Generalizations of the Strong Arnold Property and the minimum number of distinct eigenvalues of a graph

For a given graph G and an associated class of real symmetric matrices whose off-diagonal entries are governed by the adjacencies in G, the collection of all possible spectra for such matrices is considered. Building on the pioneering work of Colin de Verdiere in connection with the Strong Arnold Property, two extensions are devised that target a better understanding of all possible spectra and their associated multiplicities. These new properties are referred to as the Strong Spectral Property and the Strong Multiplicity Property. Finally, these ideas are applied to the minimum number of distinct eigenvalues associated with G, denoted by q(G). The graphs for which q(G) is at least the number of vertices of G less one are characterized.


Introduction
Inverse eigenvalue problems appear in various contexts throughout mathematics and engineering. The general form of an inverse eigenvalue problem is the following: given a family F of matrices and a spectral property P, determine if there exists a matrix A ∈ F with property P. Examples of families are tridiagonal, Toeplitz, or all symmetric matrices with a given graph. Examples of properties include: having a prescribed rank, a prescribed spectrum, a prescribed eigenvalue and corresponding eigenvector, or a prescribed list of multiplicities. Our focus is on inverse eigenvalue problems where F is a set of symmetric matrices associated with a graph. These have received considerable attention, and a rich mathematical theory has been developed around them (see, for example, [13]).
All matrices in this paper are real. Let G = (V (G), E(G)) be a (simple, undirected) graph with vertex set V (G) = {1, . . . , n} and edge set E(G). The set S(G) of symmetric matrices described by G consists of the set of all symmetric n × n matrices A = (a ij ) such that for i = j, a ij = 0 if and only if ij ∈ E(G). We denote the spectrum of A, i.e., the multiset of eigenvalues of A, by spec(A). The inverse spectrum problem for G, also known as the inverse eigenvalue problem for G, refers to determining the possible spectra that occur among the matrices in S(G). The inverse spectrum problem for G seems to be difficult, as evidenced by that fact that it has been completely solved for only a few special families of graph, e.g. paths, generalized stars, double generalized stars, and complete graphs [7,9,21].
To gain a better understanding of the inverse eigenvalue problem for graphs, other spectral properties have been studied. For example, the maximum multiplicity problem for G is: Determine M(G), where M(G) = max{mult A (λ) : A ∈ S(G), λ ∈ spec(A)}, and mult A (λ) denotes the multiplicity of λ as an eigenvalue of A. A related invariant is the minimum rank of G, which is defined by mr(G) = min{rank A : A ∈ S(G)}.
The minimum rank problem is: Given a graph G, determine mr(G). As mr(G) + M(G) = |G|, where |G| denotes the number of vertices of G, the maximum multiplicity and minimum rank problems are essentially the same. These problems have been extensively studied in recent years; see [13,14] for surveys. If the distinct eigenvalues of A are λ 1 < λ 2 < · · · < λ q and the multiplicity of these eigenvalues are m 1 , m 2 , . . . , m q respectively, then the ordered multiplicity list of A is m = (m 1 , m 2 , . . . , m q ). This notion gives rise to the inverse ordered multiplicity list problem: Given a graph G, determine which ordered multiplicity lists arise among the matrices in S(G). This problem has been studied in [6,7,21]. A recently introduced spectral problem (see [1]) is the minimum number of distinct eigenvalues problem: Given a graph G, determine q(G) where q(G) = min{q(A) : A ∈ S(G)} and q(A) is the number of distinct eigenvalues of A.
For a specific graph G and a specific property P, it is often difficult to find an explicit matrix A ∈ S(G) having property P (e.g., consider the challenge of finding a matrix whose graph is a path on five vertices and that has eigenvalues 0, 1, 2, 3, 4).
In this paper, we carefully describe the underlying theory of a technique based on the implicit function theorem, and develop new methods for two types of inverse eigenvalue problems. Suppose G is a graph and P is a spectral property (such as having a given spectrum, given ordered multiplicity list, or given multiplicity of the eigenvalue 0) of a matrix in S(G). The theory is applied to determine conditions (dependent on the property P) that guarantee if a matrix A ∈ S(G) has property P and satisfies these conditions, then for every supergraph G of G (on the same vertex set) there is a matrix B ∈ S( G) satisfying property P. (A graph G is a supergraph of G if G is a subgraph of G.) In Section 2, the technique is developed, and related to both the implicit function theorem and the work of Colin de Verdière [10,11]. The technique is then applied to produce two new properties for symmetric matrices called the Strong Spectral Property (SSP) and the Strong Multiplicity Property (SMP) that generalize the Strong Arnold Property (SAP).
In Section 3, we establish general theorems for the inverse spectrum problem, respectively the inverse multiplicity list problem, using matrices satisfying the SSP, respectively the SMP. In Section 4, we use the SSP and SMP to prove properties of q(G).
In particular we answer a question raised in [1] by giving a complete characterization of the graphs G for which q(G) ≥ |G| − 1.

