Generic maximum nullity of a graph

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1 Generic maximum nullity of a grap Leslie Hogben Bryan Sader Marc 5, 2008 Abstract For a grap G of order n, te maximum nullity of G is defined to be te largest possible nullity over all real symmetric n n matrices A wose (i, j)t entry (for i j) is nonzero wenever {i, j} is an edge in G and is zero oterwise. Maximum nullity and te related parameter minimum rank of te same set of matrices ave been studied extensively. A new parameter, generic maximum nullity, is introduced. Generic maximum nullity provides insigt into te structure of te null-space in a matrix realizing maximum nullity of a grap. It is sown tat generic maximum nullity is bounded above by edge connectivity. Keywords: minimum rank, maximum nullity, maximum corank, generic maximum nullity, grap, rank, nullity, corank, symmetric matrix. AMS Classification: 05C50, 5A03, 5A8 Introduction Te (real symmetric) minimum rank problem for a simple grap asks us to determine te minimum rank among real symmetric matrices wose zero-nonzero pattern of off-diagonal entries is described by a given simple grap G, or equivalently to determine of te maximum nullity (or maximum multiplicity of an eigenvalue) among te same family of matrices. All graps discussed in tis paper are simple, meaning no loops or multiple edges, undirected, finite, and ave nonempty vertex sets. Te order of a grap G, denoted G, is te number of vertices of G. Te set of n n real symmetric matrices will be denoted by S n. For A S n, te grap of A, denoted G(A), is te grap wit vertices {,..., n} and edges {{i, j} : a ij 0, i < j n}. Note tat te diagonal of A is ignored in determining G(A). Te set of real symmetric matrices of a grap G is S(G) = {A S n : G(A) = G}. Te minimum rank of a grap G is Te maximum nullity of G is mr(g) = min{rank(a) : A S(G)}. M(G) = max{null(a) : A S(G)} were null(a) is te dimension of te null space ker(a) of A. Clearly M(G) + mr(g) = G. If A S(G) and α R, ten A + αi S(G), so te maximum multiplicity of any eigenvalue is te same as maximum multiplicity of eigenvalue 0, i.e., te maximum nullity. See [3] for a survey Department of Matematics, Iowa State University, Ames, IA 500, USA (logben@iastate.edu) and American Institute of Matematics, 360 Portage Ave, Palo Alto, CA (ogben@aimat.org). Department of Matematics, University of Wyoming, University of Wyoming, Laramie, WY 8207, USA (bsader@uwyo.edu).

