Mathematical Computation June 2014, Volume 3, Issue 2, PP.38-43
The Minimum Ranks of n n (n=6, 7) ZeroNonzero Patterns Gufang Mou 1. School of Mathematical Sciences/University of Electronic Science and Technology of China / University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, P. R. China 2. Department of Mathematics/Chengdu Normal University/ Chengdu Normal University, Chengdu, Sichuan, 611130, P. R. China #Email: mougufang1010@163.com
Abstract For a not necessarily symmetric matrix, the zero-nonzero pattern P of entries can be described by a digraph ( P) which has an arc if an entry is nonzero. The minimum rank of not necessarily symmetric matrix zero-nonzero patterns is defined to be the smallest possible rank over all real matrices having the given zero-nonzero patterns. In this paper, we study the problem for the minimum rank of an n n (n=6,7) not necessarily symmetric matrix zero-nonzero pattern whose graph is an oriented graph, and show that mr((P))=tri((P)) for an n n (n=6,7) zero-nonzero pattern. Keywords: Minimum Rank; Directed Graph; Oriented Graph; Zero-Nonzero Pattern
1 INTRODUCTION A not necessarily symmetric zero-nonzero pattern P can be described by a digraph ( P) . The minimum rank problem, which asks us to determine the minimum rank among all real matrices whose zero-nonzero pattern of entries is described by a given digraph or zero-nonzero pattern or sign pattern. The minimum rank problems for digraphs (see [1], [3]), zero-nonzero patterns (see [2]) and sign patterns (see [3]) have been recently considered. In this paper, we study the problem for the minimum rank of an n n (n=6, 7) not necessarily symmetric matrix zero-nonzero pattern whose graph is a directed graph, and show that mr ((P)) =tri ((P)) for an (n=6, 7) zero-nonzero pattern. A graph is a pair G= (V, E), where the set of vertices V is {1, 2, , n} , and E is the edge or arc. The order of G, denoted by |G|, is the number of vertices of G. A simple graph is a graph that does not have multiple edges or loops. A digraph allows loops (but not multiple copies of the same arc) and is denoted by (V , E ) where V and E are the sets of vertices and arcs of . A digraph is symmetric if whenever (u, v) is an arc of , then (v, u) is an arc of as well. If a digraph has the property that for each pair (u, v) of distinct vertices of , at most one of (u, v) and (v, u) is an arc of , then is an oriented graph. A tournament denoted T is defined as a digraph such that for every pair (u, v) of distinct vertices, exactly one of (u, v) and (v, u) is arc. Therefore, a tournament is an orientation of a complete graph. A tournament T is transitive if whenever (u, v) and (v, w) are arcs of $T$ then (u, v) is also an arc of T. A zero-nonzero pattern is a matrix having entries in {*, 0}, where * indicates a nonzero entry. For a n n symmetric zero-nonzero pattern A, the undirected simple graph of A is G (A) with vertex set {1, 2, , n} . In G (A), (i, j) is an edge if and only if aij 0 , where (i, j) and (j, i) are regarded as the same edge. Whereas the not necessarily symmetric zero-nonzero pattern P is described by a digraph that has the corresponding arc if an entry is nonzero. Note that a digraph may have loops and the diagonal entries of P determine the presence or absence of loops in . Specially, all diagonal entries of P are nonzero when each vertex of has a loop. Example 1.1 Assume - 38 www.ivypub.org/mc
a11 0 0 P 0 0 0
a12 a22
a13 a23
a14 a24
0 0
0 0 a52
a33 0 0
a34 a44 0
a35 a45 a55
a62
a63
0
0
0 0 0 a46 a56 a66
whose digraph is (V , E ) with a loop at each vertex (see Fig. 1).
FIG. 1
The zero-nonzero pattern of is
0 0 0 0 0
0 0 0 0 0 0 . 0 0 0 0 0 0
Let ( P) denote the collection of not necessarily symmetric zero-nonzero pattern P, The minimum rank of a digraph is mr ( )=min{rank(P): P P() }, and the maximum nullity of a digraph is defined to be M ( ) =max {null (P): P P() }. Clearly mr( )+M( )=| |. Recall that the minimum rank of a zero-nonzero pattern P does not change under permutation similarity (denoted P ). Definition 1.2 [2] A k-triangle of a n n zero-nonzero pattern P is a k k subpattern that is permutation equivalent to a (lower) triangle pattern whose diagonal is all *, s . The maximum size of a triangle in P is called the triangle number of pattern P, denoted tri (P).
2 THE MINIMUM RANK PROBLEM FOR AN n n (n 6,7) ORIENTED GRAPH In this Section, we obtain that mr ( ( P) ) =tri ( ( P) ) for an n n (n=6, 7) oriented graph. Throughout this paper, we assume that all diagonal entries of zero-nonzero pattern P are nonzero and each vertex in ( P) has a loop. Lemma 2.1 [2] A semi-standard pattern contains a (1) 2-triangle iff it contains a 0; (2) 3-triangle iff it contains a line with two 0, s iff contains a (2:1) pattern with just one *; (3) 4-triangle iff it contains a 2-by-2 zero block; - 39 www.ivypub.org/mc
(4) 5-triangle iff contains a (3:1) pattern with just one *; (6) 6-triangle iff contains a pattern permutation equivalent to
0 0 0 0
0 0 0 0 0 0 0 0 . 0 ?
