On the inverse eigenvalue problem for block graphs

The inverse eigenvalue problem of a graph $G$ aims to find all possible spectra for matrices whose $(i,j)$-entry, for $i\neq j$, is nonzero precisely when $i$ is adjacent to $j$. In this work, the inverse eigenvalue problem is completely solved for a subfamily of clique-path graphs, in particular for lollipop graphs and generalized barbell graphs. For a matrix $A$ with associated graph $G$, a new technique utilizing the strong spectral property is introduced, allowing us to construct a matrix $A'$ whose graph is obtained from $G$ by appending a clique while arbitrary list of eigenvalues is added to the spectrum. Consequently, many spectra are shown realizable for block graphs.


Introduction
Let G = (V, E) be a simple graph on n vertices. Define S(G) as the set of all n × n real symmetric matrices A = (a ij ) such that for i = j, a ij = 0 if and only if {i, j} ∈ E. Note that there is no restriction on the diagonal entries of matrix A. The inverse eigenvalue problem of a graph G (IEP-G) asks what possible spectra occur among matrices in S(G). The IEP-G is motivated from the theory of vibrations [10,11] and can be viewed as a discrete version of the question: What kind of vibration behaviors (spectra) are allowed on a given structure (graph)? On the other hand, the IEP-G is a fundamental question in matrix theory and studies the possible spectra of a matrix with a given zero-nonzero pattern.
The IEP-G is solved for only a handful of families of graphs. In particular, the IEP-G for paths has been solved and many additional properties of tridiagonal matrices were studied [12,14,15]. The solutions of the IEP-G for generalized star graphs [17] and cycles [8] are also known. The IEP-G for complete graphs and small graphs (up to 4 vertices) was solved by explicit matrix construction in [5,6]. Recently, Barrett et al. introduced new techniques to the problem based on the strong spectral property (SSP), and solved the IEP-G for graphs with up to 5 vertices [3].
This paper introduces two techniques for constructing matrices of a given graph. Section 2 considers the operation of duplicating a vertex into a clique with the same neighborhood; see, e.g., Figure 1 or 2. For a graph G of order n and a graph H on m vertices obtained from G by a series of such duplications, Corollary 2.5 shows that any list of m real numbers having at least n distinct elements is realizable by some matrix in S(H). As a consequence, the IEP-G for lollipops and barbell graphs is solved.
Section 3 considers a generalization of the SSP and establishes a cliqueappending lemma. Suppose a graph H is obtained from a graph G by appending a leaf. It is known that if Λ is a spectrum realizable by a matrix with the SSP in S(G), then Λ ∪ {λ} is realizable by a matrix with the SSP in S(H) for any λ / ∈ Λ [3]. We generalize this behavior to the case where H is obtained from G by appending a clique K k while at the same time an eigenvalue of multiplicity k is added to the spectrum of the corresponding matrix in S(G).
Utilizing the tools developed in Sections 2 and 3, we provide partial solution to the IEP-G for block graphs in Section 4.

Preliminaries
Below we define the main notation used in the paper, that is predominantly standard.
By S n (R) we will denote the set of all n × n symmetric matrices. For a simple graph G = (V (G), E(G)), |G| = |V (G)| will denote the order of G, and recall that S(G) denotes the set of all A = (a ij ) ∈ S |G| (R) such that for i = j, a ij = 0 if and only if {i, j} ∈ E(G). Moreover, S cl (G) will denote the topological closure of S(G), i.e., the set of A = (a ij ) ∈ S |G| (R) with (i, j)-entry nonzero only when i = j or {i, j} ∈ E(G). Note that S cl (G) is a linear subspace in S n (R) of dimension n + |E(G)|.
For a matrix A, A(i) will denote the submatrix of A with the ith row and the ith column removed, and A ⊕ B will denote the direct sum of matrices A and B. The n × n identity matrix will be denoted by I n , and the m × n zero matrix by O m,n . In both cases the indices will be omitted if they are clear from the context.
