma 3.4. Let A 2 B n ( 3) be p;irreducible, ;A = fpg phg (g <h), andq = g(h ; 1). If every i is nonsymmetric and g + h>, then pq index(a) pq + p( ; g)+ is a sharp inequality. Proof. If p = 1, then A is primitive and the lemma holds by the results of Lewin and Vitek [7]. From their results, if A is symmetric, then index(a)! +1. For each h;cycle of G(A), there is an arc (s t) in with d(t s) =h ; 1, and g(h ; 1) index(a) g(h ; 1) + ( ; g). Thus in the following proof, we will assume that p 2. We also let u = pg and v = ph. (Upper bound) Since A is a p;irreducible matrix, without loss of generality, let A be of the form (in Denition 1.2). If the order of i is m for some i, then index( i ) q +(m ; g). If m <, then index( j ) q +( ; g) for any j, and index(a) pq + p( ; g) in this case. Now let the minimum of forder of i g be. Then there are at most many i 's such that the order of each i is greater than. Therefore by the structure of A(= ) and Lemma 1.3, index(a) pq + p( ; g)+. (Lower bound) Consider a v;cycle of G(A) labeled as (v 1 v 2 v v v 1 ). Without loss of generality let d(v p+1 v 1 ) be v ; p (so there is no path of length pq ; p from v p+1 to v 1 ). If index(a) pq ; 1, then there is a path j of length pq ; 1 from v p+1;j to v p;j for 0 j <p.

7 Indices of Irreducible Boolean Matrices 273 Note that the length of i can be expressd as a sum of the length of a simple path from v p+1;j to v p;j and some circumferences of A. Also note that p(g ; 1)(h ; 1) ; p(= pgh ; pg ; ph) cannot be expressed as a nonnegative linear combination of u and v. So there is a path of length u ; 1 from v p+1;j to v p;j for 0 j < p since pq ; 1 = p(gh ; g) ; 1=pgh ; pg ; 1=pgh ; pg ; ph +(v ; 1). By the similar reason, we can construct a path of length k(u ; 1) from v p+1 to v p+1;k for 0 < k p. Then there is a path of length pq ; p from v p+1 to v 1 since pq ; p = u(h ; 1) ; p = u(h ; p ; 1) + p(u ; 1), contradiction. Thus index(a) pq. (Sharpness) By the Lemma 3.2. Theorem 3.5. For each ( 3) and n = p + with = [ n p ]( 3) [p( 2 ; +[( +1) 2 =4]) + 1 p( 2 ; ( ; 1) +( ; 2)) ; 1] is a gap interval greater than p(! +1)+ in the index set E p n. Proof. Let Mn[" p ] denote the set fa 2 Mn p j;a = fpg phg(g <h) such that " = h ; g +1 and = ; hg. For a positive integer, M p n[] denotes the set S "+2= M p n[" ]. Now choose any A 2 M p n. Then we may assume that ;A = fpg phg(g < h) with g + h > by Lemma 3.3, and A 2 M p n[] for some. From Lemma 3.4 we can obtain an inequality p( 2 ;+[; 2 +( ;2) +( ;1)] index(a) p( 2 ;+[; 2 +( ;3)+2( ;1)]). Since =( ;")=2 the minimum possible value of [; 2 +( ;2) +( ;1)] is ;1 when =0by simple calculation, and the maximum possible value of [; 2 +(;3)+2(;1)] is [( +1) 2 =4] when is ( ; 3)=2. Therefore there does not exist any matrix in M p n whose index lies between p( 2 ; +[( +1) 2 =4]) + 1 and p( 2 ; ( ; 1) +( ; 2)) ; 1. It is our belief that if 14 or 8, then there is no gap less than p(! +1)+ in the index set E p n. References 1. A.Berman and R.J. Plemmons, Nonnegative Matreces in the Mathematical Sciences, Academic Press, New York, R.A. Brualdi and B. Liu, Generalized exponents of primitive directed graphs, J. Graph Theory (4) 14 (1990),

8 274 Han Hyuk Cho 3. H.H. Cho, On the indices of the Boolean matrices, Proc. Workshops Pure Math. 9 (1989), A.L. Dulmage and N.S. Mendelsohn, Gaps in the exponent set of primiteve matrices, Illinois J. Math. 8 (1964), K.H. Kim, Boolean Matrix Theory and Applications, Pure and Applied Mathematics, 70, Marcel Dekker,New York, K.H. Kim, An extension of the Dulmage-Mendelsohn theorem, Linear Algebra and Appl. 27 (1979), M. Lewin and Y. Vitek, A system of gaps in the exponect set of primitive matrices, Illinois J. Math. 25 (1981), G. Markowsky, Bounds on the index and period of a binary relation on a nite set, Semigroup Forum. 13 (1977), Z.K.Min, On Lewin and Vitek's conjecture about the exponent set of primitive matrices, Linear Algebra and Appl. 96 (1987), H.Minc, The structure of irreducible matrices, Lin. and Multilinear Algebra 2 (1974), J. Shao, On a conjecture about the exponent set of primitive matrices, Linear Algebra and Appl. 65 (1985), Y.Vitek, Bounds for a linear diophantine problem of Frobenius, J.London Math. Soc. 10 (1975), Department of Mathematics Ed. Seoul National University Seoul , Korea

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Nonnegative Matrices I Daisuke Oyama Topics in Economic Theory September 26, 2017 References J. L. Stuart, Digraphs and Matrices, in Handbook of Linear Algebra, Chapter 29, 2006. R. A. Brualdi and H. J.