Motivation and fundamental results
We begin by recalling an inverse problem due to Colin de Verdière. In his study of certain Schrödinger operators, Colin de Verdière was concerned with the maximum nullity, µ(G), of matrices in the class of matrices A ∈ S(G) satisfying: (i) all off-diagonal entries of A are non-positive; (ii) A has a unique negative eigenvalue with multiplicity 1; and Here • denotes the Schur (also known as the Hadamard or entrywise) product and O denotes the zero matrix. Condition (iii) is known as the Strong Arnold Property (or SAP for short). Additional variants, called Colin de Verdière type parameters, include ξ(G), which is the maximum nullity of matrices A ∈ S(G) satisfying the SAP [8], and ν(G), which is the maximum nullity of positive semidefinite matrices A ∈ S(G) satisfying the SAP [11]. The maximum multiplicity M(G) is also the maximum nullity of a matrix in S(G), so ξ(G) ≤ M(G). Analogously, ν(G) ≤ M + (G), where M + (G) is the maximum nullity among positive semidefinite matrices in S(G).
Colin de Verdière [10] used results from manifold theory to show conditions equivalent to (i)-(iii) imply that if G is a subgraph of G, then the existence of A ∈ S(G) satisfying (i)-(iii), implies the existence of A ∈ S( G) having the same nullity as A and satisfying (i)-(iii). 1 The formulation of the SAP in (iii) is due to van der Holst, Lovász and Schrijver [19], which gives a linear algebraic treatment of the SAP and the Colin de Verdière number µ. Using a technique similar to that in [19], Barioli,Fallat,and Hogben [8] showed that if there exists A ∈ S(G) satisfying the SAP and G is a subgraph of G, then there exists A ∈ S( G) such that A has the SAP and A and A have the same nullity. 2 The Strong Arnold Property has been used to obtain many results about the maximum nullity of a graph. Our goal in this section is to first describe the general technique behind Colin de Verdière's work, and then develop analogs of the SAP for the inverse spectrum problem and the inverse multiplicity list problem. For convenience, we state below a version of the Implicit Function Theorem (IFT) because it is central to the technique; see [12,22].
has rank t, then there exist an open neighborhood V containing a and an open neighborhood W containing b such that V × W ⊆ U and a continuous function φ : The IFT concerns robustness of solutions to the system F (x, y) = c. Namely, the existence of a "nice" solution x = a to F (x, b) = c guarantees the existence of solutions to all systems F (x,b) = c withb sufficiently close to b. Here, "nice" means that the columns of the matrix in (1) span R t .

Remark 2.2.
More quantitative proofs of the implicit function theorem (see [22,Theorem 3.4.10]) show that there exists an ǫ > 0 that depends only on U and the Jacobian in (1) such that there is such a continuous function φ with F (φ(y), y) = c for all y with y − b < ǫ.
A useful application of the IFT arises in the setting of smooth manifolds. We refer the reader to [22,23] for basic definitions and results about manifolds. Given a manifold M embedded smoothly in an inner product space and a point x in M, we denote the tangent space to M at x by T M.x and the normal space 3 to M at x by N M.x . For the manifolds of matrices discussed here, the inner product of n × n matrices A and B is A, B = tr(A ⊤ B), or equivalently, the Euclidean inner product on R n 2 (with matrices viewed as n 2 -tuples).
Many problems, including our inverse problems, can be reduced to determining whether or not the intersection of two manifolds M 1 and M 2 is non-empty. There is a condition known as transversality such that if M 1 and M 2 intersect transversally at x, then any manifold "near" M 1 and any manifold "near" M 2 intersect non-trivially. In other words, the existence of a nice solution implies the existence of a solution to all "nearby" problems. More precisely, let M 1 and M 2 be manifolds in some R d , and x be a point in M 1 ∩ M 2 . The manifolds M 1 and M 2 intersect transversally at x provided T M 1 By a smooth family M(s) (s ∈ (−1, 1)) of manifolds in R d we mean that M(s) is a manifold in R d for each s ∈ (−1, 1), and M(s) varies smoothly as a function of s. Thus, if M(s) is a smooth family of manifolds in R d , then (i) there is a k such that M(s) has dimension k for all s ∈ (−1, 1); (ii) there exists a collection {U α : α ∈ I} of relatively open sets whose union is ∪ s∈(−1,1) M(s); and Note that if x 0 ∈ (−1, 1) k , s 0 ∈ (−1, 1), y 0 ∈ M(t 0 ) and F α (x 0 , s 0 ) = y 0 , then the tangent space to M(s 0 ) at y 0 is the column space of the d × k matrix The following theorem can be viewed as a specialization of Lemma 2.1 and Corollary 2.2 of [19] to the case of two manifolds. Theorem 2. 3. Let M 1 (s) and M 2 (t) be smooth families of manifolds in R d , and assume that M 1 (0) and M 2 (0) intersect transversally at y 0 . Then there is a neighborhood W ⊆ R 2 of the origin and a continuous function f : W → R d such that for each ǫ = (ǫ 1 , ǫ 2 ) ∈ W , M 1 (ǫ 1 ) and M 2 (ǫ 2 ) intersect transversally at f (ǫ).
Choose α so that y 0 ∈ U α and β so that y 0 ∈ V β . Define where u ∈ (−1, 1) k , v ∈ (−1, 1) ℓ , and s, t ∈ (−1, 1). There exists u(0) and v(0) such that F α (u(0), 0) = y 0 and G β (v(0), 0) = y 0 . Hence H(u(0), 0, v(0), 0) = 0. Since F and G are diffeomorphisms, the Jacobian of the function H restricted to s = 0 and t = 0 and evaluated at u = u(0) and v = v(0) is the d by k + ℓ matrix The assumption that M 1 (0) and M 2 (0) intersect transversally at y(0) implies that the column space of Jac is all of R d . The result now follows by applying the implicit function theorem (Theorem 2.1) and the fact that every matrix in a sufficiently small neighborhood of a full rank matrix has full rank.