Suppose that A ∈ S n (R) has distinct eigenvalues λ 1 , . . . , λ q with multiplicities m 1 , . . . , m q , respectively. The spectrum of A will be denoted by spec(A) = {λ (m1) 1 , . . . , λ (mq) q }, where λ (k) denotes k copies of λ. The multiplicity list of A is defined to be a list of multiplicities {m 1 , . . . , m q }, in no particular order. We say that the multiplicity list {m 1 , . . . , m q } is spectrally arbitrary in S(G) if for any λ 1 , . . . , λ q , the spectrum {λ (m1) 1 , . . . , λ (mq) q } is realizable by a matrix in S(G). Note that this definition of spectral arbitrariness is stronger than the one in some other works (see e.g., [2,3]), where it is assumed that the multiplicity lists are ordered.
A matrix A ∈ S n (R) has the strong spectral property (SSP ) if the zero matrix X = O is the only symmetric matrix X satisfying A • X = I • X = O and [A, X] = O, where • denotes the entry-wise product of matrices and [A, X] = AX − XA. The strong spectral property of a matrix was first defined in [4] and has been proven useful in expanding any information on the IEP-G for a given graph to information for its supergraph.
Theorem 1.1. [4] Let H be a graph and G a spanning subgraph of H. Suppose A ∈ S(G) is a matrix with spectrum Λ and the SSP. Then for any ǫ > 0 there is a matrix A ′ ∈ S(H) with the SSP such that spec(A) = spec(A ′ ) and A− A ′ < ǫ.
We follow standard notation for basic graphs encountered in this work, i.e., K n denotes the complete graph on n vertices, P n denotes the path on n vertices and S n the star on n vertices.
If v ∈ V (G), let G−v denote the subgraph of G obtained from G by removing the vertex v and all edges incident to it. A connected graph G, |G| ≥ 2, is called 2-connected if G−v is connected for any v ∈ V (G). Let G and H be two graphs. We denote the disjoint union of G and H by G ⊕ H. If each of G and H has a vertex labeled as v, then the vertex-sum G ⊕ v H of G and H at v is the graph obtained from G ⊕ H by identifying the two vertices labeled by v.

Vertex duplication
In this section we will develop a method that will allow us to replace a vertex v ∈ V (G) in a graph G with k mutually adjacent vertices whose neighborhood in G − v is the same as that of v, while preserving some control on the eigenvalues of A ∈ S(G). Note that the resulting graph H has |G| + k − 1 vertices. We will call this operation k-duplication of v in G. In particular, for A = (a ij ) ∈ S(G), we will apply the following two lemmas to create a matrix C ∈ S(H).
Choosing B ∈ S(K k ) in Lemma 2.1 results in k-duplication in the associated graph. While this lemma can be applied more generally, we will take particular advantage of the fact that the IEP-G for complete graphs is solved (see e.g., [5]). Furthermore, we will need information on possible patterns of the eigenvectors of matrices in S(K k ), as outlined in the following Lemma.
Lemma 2.2. [22] For any given list of real numbers σ = {µ 1 , µ 2 , . . . , µ k }, Furthermore, given any zero-nonzero pattern of a vector in R k that contains at least two nonzero elements, B can be chosen so that it has an eigenvector corresponding to µ 1 with that given pattern.
