A variant on the graph parameters of Colin de Verdiere: Implications to the minimum rank of graphs

Electronic Journal of Linear Algebra Volume 3 Volume 3 (005) Article 4 005 A variant on the graph parameters of Colin de Verdiere: Implications to the minimum rank of graphs Francesco Barioli Shaun Fallat Leslie Hogben hogben@aimath.org Follow this and additional works at: http://repository.uwyo.edu/ela Recommended Citation Barioli, Francesco; Fallat, Shaun; and Hogben, Leslie. (005), "A variant on the graph parameters of Colin de Verdiere: Implications to the minimum rank of graphs", Electronic Journal of Linear Algebra, Volume 3. DOI: https://doi.org/0.300/08-380.70 his Article is brought to you for free and open access by Wyoming Scholars Repository. It has been accepted for inclusion in Electronic Journal of Linear Algebra by an authorized editor of Wyoming Scholars Repository. For more information, please contact scholcom@uwyo.edu.

A VARIAN ON HE GRAPH PARAMEERS OF COLIN DE VERDIÈRE: IMPLICAIONS O HE MINIMUM RANK OF GRAPHS FRANCESCO BARIOLI, SHAUN FALLA, AND LESLIE HOGBEN Abstract. For a given undirected graph G, the minimum rankof G is defined to be the smallest possible rankover all real symmetric matrices A whose (i, j)th entry is nonzero whenever i j and {i, j} is an edge in G. Building upon recent workinvolving maximal coranks (or nullities) of certain symmetric matrices associated with a graph, a new parameter ξ is introduced that is based on the corankof a different but related class of symmetric matrices. For this new parameter some properties analogous to the ones possessed by the existing parameters are verified. In addition, an attempt is made to apply these properties associated with ξ to learn more about the minimum rankof graphs the original motivation. Key words. Graphs, Minimum rank, Graph minor, Corank, Strong Arnold property, Symmetric matrices. AMS subject classifications. 5A8, 05C50.. Introduction. Recent work and subsequent results have fueled interest in important areas such as spectral graph theory and certain types of inverse eigenvalue problems. Of particular interest here is to bring together some of the pioneering work of Y. Colin de Verdière (specifically his parameter related to planarity of graphs) and the minimum rank of graphs. All matrices discussed in this paper are real and symmetric. If A M n is a fixed symmetric matrix, the graph of A denoted by G(A), has {,..., n} as vertices, and as edges the unordered pairs {i, j} such that a ij 0withi j. Graphs G of the form G = G(A) do not have loops or multiple edges, and the diagonal of A is ignored in the determination of G(A). Similarly, for a given graph G, welet S(G) ={A M n A = A,G(A) =G}. Finally, for any symmetric matrix A M n,welets A = S(G(A)). Suppose that G is a graph on n vertices. hen the minimum rank of G is given by mr(g) = min rank A. A S(G) Received by the editors 7 May 005. Accepted for publication 9 November 005. Handling Editor: Richard A. Brualdi. Department of Mathematics, University of ennessee at Chattanooga, Chattanooga N, 37403 (francesco-barioli@utc.edu). Department of Mathematics and Statistics, University of Regina, Regina, SK, Canada (sfallat@math.uregina.ca). Research supported in part by an NSERC research grant. Department of Mathematics, Iowa State University, Ames, IA 500, USA (lhogben@iastate.edu). 387

388 F. Barioli, S. Fallat, and L. Hogben It is not difficult to verify that mr(g) =n M(G), where M(G) isthemaximum multiplicity of G, and is defined to be M(G) = max {mult A(λ) :λ σ(a)}. A S(G) Furthermore it is easy to see that for any graph, max{corank A A S(G)} = M(G), where corank A is defined to be the nullity of A. Here σ(a) denotes the spectrum of A and mult A (λ) is the multiplicity of λ σ(a). Also, if W {,,...,n} and A M n,thena[w] means the principal submatrix of A whose rows and columns are indexed by W, anda(w ) is the complementary principal submatrix obtained from A by deleting the rows and columns indexed by W. In the special case when W = {v} a singleton, we let A(v) =A(W ). For a fixed m n matrix A, R(A) and Null(A) denote the range and the null space of A, respectively. An interesting and still rather unresolved problem is to characterize mr(g) for a given graph G. Naturally, there have been a myriad of preliminary results, which take on many different forms. For example, if P n, C n, K n, E n, denote the path on n vertices, the cycle on n vertices, the complete graph on n vertices, and the empty (edgeless) graph on n vertices, respectively, then mr(p n )=n, mr(c n )=n, mr(k n )=, mr(e n )=0. Further it is well known that for any connected graph G on n vertices that mr(g) = if and only if G is K n. Fiedler [8] established that mr(g) =n if and only if G is P n. Barrett, van der Holst, and Loewy [4] have characterized all of the graphs on n vertices that satisfy mr(g) =. Other important work pertaining to the class of trees [], states that mr( )= n P ( ), where P ( )isthepath cover number, namely, the minimum number of vertex disjoint paths occurring as induced subgraphs of, that cover all the vertices of. More recently, some modest extensions along these lines have been produced for graphs beyond the class of trees. Namely, for vertex sums and edge sums of so-called non-deficient graphs (which include trees), and for the case of unicyclic graphs, i.e., graphs that contain a unique cycle [, 3]. On a related topic there has been some extremely interesting and exciting work on spectral graph theory that is connected to certain aspects of planarity. For a given graph, a matrix L =[l ij ] S(G) is called a generalized Laplacian matrix of G if for i j, l ij < 0 whenever i, j are adjacent in G and l ij = 0 otherwise. Colin de Verdière introduced the parameter µ(g) associated with the nullity of certain generalized Laplacian matrices in S(G) (see [5, 9, 0] for more specific details). he paper [0] provides a clear exposition and survey of these results, and we will follow much of the notation and treatment given in that paper. he actual definition of µ(g) will be presented below. We now turn our attention to the so-called Strong Arnold Property, which will be shortened to SAP throughout. We will see that it plays a crucial role in monotonicity, such as the subgraph monotonicity of µ. We say two matrices are orthogonal if, when viewed as n -tuples in R n,they are orthogonal under the ordinary dot product. Equivalently, B is orthogonal to A