The Strong Arnold Property
We now use Theorem 2.3 to describe a tool for the inverse rank problem. This tool is described in [19] and used there and in [4,5,8]. We use this problem to familiarize the reader with the general technique.
Let G be a graph of order n, and A ∈ S(G). For this problem the two manifolds that concern us are S(G) and the manifold consisting of all n × n symmetric matrices with the same rank as A. We view both of these as subsets of S n (R), the set of all n × n symmetric matrices endowed with the inner product V, W = tr(V W ). Thus S(G) and R A can be thought of as submanifolds of R n(n+1)/2 . It is easy to see that For N R A .A we have the following result [19].
Lemma 2. 4. Let A be a symmetric n × n matrix of rank r. Then Proof. There exists an invertible r×r principal submatrix of A, and by permutation similarity we may take this to be the leading principal submatrix. Hence A has the form  for some invertible r × r matrix A 1 and some r × (n − r) matrix U. Let B(t) (t ∈ (−1, 1)) be a differentiable path of symmetric rank r matrices such that B(0) = A. For t sufficiently small, the leading r × r principal submatrix of B(t) is invertible and B(t) has the form  where B 1 (t) and U(t) are differentiable, B 1 (0) = A 1 and U(0) = U. Differentiating with respect to t and then evaluating at t = 0 giveṡ It follows that T R A .A = T 1 + T 2 , where Consider an n × n symmetric matrix It is easy to verify that this is precisely that set of symmetric matrices X such that AX = O.

Lemma 2.4 implies that R A and S(G) intersect transversally at A if and only if
A satisfies the SAP. Now that we know the pertinent tangent spaces for the inverse rank problem for G, we can apply Theorem 2.3 to easily obtain the following useful tool, which was previously proved and used in [19].
Theorem 2. 5. If A ∈ S(G) has the SAP, then every supergraph of G with the same vertex set has a realization that has the same rank as A and has the SAP. Proof. Let G be a supergraph of G with the same vertex set, and define , and A has the SAP, R A and M(0) intersect transversally at A. Therefore Theorem 2.3 guarantees a continuous function f such that for ǫ sufficiently small, R A and M(ǫ) intersect transversally at f (ǫ), so f (ǫ) has the same rank as A and f (ǫ) has the SAP. For ǫ > 0, f (ǫ) ∈ S( G).