2 of known results and discussion of te motivation for te minimum rank problem; an extensive bibliograpy is also provided tere. If W, U {, 2,..., n} and B S n, ten B[W, U] denotes te submatrix of B aving rows indexed by W and columns indexed by W. In case W = U, tis is a principal submatrix and is denoted by B[W ]; te complementary principal submatrix obtained from B by deleting te rows and columns indexed by W is denoted B(W ). In te special case wen W = {k}, we use B(k) to denote B(W ). A grap G = (V, E ) is a subgrap of grap G = (V, E) if V V, E E. Te subgrap G[W ] of G = (V, E) induced by W V is te subgrap wit vertex set W and edge set {{i, j} E i, j W }; G(W ) is used to denote G[V \ W ], obtained from G by deleting all te vertices in W and all edges incident wit tese vertices. If S E, te subgrap G S is te subgrap obtained by deleting te edges in S, i.e., te grap wit vertex set V and edge set E \ S. A pat on n vertices, a cycle on n vertices, and a complete grap on n vertices will be denoted by P n, C n, and K n respectively. A grap is connected if tere is a pat from any vertex to any oter vertex. A component of a grap is a maximal connected subgrap. A set W of vertices of G is a separating set or vertex cut if G(W ) as more tan one component. Te vertex connectivity of G, denoted κ v (G) is te minimum size of a separating set of G. A set S of edges of a grap G (wit G > ) is a disconnecting set if G S as more tan one component. Te edge connectivity of G, denoted κ e (G) is te minimum size of a disconnecting set of G. Given W, U V (G), te set of edges of G aving one endpoint in W and te oter in U is denoted [W, U]. An edge cut is a set of edges of te form [W, V (G) \ W ] for some W V (G). Every edge cut is a disconnecting set but not every disconnecting set is an edge cut. However, a minimal disconnecting set is an edge cut (cf. [5, p. 52]). Te degree of a vertex is te number of edges incident wit te vertex, and te minimum degree over all vertices of a grap G will be denoted by δ(g). Since te edges incident wit a vertex of minimal degree constitute a disconnecting set, and since te endpoints in W of eac edge in an edge cut [W, V (G) \ W ] constitutes a separating set, κ v (G) κ e (G) δ(g). Tese inequalities can be strict (e.g., see [5, p. 53]) A n k real matrix X is generic if every square submatrix of X is nonsingular. A generic matrix could be called a totally nonsingular in analogy wit te definition of a totally positive matrix as a matrix all of wose minors are positive. Clearly a totally positive matrix is generic. Notice tat any submatrix of a generic matrix is generic. Te generic nullity of a nonzero matrix A R n n is GN(A) = max{k : X R n k, AX = 0, and X is generic} (te generic nullity of an n n zero matrix is n). Te maximum generic nullity of a grap G is GM(G) = max{gn(a) : A S(G)}. Te maximum generic nullity of a grap can be strictly less tan te maximum nullity. In tis case, te null space of an optimal matrix is often igly structured. Example.. Let G = G30 be te grap sown in Figure (te numbering of graps is taken from [4]). Since G can be covered by two copies of K 3 and one K 2, mr(g) 3 and since G as an induced P 4, mr(g) 3. Tus M(G) = 6 3 = 3. We claim tat GM(G). Suppose to te contrary tat tere is a generic 6 2 matrix X = [ ] x x 2 wose columns xi are in te nullspace of A S(G). Te nonzero pattern of 2

3 Figure : Te grap G = G30 in Example. A S(G) is? 0 0 0? 0 0 0? ?, 0 0 0? 0 0 0? were denotes a nonzero entry and? denotes an entry about wic noting is known. Columns 2, 3 and 4 (and columns 3, 4, and 5) are clearly independent. If rank(a) = 3 ten te first two columns ave rank and te last two columns ave rank. So tere are vectors in te null space of A of te forms y = [,, 0, 0, 0, 0] T and z = [0, 0, 0, 0,, ] T. Any linear combination of y and z as te 3rd and 4t coordinates equal to 0. Since tere cannot be a relationsip between te rows of X, x, x 2, y, z are linearly independent, wic implies rank(a) 2, a contradiction. If rank(a) = 4 ten eiter te first two rows ave rank and te last two rows ave rank ; assume te first case. Ten tere is a vector in te null space aving all te last four entries 0, ence independent of x and x 2, and again a contradiction is obtained. Te matrix in S(G) wit nonzero off-diagonals equal to and row sums 0 as nullspace spanned by te vector of all s, and ence GM(G) =. Our main result about maximum generic nullity is tat for any connected grap G, GM(G) κ e (G). Tis will be establised in Section 2 using metods based on te ideas in Example.. Using te metods of [], it is easy to sow tat GM(G) δ(g), but we do not include tat proof since κ e (G) δ(g). 2 Maximum generic nullity and edge connectivity A nonzero pattern C = [c ij ] is a m n matrix wose entries c ij are elements of {, 0}. Te number of (nonzero entries) in C is denoted by nz(c). Given a pattern C = [c ij ], we let Q(C) denote te set of all matrices A = [a ij ] R m n suc tat a ij 0 if and only if c ij =. Note tat (unlike te set of symmetric matrices described by a grap), ere te diagonal is constrained by te nonzero pattern. Te minimum rank of a nonzero pattern C is mr(c) = min{rank(a) : A Q(C)}. Teorem 2.. If C is an m n nonzero pattern tat does not ave any zero row or zero column, mr(c) m + n nz(c). Proof. Note tat arbitrary permutation of rows or columns of C does not affect mr(c). For fixed m and n, te proof is by induction on nz(c). Te base case is any C (witout zero row or column) suc tat for every nonzero entry, it is te only nonzero in its row or te only nonzero in its column. 3