A Variant on the Graph Parameters of Colin de Verdière 389 if and only if trace(a B) = 0. he matrix B is orthogonal to the family of matrices F if B is orthogonal to every matrix C F. A family we will consider frequently is F = S(G) whereg is a graph; X orthogonal to S(G) requires that every diagonal entry of X is 0 and for every edge of G, the corresponding off-diagonal entry of X is 0. Recall that, if A =[a ij ]andb =[b ij ] are matrices in M n,thematrixa B defined by [A B] ij = a ij b ij is called the Hadamard product of A and B. Definition.. Let A, X be symmetric n n matrices. We say that X fully annihilates A if. AX =0;. A X =0; 3. I n X =0. In other words, X fully annihilates A if X is orthogonal to S A and AX =0. Definition.. he matrix A has the Strong Arnold Property (SAP) if the zero matrix is the only symmetric matrix that fully annihilates A. We begin with a basic, yet useful, observation concerning low corank. Lemma.3. If corank A, thena has SAP. Proof. If coranka = 0, thena is nonsingular, and the only matrix X that fully annihilates A is the zero matrix. Suppose now corank A =,and let X fully annihilate A. herefore, the diagonal of X is 0. Since X is symmetric, this implies X is not a rank matrix. hus if X 0,thenrankX, and AX =0wouldimply corank A. hus X =0andA has SAP. We are now in a position to formally define the Colin de Verdière parameter, µ(g). For a given graph G, µ(g) is defined to be the maximum multiplicity of 0 as an eigenvalue of L, wherel satisfies:. L S(G), and is a generalized Laplacian matrix;. L has exactly one negative eigenvalue (with multiplicity one); 3. L has SAP. In other words µ(g) isthemaximumcorank among the matrices satisfying ()-(3) above. Further observe that µ(g) M(G) =n mr(g). Hence there is an obvious relationship between µ(g) andmr(g). Colin de Verdière and others ([5, 9, 0]) have shown that µ(g) if and only if G is a disjoint union of paths, µ(g) if and only if G is outerplanar, µ(g) 3 if and only if G is planar, µ(g) 4 if and only if G is linklessly embeddable. A related parameter, also introduced by Colin de Verdière [6] is denoted by ν(g), and is defined to be the maximum corank among matrices A that satisfy:. A S(G);. A is positive semidefinite; 3. A has SAP. Properties analogous to µ(g) have been established for ν(g). For example, ν(g) if the dual of G is outerplanar, see [6]. Furthermore, ν(g), like µ(g) is graph minor monotone we will come back to this issue later. One of the motivating issues for this work is an attempt to learn more about the minimum rank of graphs by studying a variant of µ(g) andν(g). Consequently, we

390 F. Barioli, S. Fallat, and L. Hogben introduce the following new parameter, which we denote by ξ(g). Definition.4. For a graph G, ξ(g) is the maximum corank among matrices A S(G) having SAP. For a graph parameter ζ defined to be the maximum corank over a family of matrices associated with graph G, wesaya is ζ-optimal for G if A is in the family and corank A = ζ(g). Remark.5. Since for any graph G, any µ-optimal or ν-optimal matrix for G is in S(G) and has SAP, µ(g) ξ(g) andν(g) ξ(g). In Example 3. we determine a graph G such that µ(g) <ξ(g) andν(g) < ξ(g). For motivational purposes and completeness, we give several examples and observations on the evaluation of ξ. Observation.6. ξ(g) = exactly when G is a disjoint union of paths. Indeed, if ξ(g) =,wehaveµ(g), so that G is a disjoint union of paths. On the other hand, since M(P n ) =, and any corank matrix has SAP, ξ(p n )=. hen,the converse will follow easily from heorem 3., in which we show that ξ of a disjoint union is the maximum value of ξ on the components. Observation.7. If n>, ξ(k n )=n, because J, the all s matrix, is in S(K n ), has corank n, and has SAP (any matrix in S(K n )hassapsince a matrix orthogonal to S(K n ) is necessarily 0). ξ(k ) = because any corank matrix has SAP. Conversely, it is well known that the only connected graph having M(G) = G isk n, so (again using heorem 3.) ξ(g) = G implies G = K n or G = K, the complement of K, also denoted by E. Example.8. ξ(c n ) =, because M(C n ) = (see for example [, 3]) so ξ(c n ), but C n is not a disjoint union of paths so ξ(c n ). he next example shows that it is possible to have a matrix A that is ξ-optimal for graph G, and another matrix B S(G) with corank A = corank B but B does not have SAP. It also illustrates how computations to establish SAP (or find a matrix X showing failure to have SAP) can be performed. 5 6 3 4 Fig... A graph with B S(G), corank B = ξ(g), butb does not have SAP Example.9. Let G be the graph in Figure.. Since G is a tree, we can easily compute the maximum corank possible for a matrix in S(G): M(G) =P (G) =.