The Strong Spectral Property
We now follow the general method outlined in the previous subsection to derive an analog of the SAP and Theorem 2.5 for the inverse spectrum problem. Certain aspects of this section were in part motivated by discussions with Dr. Francesco Barioli in connection with the inverse eigenvalue problem for graphs [3].
Given a multiset Λ of real numbers that has cardinality n, the set of all n × n symmetric matrices with spectrum Λ is denoted by E Λ . Thus if A ∈ E Λ , then E Λ is all symmetric matrices cospectral with A. It is well known that E Λ is a manifold [2]. A comment on notation: The notation E Λ for the constant spectrum manifold was chosen because this manifold is determined by Λ. Then a symmetric matrix A is in E spec(A) . In conformity with this, the constant rank manifold containing A should be denoted R rank A , but we follow the literature in denoting it by R A .
Let A be a symmetric n × n matrix. The centralizer of A is the set of all matrices that commute with A, and is denoted by C(A). The commutator, AB − BA of two matrices is denoted by [A, B]. The next result is well known but we include a brief proof for completeness.
Throughout this section we assume that λ 1 , . . . , λ q are the distinct eigenvalues of A and . the E i are mutually orthogonal idempotents that sum to I).
Clearly F i (0) = E i for i = 1, 2, . . . , q. As F i (t) is given by a polynomial in B(t) with coefficients that depend only on the spectrum of B(t), i.e. on Λ, [18, Corollary to Theorem 9 (Chapter 9)], F i (t) is a differentiable function of t. Since the F i (t) are mutually orthogonal idempotents, we have thaṫ Post-multiplying (3) by E j gives Pre-multiplying (4) by E j gives Post-multiplying (3) Equation (5) implies that the image ofḞ Consider eigenvectors x and y of A. First suppose that they correspond to the same eigenvalue, say λ i . Since x = E i x and y = E i y, (6) and (7) imply that y ⊤Ḟ j (0)x = 0 for all j. Thus Therefore xy ⊤ + yx ⊤ ∈ N E Λ .A for all such choices of x and y corresponding to the same eigenvalue. Now suppose that x and y are unit eigenvectors of A corresponding to distinct eigenvalues, say λ 1 and λ 2 . Let µ 3 , . . . , µ n be the remaining eigenvalues of A (with no assumption that they are distinct from each other or λ 1 or λ 2 ) and z 3 , . . . , z n a corresponding orthonormal set of eigenvectors. Let It follows that xy ⊤ + yx ⊤ ∈ T E Λ .A for any eigenvectors x and y corresponding to distinct eigenvalues of A.
Let v 1 , . . . , v n be a basis of eigenvectors of A. . We have shown that if v i and v j correspond to distinct eigenvalues, A . It follows that: Definition 2. 8. The symmetric matrix A has the Strong Spectral Property (or A has the SSP for short) if the only symmetric matrix X satisfying Lemma 2.7 asserts that A has the SSP if and only if the manifolds S(G) and E Λ intersect transversally at A, where G is the graph such that A ∈ S(G) and Λ = spec(A). A proof similar to that of Theorem 2.5 yields the next result.
Theorem 2. 10. If A ∈ S(G) has the SSP, then every supergraph of G with the same vertex set has a realization that has the same spectrum as A and has the SSP.
For every A ∈ S(K n ), A • X = O and I • X = O imply X = O, so trivially A has the SSP. Next we discuss some additional examples.
The fact that X commutes with A implies that X has all row sums and column sums equal to zero, which in turn implies X = O. Thus, A has the SSP, and by Theorem 2.10, every supergraph of K 1,3 has a realization with spectrum {0, 0, Example 2. 12. Let G be the star on n ≥ 5 vertices having 1 as a central vertex, let A ∈ S(G), and let λ be an eigenvalue of A of multiplicity at least 3. From a theorem of Parter and Wiener (see, for example, [13, Theorem 2.1]), λ occurs on the diagonal of A(1) at least 4 times. 4 Without loss of generality, we may assume that a 22 = a 33 = a 44 = a 55 = λ.
Thus, A does not have the SSP. Therefore, no matrix in S(G) with an eigenvalue of multiplicity at least 3 has the SSP.
Example 2. 13. Let G be as in the previous example. Let A ∈ S(G) such that no eigenvalue of A has multiplicity 3 or more. Without loss of generality we may assume that A(1) = ⊕ k j=1 λ j I n j for some distinct λ 1 , . . . , λ k and positive integers n 1 , . . . , n k with n − 1 = n 1 + n 2 + · · · + n k . As every eigenvalue of A has multiplicity 2 or less, each n j ≤ 3 for j = 1, 2, . . . , k.
Let X be a symmetric matrix with [A, X] = O, A • X = O and I • X = O. The last two conditions imply that all entries in the first row, the first column, and on the diagonal of X are zero. This and the first condition imply that X(1) is in C(A(1)). By the distinctness of the λ j we conclude that X(1) = ⊕ k j=1 X j where X j is a symmetric matrix of order n j with zeros on the diagonal. Partition A as A = α a ⊤ a A(1) and the partition a = (a ⊤ 1 , . . . , a ⊤ k ) ⊤ conformally with X(1). Then [A, X] = O implies that X j a j = 0. As n j ≤ 3, X j is symmetric and has zeros on its diagonal and every entry of a j is nonzero, this implies X j = O. Thus, X = O and we conclude that A has the SSP.
Observation 2.14. If the diagonal matrix D = diag(µ 1 , µ 2 , . . . , µ n ) has a repeated eigenvalue, say µ i = µ j , then D does not have the SSP as validated by the matrix X with a 1 in positions (i, j) and (j, i) and zeros elsewhere.
Remark 2. 15. Note that a diagonal matrix D = diag(λ 1 , λ 2 , . . . , λ n ) with distinct eigenvalues has the SSP, because DX = XD implies all off-diagonal entries of X are zero. Therefore, by Theorem 2.10, every graph on n vertices has a realization that is cospectral with D and has the SSP. The existence of a cospectral matrix was proved in [24] via a different method.
However, not every matrix with all eigenvalues distinct has the SSP.