Thus, inserting B ∈ S(K k ) with an eigenvalue µ 1 in Lemma 2.1 will show that any spectrum of the form spec(A) ∪ {µ 2 , . . . , µ k } is realizable in S(H). Special attention needs to be paid to the case when a vv = µ 1 = µ 2 = · · · = µ k . In this case, we necessarily have B = a vv I k , which is not in S(K k ) whenever k ≥ 2. Typically, this situation can be avoided by replacing A with a matrix of the same pattern and eigenvalues, but different diagonal elements. In particular, Lemma 2.3 states that if we require A to have distinct eigenvalues, then we can avoid any prescribed finite set of real numbers on the diagonal. Proof. First we claim that the lemma is true for stars with at least two vertices. Let G be a star on n vertices with vertex 1 as its center. Let µ 1 , . . . , µ n−1 be real numbers such that λ 1 < µ 1 < λ 2 < · · · < λ n−1 < µ n−1 < λ n , Since F is assumed to be finite, such numbers exist. By [21, Lemmas 2.1 and 2.2], there is a matrix A = (a ij ) ∈ S(G) with the SSP such that spec(A) = {λ 1 , . . . , λ n } and spec(A(1)) = {µ 1 , . . . , µ n−1 }. Notice that spec(A(1)) is also equal to {a 22 , . . . , a nn }, so a ii / ∈ F for i = 2, . . . , n, by construction. Also, As an intermediate step, we claim that every connected graph G on n ≥ 2 vertices has a spanning subgraph H such that each component of H is a star with at least two vertices. This can be seen by induction on n. For n = 2, the claim is obviously true for the only connected graph K 2 . Now suppose the statement is true for all graphs satisfying 2 ≤ |V (G)| ≤ n − 1. Let G be a connected graph on n vertices, and let v ∈ V (G) be such vertex that G − v remains connected. By the induction hypothesis, there is a spanning subgraph H ′ of G − v whose components are stars with at least two vertices. Pick a neighbor of v in G, say w, and let S be the connected component of H ′ that contains w. If S along with the edge {v, w} is a star, then let H be obtained from H ′ by adding the vertex v and the edge {v, w}. If S along with the edge {v, w} is not a star, then S − w is still a star with at least two vertices. Thus, let H be obtained from H ′ by removing w and adding the K 2 induced on the vertices v and w. In either case, H is the desired spanning subgraph of G.
To complete the proof, let G be a connected graph on n ≥ 2 vertices, σ = {λ 1 , . . . , λ n } a set of distinct real numbers, and H a spanning subgraph of G whose components are stars with at least two vertices. We may write H as a disjoint union of stars S 1 , . . . , S ℓ of orders k 1 , . . . , k ℓ , respectively. Partition σ into ℓ parts σ 1 , . . . , σ ℓ of orders k 1 , . . . , k ℓ , respectively. Thus, we have already proved above that for each i = 1, . . . , ℓ, there exists a matrix A i ∈ S(S i ) with the SSP such that spec(A i ) = σ i and the diagonal entries of Clearly, spec(A) = σ and the diagonal entries of A avoid F . Also, by [4, Theorem 34] A has the SSP since the spectra of A i 's are mutually disjoint. By Theorem 1.1, for any ǫ > 0, there is a matrix A ′ ∈ S(G) with the SSP such that spec(A ′ ) = σ and A − A ′ ≤ ǫ. When ǫ is chosen small enough, the diagonal entries of A ′ remain disjoint from F .
Let G be a graph on vertices V (G) = {v 1 , . . . , v n }. Define the (closed) blowup of G with respect to n positive integers m 1 , . . . , m n by a graph obtained from G by m i -duplication of v i for i = 1, . . . , n sequentially. Lemma 2.4. Let G be a graph with |V (G)| = n and H a blowup of G with |V (H)| = m. If σ ′ is a multiset of m − n real numbers and A ∈ S(G) is a matrix whose diagonal entries avoid elements in σ ′ , then there is a matrix Theorem 2.5. Let G be a connected graph on n ≥ 2 vertices and H a blowup of G with |V (H)| = m. Suppose σ is a multiset with n distinct real numbers, and σ ′ is any multiset with m − n real numbers. Then σ ∪ σ ′ is the spectrum of some matrix in S(H).
Proof. This follows from and Lemmas 2.3 and 2.4.
The (k, p)-lollipop graph L k,p is the graph on k + p vertices obtained by adding an edge between a vertex in a complete graph K k and a leaf of a path graph P p . See an example in Figure 1(a). Since L k,p can also be viewed as a blowup of P p+2 by (k − 1)-duplication of one of its leaves, Corollary 2.5 resolves the IEP-G for lollipop graphs.
Corollary 2.6. Let σ be a multiset with k + p elements, where k ≥ 2. Then σ is a spectrum of a matrix in S(L k,p ) if and only if σ contains at least p + 2 distinct elements.