A Variant on the Graph Parameters of Colin de Verdière 39 Let A = 0 0 8 0 0 3 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 ; B = 0 0 8 0 0 3 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0. Clearly A, B S(G), and direct computation shows corank A = corank B =. he matrix 0 0 0 0 0 0 0 0 0 0 0 0 X = 0 0 0 0 3 0 0 0 0 3 0 0 3 3 0 0 0 0 0 0 fully annihilates B, sothatb does not have SAP. o show A does have SAP (and thus is ξ-optimal for G), compute AX for an arbitrary symmetric matrix X orthogonal to S(G). An examination of the last two columns of AX shows x 5 = x 6 = x 56 =0. he second, third and fourth entries of the first row are x 3 +x 4, x 3 +x 34, x 4 +x 34, forcing x 3 = x 4 = x 34 = 0. After substituting these values, we see from the third and fourth columns that x 35 = x 36 = x 45 = x 46 =0. Inotherwords,X =0. Example.0. If G is regular of degree G, then ξ(g) =M(G), since any matrix A S(G) hassap(ifx is orthogonal to S(G), at most one entry in any row is nonzero, which implies easily X =0). Example.. If isatreethatisnotapath,thenξ( ) =. his will be proved in heorem 3.7. Example.. Let K p,q denote the complete bipartite graph on sets of p and q vertices, p q. Observethatξ(K, )=ξ(p ) =, and that ξ(k, )=ξ(p 3 )=. If q 3, then ξ(k p,q )=p +, as will be proved in Corollary.8, while ξ(k, )= ξ(c 4 )=. he tools we use to exploit SAP come from manifold theory. As in [0], let M,...,M k be open manifolds embedded in R d,andletxbeapoint in their intersection. We say M,...,M k intersect transversally at x if their normal spaces at x are independent. hat is, if n i is orthogonal to M i for i =,...,k and n + + n k =0, then n i =0foralli =,...,k. A smooth family of manifolds M(t) inr d is defined by a smooth function f: U (, ) R d,whereu is an open set in R s (s d ), and for each <t<, the function f(,t) is a diffeomorphism between U and the manifold M(t). For a given n n matrix A, letr A be the set of all n n matrices B such that rank B =ranka. he next lemma is from [0]. Lemma.3. he matrix A has SAP if and only if the manifolds R A and S A intersect transversally at A.

39 F. Barioli, S. Fallat, and L. Hogben he next lemma is a slightly simplified version Lemma. of [0]. Lemma.4. Let M (t),...,m k (t) be smooth families of manifolds in R d and assume that M (0),...,M k (0) intersect transversally at x. hen there exists ɛ> 0 such that for any t such that t < ɛ, the manifolds M (t),...,m k (t) intersect transversally at a point x(t) so that x(0) = x and x(t) depends continuously on t. Lemma.5. [0, Cor..] Assume that M,...,M k are manifolds in R d that intersect transversally at x, and assume that they have a common tangent vector v at x with v =. hen for every ɛ>0 there exists a point x x such that M,...,M k intersect transversally at x,and (x x ) x x v <ɛ. In Section we establish a graph monotonicity property for ξ, which is also possessed by both µ and ν. In Section 3, from the results in Section, we build up many useful tools and facts about ξ and use them to learn more about ξ, and apply these results to mr(g).. Minor Monotonicity and Consequences. Following the previous works of Colin de Verdière as described in [0], we prove that the parameter introduced here, ξ, is also graph minor monotone. We begin with a preliminary result, which also follows from the results in Section 3. Observation.. ξ is monotone for deletion of an isolated vertex, i.e., if G is obtained from G by deleting an isolated vertex of G, thenξ(g ) ξ(g). Proof. LetG be obtained from G by deleting an isolated vertex v of G. Choosea ξ-optimal matrix A for G. It is sufficient to construct a matrix A S(G) such that corank A = corank A and A has SAP, for then ξ(g) corank A = corank A = ξ(g ). Let A be the matrix obtained from A by adding (in position v) arowandcolumn consisting of 0s except A v,v =. hen clearly A S(G), corank A = corank A,and a simple computation shows A has SAP. heorem.. ξ is edge deletion monotone, i.e., if G is obtained from G by deleting an edge of G, thenξ(g ) ξ(g). Proof. Let G be obtained from G by deleting edge {u, w}. Proceeding as in Observation., we choose a ξ-optimal matrix A for G and construct the required matrix A S(G). Since A has SAP, the two manifolds R A and S A intersect transversally at A.LetS(t) be the manifold obtained from S A by replacing (in each matrix in S A ) the 0s in positions (u, w) and(w, u) byt. Let R(t) =R A. hen by Lemma.4, for a sufficiently small positive t, R(t) ands(t) intersect transversally at A = A(t). hus A has SAP. Since A R(t) =R A, we have corank A = corank A, and since A S(t), A S(G). Corollary.3. ξ is subgraph monotone, i.e., if G is a subgraph of G, then ξ(g ) ξ(g). Recall that for a given edge e = {u, v} of G we say contract e in G to mean delete e from G, identify its ends u, v in such a way that the resulting vertex is adjacent to exactly the vertices that were originally adjacent to u or v. Acontraction of G is then defined as any graph obtained from G by contracting an edge.