The Strong Multiplicity Property
Let m = (m 1 , . . . , m q ) be an ordered list of positive integers with m 1 + m 2 + · · · + m q = n. We let U m denote the set of all symmetric matrices whose ordered multiplicity list is m. Thus, if A has multiplicity list m, then U m = {B ∈ S n (R) : B has the same ordered multiplicity list as A}.
It follows from results in [2] that U m is a manifold. In the next lemma we determine N Um.A .
Lemma 2. 17. Let A be an n × n symmetric matrix with exactly q distinct eigenvalues, spectrum Λ, and ordered multiplicity list m. Then Proof. Let A be an n × n symmetric matrix with spectrum given by the multiset Λ, and let B(t) (t ∈ (−1, 1)) be a differentiable path of matrices having the same ordered multiplicity list as A. Let A and B(t) have spectral decomposition A = q j=1 λ j E j , and Theorem 2. 20. If A ∈ S(G) has the SMP, then every supergraph of G with the same vertex set has a realization that has the same ordered multiplicity list as A and has the SMP. Remark 2. 22. In contrast to Example 2.16, every symmetric matrix whose eigenvalues are distinct has the SMP. To see this, let A ∈ S n (R) have n distinct eigenvalues and let its spectral decomposition be and tr(A i X) = 0 for i = 0, 1, . . . , n − 1. Since A and X commute, each eigenspace of A is an invariant subspace of X. These conditions and the distinctness of eigenvalues imply that each y j is an eigenvector of X, and tr(p(A)X) = 0 for all polynomials p(x). The distinctness of eigenvalues implies that y j y ⊤ j is a polynomial in A for j = 1, . . . , n. Hence 0 = tr(y j y ⊤ j X) = y ⊤ j Xy j , so the eigenvalue of X for which y j is an eigenvector of X is 0 for j = 1, . . . , n. Thus X = O, and we conclude that A has the SMP.
Observation 2. 23. Clearly the SMP implies the SAP.
The next example shows that the SMP and the SAP are distinct. Observation 2. 25. If A has the SSP (SMP), then A + λI has the SSP (SMP) for all λ ∈ R.

Example 2.24. Consider the matrices
Remark 2. 26. If λ is the only multiple eigenvalue of A, then A has the SMP if and only if A − λI has the SAP. To see this assume that λ 1 is the only multiple eigenvalue of A, A − λ 1 I has the SAP, λ 2 , . . . , λ q are the remaining eigenvalues of A, and y j is a unit eigenvector of A corresponding to λ j (j = 2, . . . , q). Then the spectral decomposition of A has the form A = λ 1 E 1 + q j=2 λ j y j y ⊤ j . Assume X is a symmetric matrix such that A•X = O, I •X = O, [A, X] = O, and tr(A k X) = 0 for all k. As in Remark 2.22, each y j is an eigenvector of X and y j y ⊤ j is a polynomial in A, so 0 = tr(y j y ⊤ j X) = y ⊤ j Xy j . Thus we conclude that y j is in the null space of X for j = 2, . . . , q. Therefore, AX = λ 1 E 1 X = λ 1 X (with the latter equality coming from E 1 + q j=2 y j y ⊤ j = I). Thus, (A − λ 1 I)X = O. Since A − λ 1 I has the SAP, X = O and we conclude that A has the SMP.
3 Properties of matrices having the SSP or SMP Section 3.1 presents characterizations of the tangent spaces T R A .A , T E Λ .A , and T Um.A and applies these to obtain lower bounds on the number of edges in a graph where a matrix has the associated strong property. Section 3.2 describes a computational test for determining whether a matrix has the SSP or the SMP. Section 3.3 presents the Gershgorin intersection graph and uses it to test for the SSP. Section 3.4 characterizes when block diagonal matrices have the SSP or the SMP in terms of the diagonal blocks.

Tangent spaces for the strong property manifolds
We begin by giving equivalent, but more useful, descriptions of the tangent spaces T R A .A , T E Λ .A , and T Um.A . The set of all n × n skew-symmetric matrices is denoted by K n (R).  To prove (a), consider an n × n matrix Y and X ∈ N R A .A . First note that Lemma 2.4 asserts that X ∈ S n (R) and AX = O. Thus XA = O and Define Ω to be the set of pairs (i, j) with 1 ≤ i ≤ j ≤ n and i ≤ r. Thus It is easy to verify that {v i v ⊤ j + v j v ⊤ i : (i, j) ∈ Ω} is an orthogonal set and hence .
To prove (b) consider K ∈ K n (R) and B ∈ N E Λ .A . Note that Lemma 2.7 asserts that B ∈ C(A) ∩ S n (R). Thus where the last equality is (8). Thus, equality holds throughout (12). The dimension of Statement (c) follows from (b) and (10), and the facts that

Remark 3.2.
Let M be a manifold in S n (R) and G be a graph of order n such that M and S(G) intersect transversally at A. Then Since dim T S(G).A = n + |E(G)| and n+1 2 = n 2 + n, Corollary 3. 3. Let G be a graph on n vertices and let A ∈ S(G) with spectrum Λ, ordered multiplicity list m = (m 1 , . . . , m q ), and rank r. Assume that A is not a scalar matrix. Then (a) [17] If R A and S(G) intersect transversally at A, then Statement (a) can be established using Remark 3.2 and part (a) of Theorem 3.1 but the argument is more complicated. Since it was previously established in in [17] (see Theorem 6.5 and Corollary 6.6) we refer the reader there.