The zero-nonzero subpattern of a pattern P of size n n lying in rows and columns , , {1, 2, denoted by P[ | ] , and the principal submatrix P[ | ] is abbreviated to P[ ] .
, n} is
P Q Let the partitioned (block) matrix M where the matrix P is nonsingular; the matrix M need not be square. R S Then M/P=S-RP-1Q is the Schur complement of P in the partitioned matrix M. In addition, we refer to the Guttman rank formula: rank (M) =rank (P) +rank (M/P).
Lemma 2.2. If T is a transitive tournament with a loop at each vertex, then an n n zero-nonzero pattern P(T) of T can be obtained a permutation matrix P such that P (T)=PP(T)PT is triangle. Theorem 2.3. Let be an oriented graph of order 6. There are the eight following possibilities for the 6 6 zerononzero patterns P() , then mr( )=tri( ). Proof. 1 and 4 are transitive, then mr( 1 )=tri( 1 ) and mr( 4 )=tri( 4 ). The zero-nonzero pattern of 2 is
0 0 P ( 2 ) 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 P ( 2 ) 0 0 0
0 0 0 0 0 0 0 . 0 0 0 0 0 0
is permutation equivalent to
Then
IF THE ★ ENTRY IS 0, THEN MR ( P / P[{2,3,4,5,6}]) 0 AND MR ( P) TRI ( P) 5 . THUS, MR( P(2 ) )= TRI ( P(2 ) ). The zero-nonzero patterns of 3 is - 40 www.ivypub.org/mc
0 0 P ( 3 ) 0 0 0
0 0 0 . 0 0 0 0 0 0 0 0
0 0 P ( 5 ) 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 P ( 5 ) 0 0
0 0 0 0 . 0 0 0 0 0 0 0 0 0
Obviously, mr( P(3 ) )=tri( P(3 ) ). The zero-nonzero pattern of 5 is
is permutation equivalent to
Then
IF THE ★ ENTRY IS 0, THEN MR ( P / P[{2,3,4,5,6}]) 0 AND MR ( P) TRI ( P) 5 . THUS, MR( P(5 ) )= TRI ( P(5 ) ). The zero-nonzero pattern of 6 is
0 0 P ( 6 )
0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 P ( 6 ) 0
0 0 0 0 0 0 . 0 0 0 0 0 0 0 0
is permutation equivalent to
Then - 41 www.ivypub.org/mc
IF THE ★ ENTRY IS 0, THEN MR ( P / P[{1,2,3,4,5}]) 0 AND MR ( P) TRI ( P) 4 . THUS, MR( P(6 ) )= TRI ( P(6 ) ). The zero-nonzero patterns of 7 and 8 are
0 0 P(T7 ) 0 0
0 0 0 0 0 0 and P(T8 ) 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 . 0 0 0 0 0 0 0 0 0
Obviously, mr(P( 7 ))=tri(P( 7 )) and mr(P( 8 )=tri(P( 8 )). Remark. If a digraph is not oriented graph, mr( ) may be not equal to tri( ). Example 2.4 [2]. Consider the 7 7 zero-nonzero 0 0 P 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
is not an oriented graph of order 7. According to Lemma 2.1, tri(P)=3. And mr(P)=4≠ tri(P). Example 2.5. Let be an oriented graph of order 7, P ( ) is 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
and P is permutation equivalent to 0 0 P 0
0 0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 0
According to Lemma 2.2, tri ( P) 4 , mr ( P) mr {P[2,3,4,5]} mr {P / P[2,3,4,5]} , and mr {P[2,3, 4,5]} 4. - 42 www.ivypub.org/mc
IF THE ★ ENTRY IS 0, THEN MR ( P / P[{2,3,4,5}]) 0 AND MR ( P) 4. THUS, MR( P() )=TRI( P() ).
REFERENCES [1]
Francesco Barioli, Shaun M. Fallat, H. Tracy Hall, Daniel Hershkowitz, Leslie Hogben, Hein van der Holst, Bryan Shader, on the minimum rank of not necessarily symmetric matrices: A preliminary study, Electron. J. Linear Algebra, 18(2009): 126-145.
[2]
Charles R.Johnson, Joshua A. Link, The extent to which triangular sub-patterns explain minimum rank, Discrete AppliedMath., 156(2008) 1637-1651.
[3]
Leslie Hogben, A note on minimum rank and maximum nullity of sign patterns, Electron. J. Linear Algebra, 22 (2011) 203-213.
AUTHOR 1Gufang
Mou (1981- ) was born in Hubei province in. PhD in School of Mathematical Sciences, University of
Electronic Science and Technology of China, Chengdu. The author’s major field is properties of special matrices by applying theories and methods of graphs. She once worked as a teacher in Chengdu Normal University, University of Electronic Science and Technology of China.
- 43 www.ivypub.org/mc