A Variant on the Graph Parameters of Colin de Verdière 393 heorem.4. ξ is contraction monotone, i.e., if G is obtained from G by contracting an edge, then ξ(g ) ξ(g). Proof. Let G = n, suppose {, } E(G), and let G be obtained from G by contracting {, } (call this new vertex v and place it first in order of the vertices of G ). If vertex is adjacent only to vertex, then the result follows by subgraph monotonicity. So we may assume is adjacent to at least one vertex in addition to. Renumber if necessary so that vertex is adjacent exactly to the vertices, 3,...,r (r 3). By the edge monotonicity of ξ, without loss of generality, we may assume vertex is not adjacent to any of the vertices 3,...,r. An n n symmetric matrix will be written in the following block form b b b b b b b b B = b b, b b B 0 where b R r. In addition, let U be the 0- matrix with G(U) =G and U ii = for each i. Wethenhave u U =, u where denotes the vector all of whose r entries are equal to, while u (U 0)is a suitable 0- vector (matrix). We define three manifolds as follows. M is the set of n n symmetric matrices B such that b =0,b =0and B() U = B(), that is, G(B()) can be obtained from G by (possibly) removing some edges. M is the set of n n symmetric matrices B such that corank B = ξ(g ). M 3 is the set of n n symmetric matrices B such that rank[b b ]=. As shown in [0], if B M, the normal space of M at B is the set of symmetric matrices X such that 0 x 0 x x 0 x (.) X = x, X() U =0; 0 x x x X 0 if B M, the normal space of M at B is the set of symmetric matrices Y such that U 0 (.) BY =0;

394 F. Barioli, S. Fallat, and L. Hogben if B M 3, and b = γb, the normal space of M 3 at B is the set of symmetric matrices Z such that (.3) Z = 0 0 z 0 0 0 γz 0 z γz 0 0 0, z b =0. Define 0 0 0 0 p p p P = 0 p, 0 p P 0 where P () is a ξ-optimal matrix for G. Note that rank P = + rankp (); so corank P = ξ(g ). In addition, P is in each of the M i s (note that p 0,sothat P M 3 ). As shown in [0], the three manifolds intersect transversally at P,andthe matrix = 0 0 p 0 0 0 0 0 p 0 0 0 is a common tangent to all three manifolds at P. hus, by Lemma.5 (or [0, Cor..]), there is a matrix Q in the intersection of the M i s such that the M i s intersect transversally at Q, andq P is almost parallel to. By a judicious choice of ɛ, we can ensure that Q has nonzero entries everywhere P or has nonzero entries. In other words we can write Q = 0 q 0 q 0 0 q q q q q 0 q where G(Q()) = G. In particular, q has no zero components. Moreover, since Q M 3,thereexistsγ 0suchthatq = γq,sothatq has no zero components as well. Let S = I n γ E, anda = SQS. Easy computations show that G(A) =G, and corank A = corank Q = ξ(g ). So it is enough to show that A has SAP. Suppose, by way of contradiction, that there exists a nonzero matrix W that fully annihilates Q 0,

A Variant on the Graph Parameters of Colin de Verdière 395 A. hena W =0andI n W =0,sothat W = 0 0 0 w 0 0 w w 0 w w w In addition AW =0,thatis,SQS W = 0, and since S is invertible, we can write QS WS =0. Define 0 0 γ w w γ w Y = S WS = 0 0 w w γ w w. w W 0 γ w w W 0. If w 0, define further Z = 0 0 γ w 0 0 0 w 0 γ w w 0 0 0, and X = Y Z. By using (.), (.), and (.3), we see that X, Y,andZ are normal at Q to M, M,andM 3, respectively, so that the M i s do not intersect transversally at Q, which is a contradiction. On the other hand, if w =0,Y would be normal at Q to both M and M, which is again a contradiction. For a given graph G, wecallh a minor of G if H is obtained from G by a sequence of deletions of edges, deletions of isolated vertices, and contractions of edges. We are now in a position to state the minor monotonicity result of ξ which both µ and ν also satisfy. Corollary.5. ξ is minor monotone, i.e., if G is a minor of G, thenξ(g ) ξ(g). As noted in [0], this implies that the Robertson-Seymour graph minor theory applies to ξ, so that the graphs G that have the property ξ(g) k can be characterized by a finite set of forbidden minors. he Robertson-Seymour graph minor theorem is an extremely powerful tool; consult the last chapter [7] for further discussion. Using the results thus far, we continue to derive more properties of the parameter ξ, while at the same time adding to the list of examples in which ξ can be calculated. he first result below is a direct consequence of Corollary.3 and Observation.7. Corollary.6. If K p is a subgraph of G then ξ(g) p. Corollary.7. Suppose V (G) has disjoint subsets W i,i =,...,q such that for all i =,...,q the subgraph of G induced by W i is a path, and for all i j, thereisnoedgeing between a vertex in W i and a vertex in W j. hen ξ(g)