Equivalent criteria for the strong properties
Let H be a graph with vertex set {1, 2, . . . , n} and edge-set {e 1 , . . . , e p }, where e k = i k j k . For A = (a i,j ) ∈ S n (R), we denote the p × 1 vector whose kth coordinate is a i k ,j k by vec H (A). Thus, vec H (A) makes a vector out of the elements of A corresponding to the edges in H. Note that vec H (·) defines a linear transformation from S n (R) to R p . The complement G of G is the graph with the same vertex set as G and edges exactly where G does not have edges. In the following, E ij denotes the n × n matrix with a 1 in position (i, j) and 0 elsewhere, and K ij denotes the n × n skew-symmetric matrix E ij − E ji . Theorem 3.1 and Proposition 3.4 imply the next result.
Theorem 3. 5. Let G be a graph, let A ∈ S(G) with q distinct eigenvalues, and let p be the number of edges in G. Then (a) A has the SAP if and only if the matrix whose columns are vec G (AE ij + E ⊤ ij A) for 1 ≤ i, j ≤ n has rank p; (b) A has the SSP if and only if the matrix whose columns are vec G (AK ij − K ij A) for 1 ≤ i < j ≤ n has rank p; and (c) A has the SMP if and only if the matrix whose columns are vec G (AK ij − K ij A) for 1 ≤ i < j ≤ n along with vec G (A k ) (k = 0, 1, . . . , q − 1) has rank p.
Example 3. 6. Let Let G be the graph of A, and let M be the matrix defined in part (b) of Theorem 3.5 whose columns are vec G ([A, Since M is strictly diagonally dominant (equivalently, 0 is not in the union of Gershgorin discs of M ), M is invertible, and so rank M = 4. Therefore M has rank 4 and by Theorem 3.5, we conclude that A has the SSP.

Gershgorin discs and the SSP
Given a square matrix A = (a ij ) ∈ C n×n , the Gershgorin intersection graph of A is the graph on vertices labeled 1, . . . , n in which two vertices i = j are adjacent exactly when Gershgorin discs i and j of A intersect, that is, when the inequality is satisfied. If A has real spectrum, then Gershgorin discs intersect if and only if they intersect on the real line, and the Gershgorin intersection graph of A is an interval graph. Note that when graphs have a common vertex labeling, one of them may be a subgraph up to isomorphism of another without being identically a subgraph. The next result requires the stronger condition of being identically a subgraph.
Theorem 3. 7. Let G be a graph with vertices labeled 1, . . . , n and let A ∈ S(G). If the Gershgorin intersection graph of A is identically a subgraph of G, then A satisfies the SSP. Proof. Suppose that e k = i k j k (k = 1, 2, . . . , p) are the edges of G. Let M be the p × p matrix whose k-th column is vec The (k, k)-entry of M has absolute value |a i k ,i k − a j k ,j k | and the remaining entries of the k-th column of M are, up to sign, a subset of the entries a i k ,ℓ , ℓ = i k , and a j k ,ℓ , ℓ = j k . If the Gershgorin intersection graph of A is identically a subgraph of G, then inequality (13) is not satisfied for any k (because the e k are nonedges of G and therefore of the Gershgorin intersection graph). Thus M is strictly diagonally dominant, so M is invertible and has rank p. Therefore, by Theorem 3.5, A has the SSP.
Of course it is possible to have M strictly diagonally dominant implying the invertibility of M even when the Gershgorin intersection graph of A is not a subgraph of G, as in Example 3.6.

Block diagonal matrices
Theorem 3. 8. Let A i ∈ S n i (R) for i = 1, 2. Then A := A 1 ⊕ A 2 has the SSP (respectively, SMP) if and only if both A 1 and A 2 have the SSP (respectively, SMP) and spec(A 1 ) ∩ spec(A 2 ) = ∅.
To obtain X i = O, i = 1, 2 (and thus X = O and A has the SMP) it suffices to show that tr(A k i X i ) = 0 for i = 1, 2 and k = 2, . . . , n − 1. Consider the spectral decompositions of the diagonal blocks of A, As one application we give an upper bound on q(G) in terms of chromatic numbers.