4 [ ]? Tat is, no row and column permutation of C contains a 2 2 submatrix. By row and column permutations, any suc a C can be put into te following form: a a 2 a s }{{} b }{{} b 2 }{{} b t nz(c) = a + + a s + b + + b t m = a + + a s + t n = s + b + + b t m + n nz(c) = t + s = mr(c) [ ]? Now assume C contains a 2 2 submatrix. Consider te nonzero pattern C obtained [ ]? from C by replacing one by 0 so te 2 2 submatrix is now. Ten by te induction 0 ypotesis applied to C, mr(c) mr(c ) m + n nz(c ) = m + n (nz(c) ) = m + n nz(c). Teorem 2.2. If G is connected, ten GM(G) κ e (G). Proof. Let S be a minimal disconnecting set for G (so κ e (G) = S ). Since S is an edge cut, S = [W, W ] for some W V. Let W = W and W 2 = W. Number te vertices of G so tat te vertices of W are,..., W, all vertices of W incident wit an edge of S are last among te vertices of W, and all vertices of W 2 incident wit an edge of S are first among te vertices of W 2. Let A S(G) be suc tat GN(A) = GM(G). Let A i = A[W i ] and ten A can be partitioned as  0 0 A = D C 0 F E Â2 4

5 ] [ [ were A =, A 2 = E D rank(âi) = r i. Ten Â2 ], C is d e,  is (n d) n and Â2 is n 2 (n 2 e). Let rank(a) r + mr(c) + r 2 r + r 2 + d + e nz(c) = r + r 2 + d + e κ e (G). n + n 2 null(a) r + mr(c) + r 2 n r + n 2 r 2 mr(c) null(a) () Now consider te vectors tat must be in ker(a). Since rank(â2) = r 2, tere exist k 2 = n 2 e r 2 independent vectors ŷ i R n2 e suc tat Â2ŷ i = 0. If we let y i = 0 0 (were te first zero vector is of lengt n d and te second is of lengt d+e), ten y i ker(a), i =,..., n 2 e r 2. Since rank(â) = r, tere exist k 2 = n d r independent vectors ˆx i R n d suc tat ˆx i ˆx T i  = 0. Since A is symmetric, if we let x i = 0 (were te first zero vector is of lengt d + e 0 and te second is of lengt n 2 e), ten x i ker(a), i =,..., n d r. Any z Span(x,..., x k, y,..., y k2 ) as d + e zeros. No set of columns in a generic basis can ave a dependence relation, so eiter (a) tere are d + e additional vectors z i in te null space of A (and te generic vectors may be a subset of Span(x,..., x k, y,..., y k2, z,..., z d+e )), or (b) tere are GM(G) additional vectors in te null space of A. But (a) is impossible, because if so we would ave n r +n 2 r 2 n r +n 2 r 2 mr(c) null(a) n d r +n 2 e r 2 +d+e = n r +n 2 r 2. So tere are GM(G) additional independent vectors in te null space of A. By considering tese vectors togeter wit te x i and y i, and equation (), we ave null(a) n d r + n 2 e r 2 + GM(G) n r + n 2 r 2 mr(c) n d r + n 2 e r 2 + GM(G) mr(c) d e + GM(G) By Teorem 2., mr(c) d + e nz(c) = d + e κ e (G), κ e (G) d e d e + GM(G) κ e (G) GM(G) It is possible to ave GM(G) < κ e (G), as te next example sows. Example 2.3. Te grap H sown in Figure 2, as GM(G) = 2 < 3 = κ e (G). We assume tere ŷ i 5