Proof. Since L k,p has a unique shortest path on p + 2 vertices, every matrix in L k,p has at least p + 2 distinct eigenvalues by [1,Theorem 3.2]. Together with Theorem 2.5 the result follows.  Corollary 2.7. Let G be a graph that contains a spanning subgraph isomorphic to L k,p . Then any spectrum with at least p + 2 distinct elements is realizable by some matrix with the SSP in S(G).

Similar arguments apply for a generalized barbell graphs
where v and w are the leaves of P p+2 . Note that B k ′ ,p,k ′′ is a blowup of P p+4 by (k ′ − 1)-duplication and (k ′′ − 1)-duplication. See example on Figure 1(b).
Example 2.9. In [2, Appendix B] the authors investigate spectral arbitrariness of graphs on at most six vertices, where some of the achieved multiplicities remain unsolved.
Moreover, graphs G 117 and G 150 can be obtained as blowups of K 1,3 and P 4 , respectively, see Figure 2. Theorem 2.5 implies that (unordered) multiplicity list {3, 1, 1, 1} is spectrally arbitrary for G 117 and G 150 . Again, in particular, ordered multiplicity lists (1, 1, 3, 1) and (1, 3, 1, 1) are spectrally arbitrary for G 117 and G 150 . 3 Strong spectral property with respect to a supergraph In this section we extend the notion of the strong spectral property, that is built on the implicit function theorem for two transversally intersecting manifolds. Below we recall only a brief overview of the notions needed, and direct the reader to [4,19] for further details. By definition, two manifolds intersect transversally at a point A if their normal spaces at that point have only trivial intersection. The implicit function theorem states that if two manifolds intersect at a point transversally, then this intersection will move continuously corresponding to any small perturbation of the two manifolds. A version of the implicit function using the notion of a family of manifolds being smooth is given below. 1), and M 1 (0) and M 2 (0) intersect transversally at y 0 . Then there exists 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).
One family of manifolds that we are interested in is S cl (G), where G is a graph on n vertices. Note that S cl (G) is a subspace of S n (R), so the tangent space to S cl (G) at any of its points is equal to S cl (G). Then the normal space of S cl (G) at A ∈ S cl (G) is and it does not depend on A ∈ S cl (G).
The second family of manifolds that is relevant to our discussion is where Λ is a multiset with n real numbers. If A is an n × n matrix, then let E A = E spec(A) . By [4,Lemma 7], the normal space of E A at A is From the discussion, a matrix A in S(G) has the SSP is equivalent to E A and S cl (G) intersecting transversally at A. Here we generalize this definition to a more flexible version of the SSP. By definition, a matrix A ∈ S(G) has the SSP (in the classical sense) if and only if A has the SSP with respect to G. Also, if H ′ is a supergraph of H of the same order, then A has the SSP with respect to H implies A has the SSP with respect to H ′ . Example 3.3. For n ≥ 4, let us observe the matrix where 1 ∈ R n denotes a vector with all entries equal to 1. That A does not have the SSP, can be shown by choosing X 1 to be any n × n nonzero symmetric matrix with zero diagonal and row sums equal to 0, for example,   Here we extend those results by adding a clique instead of a vertex to a graph, while adding an eigenvalue of multiplicity higher than 1 to the spectrum. Theorem 3.6. Let G be a graph, A ∈ S(G) with the SSP, v ∈ V (G), and λ ∈ R with λ ∈ spec(A) ∪ spec(A(v)). Then for any given positive integer s, there exists a matrix A ′ ∈ S(G ⊕ v K s+1 ) having the SSP such that spec(A ′ ) = spec(A) ∪ {λ (s) }.
Moreover, A ′ can be chosen to be arbitrarily close to A⊕λI s while the (i, i)-entry of A ′ for any i ∈ V (K s+1 ) \ {v} is different from λ.