Since each projection in the spectral decomposition is a polynomial in
Theorem 3. 9. Let G be a graph and G its complement. Then q(G) ≤ 2χ(G).
Proof. The graph G contains a disjoint union of χ(G) cliques. Taking the direct sum of realizations of each clique each having at most two distinct eigenvalues and the SSP, and the eigenvalues of different cliques distinct gives a matrix having the SSP by Theorem 3. 8. The result then follows from Theorem 2. 10.
Another application gives an upper bound on the number of distinct eigenvalues required for a supergraph on a superset of vertices.
Theorem 3. 10. Let A be a symmetric matrix of order n with graph G. If A has the SSP (or the SMP) and G is a graph on m vertices containing G as a subgraph, then there exists B ∈ S( G) such that spec(A) ⊆ spec( B) (or has the ordered multiplicity list of A augmented with ones), and B has the SSP (or the SMP). Furthermore, q( G) ≤ m − n + q(A). If A has the SSP, then we can prescribe spec( B) to be spec(A) ∪ Λ where Λ is any set of distinct real numbers such that spec(A) ∩ Λ = ∅.

Consider the matrix
.
Note that the eigenvalues of diag(Λ) are distinct and distinct from the eigenvalues of A. It follows that diag(Λ) has the SSP (see Remark 2.15 or note this follows from Theorem 3.8), and thus has the SMP. By Theorem 3.8, if A satisfies the SSP (SMP), then B satisfies the SSP (SMP). By Theorem 2.10 (or Theorem 2.20), every supergraph of the graph of B on the same vertex set has a realization B that is cospectral with B and has the SSP (or has the same ordered multiplicity list and has the SMP). Hence q( G) ≤ q(B) = m − n + q(A).
Remark 3. 11. By taking a realization in S(G) with row sums 0, mr(G) ≤ |G| − 1. It is a well known result that the eigenvalues of an irreducible tridiagonal matrix are distinct, that is mr(P n ) = n − 1. A classic result of Fiedler [15] asserts that mr(G) = |G| − 1 if and only if G is a path.
Theorem 3.10 can be used to derive this characterization, as follows. If G contains a vertex of degree 3 or more, then G contains K 1,3 as a subgraph, and hence by Theorem 3.10 and Example 2.11 we conclude that mr(G) ≤ |G| − 2. Also, it is easy to see if G is disconnected, then mr(G) ≤ |G| − 2. Thus, if mr(G) = |G| − 1, then G has maximum degree 2 and is connected. Hence G is a path or a cycle. The adjacency matrix of a cycle C has a multiple eigenvalue, which implies that mr(C) ≤ |C| − 2.

Application of strong properties to minimum number of distinct eigenvalues
The SSP and the SMP allow us to characterize graphs having q(G) ≥ |G| − 1 (see Section 4.2). First we introduce new parameters based on the minimum number of eigenvalues for matrices with the given graph that have the given strong property.

New parameters q S (G) and q M (G)
Recall that ξ(G) is defined as the maximum nullity among matrices in S(G) that satisfy the SAP, and ν(G) is the maximum nullity of positive semidefinite matrices having the SAP, so ξ(G) ≤ M(G) and ν(G) ≤ M + (G). These bounds are very useful because of the minor monotonicity of ξ and ν (especially the monotonicity on subgraphs). In analogy with these definitions, we define parameters for the minimum number of eigenvalues among matrices having the SSP or the SMP and described by a given graph. In order to do this we need the property that every graph has at least one matrix with the property SSP (and hence SMP). For any set of |G| distinct real numbers, there is matrix in S(G) with these eigenvalues that has the SSP by Remark 2. 15.  One might ask why we have not defined a parameter q A (G) for the SAP. The reason is that the SAP is not naturally associated with the minimum number of eigenvalues. The Strong Arnold Property considers only the eigenvalue zero; that is, if zero is a simple eigenvalue of A (or not an eigenvalue of A), then A automatically has the SAP. The next result is immediate from Remark 2.19 and the fact that q(G) = 1 if and only if G has no edges. The next result is immediate from Theorem 3. 8.
Since any graph has a realization for any set of distinct eigenvalues (Remark 2.15), The next result is an immediate corollary of Theorem 3. 10.

High values of q(G)
In this section we characterize graphs having q(G) ≥ |G| − 1.
Proposition 4. 7. Let G be one of the graphs shown in Figure 1. Then q S (G) ≤ |G| − 2.