396 F. Barioli, S. Fallat, and L. Hogben G ( W +...+ W q )+. In particular, if G has a set of q independent vertices (i.e., each W i is a singleton), then ξ(g) G q +. Proof. Label the vertices in W i that are the endpoints of the path induced by W i as u i and v i (if W i =thenu i = v i ; otherwise they are distinct). Create a new graph G by adding the edges {v i,u i+ } for i =,...,q. Since there is no edge in G between avertexinw i and a vertex in W j, the subgraph H induced in G by q i= W i is a path on W +...+ W q vertices, so mrh = W +...+ W q. Since H is induced, mr(h) mr(g ), so M(G )= G mr(g ) G ( W +...+ W q ) +. By edge monotonicity (heorem.), ξ(g) ξ(g ) M(G ) G ( W +...+ W q )+. Since µ(g),ν(g) ξ(g), any graph G satisfying the hypotheses of Corollary.7 also has µ(g) G ( W +...+ W q )+ and ν(g) G ( W +...+ W q )+. he next corollary follows from the previous one and the facts that µ(k p,q )=p+ for q p andq 3 [0], and µ(g) ξ(g) for any graph G. Corollary.8. If q p and q 3 then ξ(k p,q )=p +. he next corollary is an immediate consequence of edge monotonicity. Note that the only distinction between matrices considered when maximizing corank for M and for ξ is SAP. Corollary.9. If it is possible to add an edge to G, obtaining graph G, and have M(G ) < M(G) then ξ(g) < M(G), i.e., any matrix A in S(G) with rank A =mr(g) does not have SAP. We have seen that SAP is sufficient for edge monotonicity. In fact, SAP also appears to be necessary for edge monotonicity; we have results for several families of graphs G that when ξ(g) <M(G), it is possible to add an edge and reduce M. For example, the proof of Corollary.7 shows how to add edges between the q independent vertices of K p,q to obtain a graph G with M(G) <M(K p,q ), provided q 4. Note that it follows from Corollary.8 that ξ(k p,q ) <M(K p,q ) is true exactly when q 4. We do not know of any examples with ξ(g) <M(G) where it is not possible to add an edge and reduce M (see also Proposition 3.8). 3. Constructions. In this section we examine the behavior of ξ under various constructions, such as disjoint union, vertex sum, joins, etc. In contrast to the previous section, where the results closely paralleled those for µ, and where the methods ofproofwereoftenthesameasthosein[0],heretheresultsforξ sometimes differ from those for µ, and in most cases even when the result is the same, the method of proof is different. We will need numerous technical lemmas. Lemma 3.. Let B be the direct sum of matrices B i, i =,...,k. hen B has SAP if and only if at most one B i is singular, and such B i has SAP. Proof. LetB have SAP, and suppose two of the B i,sayb and B, are singular. hen there exist nonzero vectors x, x such that B x = B x =0. Letˆx = [x 0 0] and ˆx =[0 x 0].henX = ˆx ˆx + ˆx ˆx fully annihilates B, sothat B does not have SAP. herefore, at most one of the B i,sayb, is singular. Suppose now that B does not have SAP. hen, there [ exists] a nonzero matrix X that fully X 0 annihilates B, so that the matrix X = fully annihilates B, thatis,b 0 0 does not have SAP.

A Variant on the Graph Parameters of Colin de Verdière 397 Conversely, if all B i are nonsingular, then B is nonsingular, and has SAP by Lemma.3. If[ exactly one ] of the B i, say B, is singular and[ has SAP, then ] we B 0 can write B = 0 B,whereB X X is nonsingular. Let X = fully X X annihilate B. hen BX = 0 implies B X = B X =0,thatis,X = X =0. hus we have B X = 0, and, finally, X =0,sinceB has SAP. In other words, X is necessarily the zero matrix, so that B has SAP. he next theorem follows immediately from Lemma 3. and the monotonicity of ξ on submatrices. he analogous result is true for µ. heorem 3.. If G is not connected, and the components of G are the graphs G,...,G k,thenξ(g) =max k i= ξ(g i). Lemma 3.3. Let β c c c 3 B = c B 0 0 c 0 B 0, c 3 0 0 B 3 and suppose Null [ c i B i ] [ c Proof. Fori =,, select x i Null i B i 0for i =,. henb does not have SAP. ], x i 0. hen, the matrix X = 0 0 0 0 0 0 x x 0 0 x x 0 0 0 0 0 0 fully annihilates B, sothatb does not have SAP. [ ] c Observation 3.4. In particular, note that the condition Null i 0is B i satisfied exactly when either corank B i, or when corank B i =andc i R(B i ). Lemma 3.5. Let α, γ R, and consider the matrices A = α b b [ ] b A 0 γ b ; à =. b b 0 A A If A has SAP, then i. A has SAP if b R(A ); ii. à has SAP for each γ such that rank à > rank A ; iii. If rank A =ranka +ranka,thenã has SAP for each γ. Proof. GivenX orthogonal to S A,define X = 0 0 0 0 X 0 0 0 0