6 Figure 2: Te grap H in Example 2.3 is a generic 8 3 matrix X = [ x x 2 x 3 ] wose columns xi are in te nullspace of A S(G) and derive a contradiction. Te nonzero pattern of A S(G) is A =? ? ? 0 0 0? ? 0 0 0? ? ? Columns 2, 3, 4 and 5, and columns 4, 5, 6, and 7 are clearly independent. If rank(a) = 4 ten te first two columns ave rank and te last two columns ave rank. As in Example. a contradiction is obtained. So assume rank(a) = 5. 5 = rank(a) rank(a[{, 2}, {, 2, 3, 4}) + rank(a[{3, 4}, {5, 6}) + rank(a[{5, 6, 7, 8}, {7, 8}). Since rank(a[{3, 4}, {5, 6}) = 2, eiter te first two rows ave rank or te last two columns ave rank. In te former case, since A is symmetric, te first two columns ave rank, and tus tere is a vector in te null space of A of te form y = [,, 0, 0, 0, 0, 0, 0] T. But since X is generic, tere is no relation among te rows of X, so y is independent of x, x 2, x 3, and a contradiction is obtained. 3 Maximum generic nullity and Vandermonde matrices In te section we develop tecniques for computation of maximum generic nullity and sow tat GM(G) = κ e (G) = δ(g) for all graps of order at most five. Wen constructing a n k matrix to sow tat te generic nullity of A is at least k, te next proposition sows tat it is enoug to construct Y suc tat AY = 0 and every k k submatrix of Y is nonsingular. Proposition 3.. For a real n k matrix Y, if all k k submatrices are nonsingular ten tere exists a real nonsingular k k matrix B suc tat X = Y B is generic. Proof. Given Y = [y ij ], let F be te field extension of te rational numbers generated by all te y ij. Coose k 2 real numbers β ij tat are algebraically independent over F and let B = [β ij ]. Ten for r k, te determinant of an r r submatrix is a nonzero polynomial over F in β ij and so is nonzero.. 6

7 In te study of maximum nullity, it is customary to consider only connected graps, since if te connected components of G are G i, i =,...,, ten M(G) = M(G i ). We can also reduce te study of maximum generic nullity to te study of te connected components, but wit a different relationsip. Proposition 3.2. If G i, i =,..., are connected disjoint graps and G i 2 for i =,...,, ten GM( G i mk ) min GM(G i). i i= Proof. Number te vertices of G first, ten G 2, etc. Let n i = G i. If A S(G), ten A = A A D, were A i S(G i ) and D is diagonal. In fact, order for A to ave a generic null vector, D = 0. Let X be a generic n k matrix suc tat AX = 0 and partition X as X = X. X X + were tere are n i rows in X i and m rows of X +. Ten A i X i = 0 for i =,...,. Since any nonempty submatrix of a generic matrix is generic and A i 0, k min GN(A i) min GM(G i). i i One migt expect tat te inequality in Proposition 3.2 sould be an equality (and we do not know of any cases of strict inequality). One way to establis equality for many graps is troug te use of Vandermonde matrices. Given k real numbers α,..., α k we define te n k Vandermonde matrix V n (α,..., α k ) = [α i j ]. If 0 < α < < α k, ten V n (α,..., α k ) is totally positive [2, p. 2-3]. Given k real numbers α,..., α k and n nonnegative integers m,..., m n, we define te n k generalized Vandermonde matrix V (α,..., α k ; m,..., m n ) = [α mi j ]. A matrix is a generalized Vandermonde matrix if and only if it is submatrix of a (larger) Vandermonde matrix. Tus, if 0 < α < < α k and 0 m < < m n, ten V (α,..., α k ; m,..., m n ) = [α mi j ] is totally positive and ence generic. Wen trying to exibit a generic matrix of maximum nullity it is often convenient to searc for a Vandermonde matrix, and we will see tat for every grap G of order n 5 it is always possible to use te Vandermonde matrix V n (, 2,..., GM(G)) as te generic matrix. Proposition 3.3. Let G = i= G i were n i = G i 2 but te G i are not assumed disjoint. If tere exist positive real numbers α < < α k suc tat for every generalized Vandermonde matrix V i = V (α,..., α k ; m,..., m ni ) tere exists A i S(G i ) suc tat A i V i = 0, ten GM(G) min i= GM(G i). Proof. If te vertices of G i are v,..., v ni {,..., n}, coose A i S(G i ) suc tat A i V i = 0 for V i = V (α,..., α k ; v,..., v ni ). Embed A i in te appropriate place an n n matrix Âi. Ten ÂiV = 0 for V = V n (α,..., α k ). Coose real numbers β,..., β so tat no subtractive cancellation occurs in A = i β iâi. Ten A S(G) and AV = 0. In fact, by coosing different exponents in te proof, te property tat for every generalized Vandermonde matrix V i = V (α,..., α k ; m,..., m n ) tere exists A S(G) suc tat AV = 0 is obtained. i= 7