Proof. Choose a matrix A = (a ij ) ∈ S(G) with the SSP, v ∈ V (G), and λ ∈ spec(A) ∪ spec(A(v)), as assumed in the theorem. The matrix λI s has the SSP with respect to K s , and A and λI s have no common eigenvalues. By Proposition 3.5 the matrixÂ = A ⊕ λI s ∈ S(G ⊕ sK 1 ) has the SSP with respect to H = G ⊕ K s .
Let α be the set of all edges of the form {v, w}, where w ∈ V (K s ), and let H ′ be a supergraph of H, obtained from H by adding all the edges from α, i.e., H ′ = G ⊕ v K s+1 . By Theorem 3.4, for any ǫ > 0, there exists a matrix A ′ = (a ′ ij ) ∈ S cl (G ⊕ v K s+1 ) with the SSP with respect to H ′ , such that spec(A ′ ) = spec(Â) = spec(A) ⊕ {λ (s) }, Â − A ′ < ǫ, and a ′ ij = 0 for all {i, j} ∈ α. Since the perturbationÂ − A ′ can be made to be arbitrarily small, all nonzero entries ofÂ remain nonzero in A ′ .
Finally, we show that the (i, j)-entry of A ′ − λI is nonzero whenever i, j ∈ V (K s ). This argument will allow us to conclude that a ′ ij = 0 (if i = j) and a ij = λ (if i = j) for any i, j ∈ V (K s ). Since A(v) − λI is nonsingular and the perturbation is small, A ′ can be chosen so that A ′ [W ] − λI is nonsingular for W = V (G) \ {v}. Pick any i, j ∈ V (K s ), and consider the (|G| + 1) × (|G| + 1) where b, c ∈ R are nonzero, a ∈ R |G|−1 and δ ij stands for the Kronecker delta function, i.e. δ ij = 0 if i = j and δ ij = 1 if i = j. Since λ is an eigenvalue of A ′ of multiplicity s, it follows that rank(B) ≤ rank(A ′ −λI) = |G|. If a ′ ij −δ ij λ = 0, then |G| ≥ rank(B) = rank(A ′ [W ] − λI) + 2 = |G| − 1 + 2 = |G| + 1, a contradiction. Therefore, a ′ ij − δ ij λ = 0. As this argument applies to any  In what follows we will want to apply Theorem 3.6 inductively. In this process we will keep track of the eigenvalues of A, but not of the eigenvalues of a submatrix A(v). However, if we assume that µ ∈ spec(A) ∪ spec(A(v)) for any v ∈ V (G) and µ = λ, then we can guarantee that µ ∈ spec(A ′ )∪spec(A ′ (u)) for any u ∈ V (G ⊕ v K s+1 ), since A ′ can be chosen to be arbitrarily close to A ⊕ λI s . Corollary 3.8. Let λ 1 , . . . , λ h be distinct real numbers, m 1 , . . . , m h positive integers, and G a graph. Let a sequence of graphs be defined as follows: Let A ∈ S(G) be a matrix with the SSP such that λ i / ∈ spec(A) ∪ spec(A(v)) for all v ∈ V (G) and i ∈ {1, . . . , h}. Then there exists a matrixÂ ∈ S(H (h) ) with the SSP and the spectrum Proof. We prove the result by induction on h. For h = 0, we take A = A (0) and there is nothing to prove.

Block Graphs
While techniques developed in this work can be applied more broadly, we use a family of graphs known as block graphs as an illustrative example.

Definition 4.1. A block of a graph is its maximal 2-connected induced subgraph.
A block graph G is a graph whose blocks are cliques. The family of block graphs with blocks of sizes m 1 , m 2 , . . . , m h will be denoted by BG(m 1 , . . . , m h ). Figure 5: Three examples of block graphs G i ∈ BG(6, 3, 2, 2). The cut-vertices are colored gray.

Definition 4.2. A block graph G is a minimal block graph if every block in
G contains at most one non-cut vertex of G. When H is a block graph, the minimal block graph corresponding to H is obtained from H by removing (if any) all but one non-cut vertices in each block.