398 F. Barioli, S. Fallat, and L. Hogben If A does not have SAP, then there exists a nonzero X that fully annihilates A. If b R(A ), then b X =0,sothat X fully annihilates A, thatis,a does not have SAP. his establishes (i). [ ] Suppose à does not have SAP. hen there exists nonzero X 0 y = that y X fully annihilates Ã. Inparticular,à X =0yields γy + b X = 0 (from the (,)-block) b y + A X = 0 (from the (,)-block) We will show that y = 0, causing these two equations to reduce to b X =0and A X =0,sothat X above contradicts the hypothesis that A has SAP. If y 0,then from the (,)-block equation multiplied by y, b = y y A X y R(A ). hus if b / R(A ), then y = 0. So now assume b R(A ), that is, there exists a vector u such that b = A u. hus, the (,)-block equation becomes A uy + A X =0and so (u A u)y + u A X = 0. From the (,)-block equation, γy + u A X =0. herefore (u A u)y = γy. Since rankã > rank A yields γ u A u, we conclude necessarily y =0,andifà does not have SAP, neither does A. his establishes (ii). o prove (iii), we first note that rank A =ranka +ranka implies b = A u, b = A u,andα = u A u + u A u for suitable vectors u, u. In addition, by part (ii), we only [ need to prove that ] à has SAP whenever [ rank ] à =ranka,that u is, when à = A u u A. Suppose A u A X 0 y = fully annihilates y X Ã. Let X = 0 y 0 y X y u. 0 u y 0 Clearly X is orthogonal to S A.Sinceà X =0,A y =0andA u y + A X =0, and then a computation shows AX =0. SinceA has SAP, X =0. hus X =0and so à has SAP. Lemma 3.6. Let A be a matrix in the form A = α b b... b k b A 0... 0 b 0 A... 0....... b k 0 0... A k where, for i =,...,k, corank A i corank A i+.ifa has SAP, then. corank A, and,ifcorank A =,thenb / R(A ) or b / R(A );. corank A 3, and,ifcorank A 3 =,thencorank A = corank A =,and b i / R(A i ) for i =,, 3; 3. corank A i =0for i 4.,

A Variant on the Graph Parameters of Colin de Verdière 399 Proof. By Observation 3.4, if corank [ A], as well if corank A = and b b R(A ), b R(A ), we have Null i 0,i =,. Lemma 3.3 implies that A i A does not have SAP. his establishes (). Concerning (), corank A 3 follows by (). If corank A 3 = and corank A >, then Observation 3.4 and Lemma 3.3, using B = A and B = A A 3,shows that A does not have SAP. Finally, if, for some i =,, 3, b i R(A i ), then again Observation 3.4 and Lemma 3.3, using B = A i, B = 3 j=,j i A j,showsthata does not have SAP. Finally, if corank A 4, it suffices to define B = A A, B = A 3 A 4 and proceed as in the previous cases, to conclude again that A does not have SAP. Lemma 3.6 has immediate application to computing ξ for trees. heorem 3.7. If is a tree that is not a path, then ξ( )=. Proof. Let beatreethatisnotapath,soξ( ). Let A be a ξ-optimal matrix for. Since corank A, by the Parter-Wiener heorem [], there is a vertex v such that coranka(v) = coranka +, and 0 is an eigenvalue of at least 3 principal submatrices A i corresponding to components of v. hen by Lemma 3.6 (renumbering the vertices if necessary so v = and the coranks are ordered as in the Lemma 3.6), the maximum possible number of singular A i is 3, and the corank of each of these principal submatrices is, i.e., corank A(v) =3. husξ( )=coranka =. Picking up from the remarks following Corollary.9, we now establish for trees that if there is a gap between ξ and M, then an edge can be added to the tree to reduce M. Proposition 3.8. If is a tree and ξ( ) <M( ), then we can add an edge to to obtain graph G such that M(G) <M( ) Proof. Let be tree such that ξ( ) < M( ) = h. hen is not a path, so ξ( ) = < M( ) = P ( ). Choose a minimal path cover P for, and let P,P,P 3 P. We claim that we can choose i, j {,, 3} such that no vertices of P i are adjacent to vertices of P j. Suppose, by way of contradiction, that there exist (not necessarily distinct) vertices u i,v i P i, i =,, 3, such that v u, v u 3,andv 3 u are edges in.hen would contain the cycle u v u v u 3 v 3 u, that is, a contradiction. herefore we can assume that the vertices of P are not adjacent to the vertices of P. Joinbyanedgee an endpoint of P to an endpoint of P to obtain a new path P. hus, P, P 3,...,P h provide a path covering of G = + e. Since G is unicyclic, [3], M(G) P (G) h <P( )=M(). We are now in a position to state and prove an important result for calculating ξ for vertex sums of graphs. Let G,...,G k be disjoint graphs. For each i, we select a vertex v i V (G i )andjoinallg i s by identifying all v i s as a unique vertex v. he resulting graph is called the vertex-sum at v of the graphs G,...,G k. heorem 3.9. Let G be vertex-sum at v of graphs G,...,G k.hen k max ξ(g i) ξ(g) max k ξ(g i)+. i= i= Proof. By subgraph monotonicity, ξ(g) max k i= ξ(g i). Again by subgraph monotonicity we may assume each of the G i v is connected. Let A be ξ-optimal for