8 Corollary 3.4. Let G i, i =,..., are connected disjoint graps and G i 2 for i =,...,, If tere exist positive real numbers α < < α k suc tat for every generalized Vandermonde matrix V i = V (α,..., α k ; m,..., m ni ) tere exists A i S(G i ) suc tat A i V i = 0, ten GM(G) = min i= GM(G I). We now establis te ypoteses of Proposition 3.3 for some families of graps. Proposition 3.5. For any generic X n (n ) matrix, tere exist a matrix A S(K n ) suc tat AX = 0. In particular, for any nonnegative integers m m n, tere exists A S(K n ) suc tat A V (, 2,..., n ; m,..., m n ) = 0 Proof. Since X is n (n ), tere exists a vector a R n suc tat a T X = 0. Since X is generic, all entries of a are nonzero. Let A = aa T. Corollary 3.6. If G is K n wit an edge deleted, ten GM(G) = M(G) = n. Proof. G is te union of two copies of K n. Proposition 3.7. GM(C n ) = M(C n ) = 2. Furtermore, for α > and any nonnegative integers m m n, tere exists A S(C n ) suc tat A V (, α; m,..., m n ) = 0. Proof. Let a i,i+ = α αm m2 α mi α and a α mi+ ii = (αm2 m )(α mi α mi+ ) (α mi+ α mi )(α mi α mi ). were te index n + is interpreted as and 0 is interpreted as n. Corollary 3.8. If G is a union of cycles ten GM(G) 2 and 2 can be realized by a Vandermonde matrix. If G is a union of copies of K r ten GM(G) r and r can be realized by a Vandermonde matrix. Corollary 3.9. If G is connected and G 5, ten GM(G) = κ e (G) = δ(g) and tis can be realized by a Vandermonde matrix. Proof. Any grap aving δ(g)= satisfies = GM(G) = κ e (G) = δ(g). Every connected grap of order at most 5 tat as δ(g) = 2 is a union of cycles and tus as 2 = GM(G) = κ e (G) = δ(g). A connected grap aving order 5 or less and δ(g) = 3 is K 4 or is one of tose sown in Figure 3. Figure 3: W 5 = G50 G5 8

9 G5 is K 5 wit an edge deleted and is tus a union of two copies of K 4. Let A = Ten A S(W 5 ) and A V 5 (, 2, 3) = 0. Order 5 and δ(g) = 4 implies G is K 5.. Te next example sows tat it is possible to ave κ v (G) < GM(G). Example 3.0. Te bowtie G42, sown in Figure 4, as GM(G42) = κ e (G42) = 2 > = κ v (G42) Figure 4: Te grap G = G42 in Example 3.0 References [] Avi Berman, Smuel Friedland, Leslie Hogben, Uriel G. Rotblum, Bryan Sader, An upper bound for minimum rank of a grap. Preprint. Available at ttp://orion.mat.iastate.edu/logben/researc/aimmrdelta.pdf. [2] Saun Fallat. Totally Positive and Totally Nonnegative Matrices. In Handbook of Linear Algebra, L. Hogben, Editor. Capman & Hall/CRC Press, Boca Raton, [3] Saun Fallat and Leslie Hogben. Te Minimum Rank of Symmetric Matrices Described by a Grap: A Survey. Linear Algebra and Its Applications 426 (2007), [4] Ronald C. Read and Robin J. Wilson. An Atlas of Graps, Clarendon, Oxford, 998. [5] Douglas B. West. Introduction to Grap Teory, 2nd Ed., Prentice Hall, New Jersey,