Proof. Let
The graphs of matrices A 1 , A 2 , A 3 and A 4 are the H tree, the campstool, the long Y tree, and the 3-sun, respectively. Also, q(A i ) = |G| − 2 for i = 1, 2, 3, 4. It is straightforward to verify that each of the matrices A 1 , A 2 , A 3 and A 4 has the SSP (see, for example, [26]).
Corollary 4. 8. If a graph G contains a subgraph isomorphic to the H tree, the campstool, the long Y tree, or the 3-sun then Proof. This follows from Corollary 4.6 and Proposition 4.7.
The parameters M(G) and M + (G) can be used to construct lower bounds on q(G). For a graph G of order n, clearly q(G) ≥ n M(G) . The next result improves on this in many cases, in particular for G = K 1,3 . Proposition 4. 9. For any graph G on n vertices, Moreover, if M + (G) < n 2 , then q(G) ≥ 3. Proof. Let A ∈ S(G) be a matrix with q(G) distinct eigenvalues. Let α, β be the smallest and the largest eigenvalues of A. Since A − αI and −A + βI are positive semidefinite, the multiplicity of α and β is no more than M + (G). Every other eigenvalue of A has multiplicity less than or equal to M(G). Therefore, A has at least 2 + n−2 M + (G) The final statement of the proposition readily follows.
Corollary 4. 10. If G contains two vertex disjoint subgraphs each of which is a K 3 or a K 1,3 , then q(G) ≤ |G| − 2.
In Ferguson [16,Theorem 4.3] it is shown that a multiset of n real numbers is the spectrum of an n × n periodic Jacobi matrix 5 A if and only if these numbers can be arranged as λ 1 > λ 2 ≥ λ 3 > λ 4 ≥ λ 5 > · · · λ n . This solves the inverse eigenvalue problem for a cycle of odd length: Suppose A ∈ S(C n ) and n is odd. If the cycle product a 12 a 23 · · · a n−1,n a n1 > 0, then A is similar (by a diagonal matrix with diagonal entries in {±1}) to a periodic Jacobi matrix; if a 12 a 23 · · · a n−1,n a n1 < 0, then −A is similar to a periodic Jacobi matrix.
In the next result we establish that a specific matrix A ∈ S(C n ) with q(A) = n 2 has the SMP.
Theorem 4. 11. Let C n be the cycle on n ≥ 3 vertices. Then q M (C n ) = n 2 . Proof. Since M(C n ) = 2, q(C n ) ≥ n 2 . Given n ≥ 3, let C = (c ij ) be the flipped-cycle matrix of order n, that is, the (non-symmetric) n × n matrix with entries c i,i+1 = 1 for i = 1, . . . , n − 1, c n,1 = −1, and 0 otherwise. Since C satisfies the equation C n = −I, the eigenvalues of C are the nth roots of −1. The matrix A = C + C ⊤ = C + C −1 is a symmetric matrix whose graph is the n-cycle C n , and whose eigenvalues are 2 cos(2π 2j−1 2n ) for j ∈ {1, . . . , n}, which occur in n 2 pairs satisfying j 1 + j 2 = n + 1, with one singleton eigenvalue (coming from 2j = n + 1) equal to −2 when n is odd. Thus q(A) = n 2 . We show that A has the SMP, implying q M (C n ) = n 2 . Assume X = (x ij ) is a symmetric matrix such that A • X = O, I • X = O, [A, X] = O, and tr(A k X) = 0 for k = 1, . . . , n. Divide the entries x ij (on or above the main diagonal) into n bands for k = 0, . . . , n − 1 of the form x i,i+k , i = 1, . . . , n − k; all the entries of X in bands 0, 1 and n − 1 are zero since I • X = O and A • X = O. The fact that AX is symmetric implies that all the entries in each band are equal, and in addition, x 1,1+(n−k) = −x 1,1+k for k ≤ n 2 . In the case that n = 2ℓ is even, this implies x i,i+ℓ = 0. Now assume X = O and let m be the smallest natural number such that band m of X contains a nonzero entry x ij (and 2 ≤ m < n 2 ). Notice that the sign pattern x i,i+m = x 1,1+m for i = 1, . . . , n − m and x i,i+(n−m) = −x 1,1+m for i = 1, . . . , m matches the sign pattern of A m , i.e., (A m ) i,i+m = 1 for i = 1, . . . , n − m and (A m ) i,i+(n−m) = −1 for i = 1, . . . , m. It is clear that (A m ) ij = 0 when the distance between i and j is greater than m, and by the choice of m, x ij = 0 when the distance between i and j is less than m. Thus tr(A m X) = 2nx 1,1+m = 0, a contradiction.
By the proof of Theorem 4.11 and Remark 2.19, the symmetric flipped cycle matrix C + C ⊤ has the SSP for n = 4. It is clear from X = C 2 + C 2 ⊤ that the symmetric flipped cycle matrix C + C ⊤ does not have the SSP for n ≥ 5 (this is implicit in the the proof of Theorem 4.11).
The next result characterizes all of the graphs G that satisfy q(G) ≥ |G| − 1 and resolves a query presented in [1] on connected graphs that satisfy q(G) = |G| − 1. One of our main motivations for considering the SSP and the SMP was for evaluating q(G). In this work, we concentrated on the case of graphs G for which q(G) was large relative to |G|. We established a number of forbidden subgraph-type results that were used to characterize the graphs G such that q(G) ≥ |G| − 1. As a consequence, we produced a new verification of Fiedler's characterization that the path is the only graph G such that q(G) = |G|, and we resolved an open problem left from the work in [1] concerning graphs G with q(G) = |G| − 1. A much clearer picture of q(G) may be obtained by using the tools and techniques derived here.