400 F. Barioli, S. Fallat, and L. Hogben G. Renumber the vertices of G and the order of the G i so that v =anda can be written as α b b... b k b A 0... 0 A = b 0 A... 0 (3.)....... b k 0 0... A k where, for each i, G(A i )=G i v, and corank A i corank A i+. his renumbering process does not affect SAP. By [3], rank A = k i= rank A i + δ with δ {0,, }, andδ =ifandonlyif there is an i such that b i / R(A i ). herefore, corank A = k i= corank A i + δ. Case : corank A 3 =. By Lemma 3.6, corank A = corank A =,andb i / R(A i )fori =,, 3. So in this case δ =,andξ(g) =coranka = 3 i= corank A i + = ξ(g )+. Case : corank A 3 =0,andb / R(A [ )orb ]/ R(A ). Again, δ =. By γ b Lemma 3.6, corank A. Define à =,whereγ is any real number b A such that rank à > rank A. By Lemma 3.5, à has SAP. So (3.) (3.3) (3.4) (3.5) (3.6) ξ(g) =coranka = corank A + corank A corank A corank à + ξ(g )+. Case 3: corank A 3 = 0, and b R(A ), b R(A ). From Lemma 3.6, corank A i =0fori, and so b i R(A [ i ) for all] i. In particular, δ, and γ b b = A u for some vector u. Let à =, γ R. If δ =,wechoose b A γ u A u,sothatrankã =ranka +, andã has SAP by Lemma 3.5ii. On the other hand, if δ =0,wechooseγ = u A u,sothatrankã =ranka,andã has SAP by Lemma 3.5iii. Note that, in any case, corank à = corank A + δ. herefore ξ(g) =coranka = corank A + δ = corank à ξ(g ). Observation 3.0. If ξ(g) =max k i= ξ(g i)+ and A is ξ-optimal A, then, with regard to (3.), if we let b =(b,...,b k ), A = k i= A i, we have b / R(A ). Furthermore, either

A Variant on the Graph Parameters of Colin de Verdière 40. G is a generalized star, namely, all G i are paths, or,.a. corank A > ; b. corank A =; c. corank A i =0for i =3,..., k; d. b / R(A ); e. ξ(g v) <M(G v). Proof. From the proof of heorem 3.9, ξ(g) =max k i= ξ(g i) + occurs only in Case and in Case, and in both cases δ =,thatis,b / R(A ). In addition, in Case we obtained ξ(g) =, so that ξ(g i )=foreachi. herefore all G i are paths, and G is a generalized star. In Case, we showed corank A i =0fori 3. Moreover, in (3.-3.6) we have all equalities. In particular, equality in (3.4) yields corank A =, while equality in (3.5) yields rank à =ranka +, so that b / R(A ). Finally, from (3.6), we get M(G v) corank A > corank à = ξ(g ) ξ(g v). Clearly a generalized star that is not a path realizes ξ(g) =max k i= ξ(g i)+. he next example also has ξ(g) =max k i= ξ(g i) +, in addition to ξ(g) >µ(g) and ξ(g) >ν(g). Example 3.. Let G be the graph shown in Figure 3.. G is the vertex sum of 7 5 4 3 6 Fig. 3.. A vertex sum G with ξ(g) greater than the maximum of ξ on vertex summands G =,, 3, 4, 5, 6 and G =, 7. Here W indicates the subgraph of G induced by the vertices W {,,...,n}. henξ(g) = 3, because the matrix 0 0 0 0 0 0 0 0 0 0 0 A = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 has corank 3 and SAP. Moreover, ξ(g ) = because G has an induced path of length 4, and ξ(g )=. husξ(g) =maxξ(g i )+. Nowµ(G) = because G is outerplanar. In addition we establish that ν(g) =. Suppose B S(G) has SAP and corank B =3. Letb =(b,b 3,...,b 7 ). Since B is ξ-optimal and ξ(g) = max ξ(g i )+, b / R(B[, 3, 4, 5, 6, 7]), so rank B[, 3, 4, 5, 6, 7] = rank B =. Since mr(, 3, 4, 5, 6 ) =, this forces b 77 = 0. But then B[, 7] is not positive semi-definite, so neither is B.