Note that a block graph and its corresponding minimal block graph have the same cut vertices, and the same number of blocks. In Figure 5 we present some examples of block graphs in BG(6, 3, 2, 2), while their corresponding minimal block graphs are given in Figure 6. Note that every block graph is a blowup of its minimal block graph.
Block graphs are also called block-clique graphs in [7, Subsection 4.1]. Following this terminology, graph G 1 in Figure 5 is an example of a clique-star graph and G 3 in Figure 5 is an example of a clique-path graph.
Complete graphs will be building blocks of our constructions. To avoid certain technical issues, we start this section by showing that realizations of spectra with two distinct eigenvalues in S(K n ) can be made generic enough. The following lemma is a special case of [9, Lemma 2.2]. We offer an alternative proof for completion. Lemma 4.3. Let x ∈ R n and y ∈ R m . The matrix has eigenvalues (µ 1 , µ 2 , β Proof. Note that Sherman-Morrison formula implies that the inverse of matrix α2yy ⊤ (β2−λ)+α2 y 2 and that det(Y λ ) = (β 2 − λ) m−1 (β 2 − λ + α 2 y 2 ). Using the Schur complement of a matrix C − λI m+n we get with some computation that the characteristic polynomial of matrix C is equal to The assertion of the lemma follows.
Then there exists A ∈ S(K n ) with eigenvalues {λ } such that A(v) has no eigenvalues in F for all v ∈ V (K n ), and A has no diagonal elements in G.
When n 1 = 1 and v = 1, the spectrum of } and can be chosen to avoid elements in F . The case for n 2 = 1 and v = n follows from a similar argument.
Assume that n 1 > 1 and n 2 > 1. To complete the proof we note that C(v) again has the form (1) where either x is replaced by x(v) or y is replaced by y(v), and the eigenvalues of C(v) can be deduced from Lemma 4.3. Thus, we may choose unit vectors x and y so that the eigenvalues of matrices . . , n, j = 1, . . . , m, avoid F , while at the same time the diagonal elements of the matrix of the form (1) avoid G for those vectors.
In Theorem 4.6 we will use matrices constructed in Corollary 4.4 to produce a family of multiplicity lists that are realizable by a matrix with SSP for block graphs.  Remark 4.7. To obtain our next result, we will apply Lemma 2.1 to the matrix A constructed in the proof above. To do that we need some information on the diagonal elements of A. Following the construction of A we can deduce that the diagonal of A can be made arbitrarily close to the diagonal of A (0) ⊕ λ 2 I m1,2 ⊕ · · ·⊕λ r I m h,r h . Moreover, by Corollary 4.4 we can choose A (0) so that its diagonal elements avoid any given finite set of real numbers G, and by Theorem 3.6 we know the rest of diagonal elements, while arbitrarily close to λ i , are not equal to λ i . When a block graph is not minimal, we can extend Theorem 4.6 using the approach from Section 2.
} be the spectrum that we want to achieve for a matrix A ∈ S(G), and let r be a refinement of {k, m 1 + 1 − k, m 2 , . . . , m h } that is covered bym 0 , . . . ,m ℓ . Thus, we may make a partition σ = σ 0 ∪ σ ′ such that σ 0 has multiplicity list r.
By Theorem 4.6 and Remark 4.7, there exists a matrix A 0 ∈ S(G 0 ) whose spectrum is σ 0 and whose diagonal entries avoid elements in σ ′ . Since G is a blowup of G 0 , Lemma 2.4 guarantees the existence of a matrix A ∈ S(G) whose spectrum is spec(A 0 ) ∪ σ ′ = σ.
Recall that Corollaries 2.6 and 2.8 solve IEP-G for special cases of cliquepath graphs, where the only cliques of size greater than two are allowed to be at the ends of the clique-path graphs, namely for lollipop and generalized barbell graphs. However, Theorem 4.8 now resolves IEP-G for clique-path graphs with an additional clique of size greater than two. Example of such graph is shown on Figure 5(c). Proof. Let G and j be as in the statement. If b j = 2, then G is a generalized barbell graph, so this case is covered by Corollary 2.8. Therefore, we may assume b j ≥ 3.