40 F. Barioli, S. Fallat, and L. Hogben It is also interesting to contrast the behaviour of ξ and µ on clique sums. If G is a clique sum on K s of G i,i =,...,k with s<max k i= µ(g i) (as is the case for a vertex sum provided at least one G i is not a path), then µ(g) =max k i= µ(g i)(see [0]). In [0, heorem.7] it is shown that µ(g) µ(g v)+, where v is an arbitrary vertex in G, andifv is connected to every other vertex and G v is connected of order greater than, then equality holds. Do the same results hold for ξ? We have some partial results. Lemma 3.. If G is connected, G >, andg is obtained by joining v to each vertex of G, thenξ(g ) ξ(g)+. Proof. Let the new vertex be n. ChooseamatrixA that is ξ-optimal for G, so A S(G), corank A = ξ(g) anda has SAP. Since G is connected and G >, there is no column of A consisting entirely of 0 s. So there is a b R(A) such that every entry[ b i is nonzero. Since ] b R(A), there exists u R n such that b = Au. Let A A Au = u A u. hen ranka Au =ranka so corank A = corank A +. [ ] X 0 Since b i 0foralli, A S(G ). Let X fully annihilate A.henX = 0 0 (since n is joined to every other vertex). In addition, A X = 0 implies AX =0, and since A has SAP, we conclude X =0,thatis,X =0. husa has SAP, so ξ(g ) corank A = corank A +=ξ(g)+. Lemma 3.3. If there exists a ξ-optimal A for G with b R(A(v)), thenξ(g) ξ(g v)+. Proof. Without loss of generality, v =. LetA be ξ-optimal for G with b R(A 0 ) [ α b where A = b A 0 ]. hen by Lemma 3.5, A 0 has SAP. rank A 0 rank A, so corank A 0 corank A =ξ(g). Since A 0 has SAP, ξ(g v) corank A 0. AgraphG is called vertex transitive if, for any two distinct vertices u, v of G, there is an automorphism (that is, a bijection from V to V that preserves adjacency) of G mapping u to v. Corollary 3.4. If G is vertex transitive, then ξ(g) ξ(g v)+. Proof. Without loss of generality, v =. LetG be vertex transitive and let A be ξ-optimal for G. SinceA is singular, there is a row, say k, that is a linear combination of other rows of A. Since G is vertex transitive, there is a graph automorphism ψ such that ψ() = k. LetP ψ be the permutation matrix for ψ, i.e., row i of P ψ is row ψ(i) of the identity matrix. hen since ψ is an automorphism of G, P ψ APψ S(G) and row of P ψ APψ is a linear combination of the other rows. hus by Lemma 3.3, ξ(g) ξ(g v)+. If G and H are two graphs, then the join of G and H, denoted by G H, is the graph obtained from the disjoint union of G and H by adding an edge from each vertex of G to each vertex of H. [ ] A B Lemma 3.5. If G = G G, A S(G) is partitioned as B with A A i S(G i ), i =,, anda i has SAP for i =,, thena has SAP.

A Variant on the Graph Parameters of Colin de Verdière 403 [ ] X 0 Proof. Suppose X fully annihilates A. henx has the form X = 0, [ ] X A X and AX = BX B =0,sothatA X A X i X i =0fori =,. Since A i has SAP, X i =0andA has SAP. heorem 3.6. If G is connected, G > and ξ(g) =M(G), then ξ(g K r )=ξ(g)+r = M(G K r ). Proof. he graph G K r can be obtained from G by adjoining one vertex at a time to all vertices of previous graph. By Lemma 3., ξ(g K r ) ξ(g)+r. Since ξ(g) =M(G), ξ(g) = G mr(g). hus ξ(g) +r ξ(g K r ) M(G K r ) G K r mr(g K r ) G + r mr(g) =ξ(g)+r. Corollary 3.7. here is a graph on n vertices with minimum rank k having n(n ) k(k ) edges. his can be obtained as G = P k+ K n k or from K n by deleting k(k ) (specific) edges. his graph satisfies ξ(g) =M(G). REFERENCES [] F. Barioli and S. M. Fallat. On two conjectures regarding an inverse eigenvalue problem for acyclic symmetric matrices. Electronic Journal of Linear Algebra, :4 50, 004. [] F. Barioli, S. Fallat, and L. Hogben. Computation of minimal rankand path cover number for graphs. Linear Algebra and its Applications, 39:89 303, 004. [3] F. Barioli, S. Fallat, and L. Hogben. On the difference between the maximum multiplicity and path cover number for tree-like graphs. Linear Algebra and its Applications, 409:3 3, 005. [4] W. W. Barrett, H. van der Holst, and R. Loewy. Graphs whose minimal rankis two. Electronic Journal of Linear Algebra, :58 80, 004. [5] Y. Colin de Verdière. On a new graph invariant and a criterion for planarity. In Graph Structure heory. Contemporary Mathematics 47, American Mathematical Society, Providence, pp. 37 47, 993. [6] Y. Colin de Verdière. Multplicities of eigenvalues and tree-width graphs. Journal of Combinatorial heory, Series B, 74: 46, 998. [7] R. Diestel Graph heory. Springer-Verlag, New York, 997. Electronic edition, 000: http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/download.html [8] M. Fiedler. A characterization of tridiagonal matrices. Linear Algebra and its Applications, :9 97, 969. [9] H. van der Holst. opological and spectral graph characterizations. Ph. D. hesis, University of Amsterdam, Amsterdam, 996. [0] H. van der Holst, L. Lovász, and A. Schrijver. he Colin de Verdière graph parameter. In Graph heory and Computational Biology (Balatonlelle, 996). Mathematical Studies, Janos Bolyai Math. Soc., Budapest, pp. 9 85, 999. [] C.R. Johnson and A. Leal Duarte. he maximum multiplicity of an eigenvalue in a matrix whose graph is a tree. Linear and Multilinear Algebra, 46:39 44, 999. [] C.R. Johnson, A. Leal Duarte, and C. Saiago. he Parter-Wiener theorem: refinement and generalization SIAM Journal on Matrix Analysis and Applications, 5:35 36, 003. [3] P.M. Nylen. Minimum-rankmatrices with prescribed graph. Linear Algebra and its Applications, 48:303 36, 996.

404 F. Barioli, S. Fallat, and L. Hogben [4] S. Parter. On the eigenvalues and eigenvectors of a class of matrices. Journal of the Society for Industrial and Applied Mathematics, 8:376 388, 960. [5] G. Wiener. Spectral multiplicity and splitting results for a class of qualitative matrices. Linear Algebra and its Applications, 6:5 9, 984.