Corrigendum to “Achievable Multiplicity partitions in the Inverse Eigenvalue Problem of a graph” [Spec. Matrices 2019; 7:276-290.]

Abstract We correct an error in the original Lemma 3.4 in our paper “Achievable Multiplicity partitions in the IEVP of a graph”’ [Spec. Matrices 2019; 7:276-290.]. We have re-written Section 3 accordingly.

In an unpublished manuscript (see [6]), it is shown that any complete multipartite graph Kp ,p ,...,p satis es q(Kp ,p ,...,p ) ≤ . The basic idea employed in the proof of this inequality is to note that the matrix B = [bu,v] with entries de ned as There are several known results for the partitions that are achievable along with the corresponding values of q for complete multipartite graphs; we list them in the following lemma.
Continuing with the complete multipartite graph, the only case that remains unresolved is when p ≤ p + · · · + p . We suspect that in this case q(Kp ,p ,...,p ) = .
To address the speci c case when all p i = k, we begin by stating a technical result that is a special case of [7, Lemma 10] (in the notation of [7], we are setting q = and p = k ≥ ).
as needed.
The next result is a technical result concerning a bound on the minimum semide nite rank of joins of graphs. This result is needed to study the case when all p i = k for determining the minimum number of distinct eigenvalues of the complete multipartite graph and establish a bound on the corresponding minimal multiplicity bipartition. In this proof s is used to denote the vector in R s with all entries equal to one. Similarly, s is used to denote the vector in R s with all entries equal to zero and O is the all zeros matrix, the size will be clear from context.
where the eigenvalue has multiplicity |V(H)| − d and the eigenvalue β has multiplicity d. Up to translation, we may assume that the two distinct eigenvalues of ZZ T are 0, and λ. Applying Lemma 3.3, we may replace Z by ZR (for some orthogonal matrix R) so that each column of ZR is an entry-wise nonzero vector. If we set Q = ZR, Proof. Using Lemma 3.1, we know that q(K k,k ) = and it is not di cult to determine that MB(K k,k ) = k (since the minimum semide nite rank of K k,k is equal to k). The graph K k,k represents the base case for an induction argument based on the number of parts. Assume for the complete multipartite G = K k,k,...,k on l − (l ≥ ) parts that q(G) = with MB(G) = k. Let H = K k,k,...,k be the complete multipartite graph with l parts.
is the closed neighbourhood of v (that is, a neighbourhood of v containing v). It turns out that cloning a vertex of a graph G with MB(G) = k results in a graph G with MB(G ) ≤ k. The following proposition is proved in Theorem 6.3 of [13], it is also implied by Corollary 4 of [2]. In [13], this is used to characterize graphs G with MB(G) = by constructing minimal such graphs (these are K , K ∪ K , K , , K , ,..., and K , ,..., , ) and constructing all the other such graphs by cloning vertices in the minimal graphs.

Proposition 3.6. Let G be a graph with G ∈ MP([n − k, k]). If H is obtained from G by cloning a vertex in G, then H ∈ MP([n − k + , k]).
Suppose G is a graph with q(G) = . If H is obtained from G by cloning a vertex, then MB(H) ≤ MB(G). It is not clear if this inequality is ever strict. By cloning vertices in K k,k,...,k , where k ≥ , we have the following consequence, reminiscent of Lemma 2.3.  By Lemma 3.2, there exists a k × k matrix R such that R T R = I k and M T RM has no zero entries. De ne C as follows: Hence C is positive semide nite and C ∈ S(G ∨ H) since M T RM is an entry-wise nonzero matrix. It is easy to note that C has rank k since [M RM ] has a full-row rank. Therefore, null(C) = n + n − k. Moreover, C = C which implies σ(C) = (n +n −k) , (k) . Hence q(G ∨ H) = since C ∈ S(G ∨ H) and q(G ∨ H) > .
In In fact, this idea can be easily generalized as follows: Suppose G is a graph with q(G) = that contains an independent set of vertices Then for any graph H with q(H) = and |H| < k, we have q(G ∨ H) > . To see this, it is enough to observe that in the graph G ∨ H we have and hence the condition of Lemma 2.4 fails to hold. We also note that the assumption of no isolated vertices in Theorem 3.8 is possibly a stronger condition than is in fact necessary; this assumption is used to ensure that the matrix M in the proof has no zero columns. For instance, in the next result, which is a consequence of Lemma 3.4, all the vertices of the second graph are isolated vertices. The proof of Lemma 3.9 is the same as the proof of Theorem 3.8, except that the matrix B is replaced with the identity matrix. We denote the graph on k vertices with no edges by K k .

Lemma 3.9. Let G be a graph with no isolated vertices and q(G)
Note that the multiplicity bipartition [n − k, k] for regular complete multipartite graphs K k,k,...,k can also be obtained from the proof of Theorem 3.8 and induction.
It is also interesting to note that the minimum number of distinct eigenvalues of the join of two graphs can be large. Proof. The eigenvalues for any matrix in S(G) interlace the eigenvalues any matrix S(G ∨ K ).
The next theorem is the main result of [15]. Theorem 3.11 (Theorem 4.3 [15]). Let G be a connected graph on n vertices and let λ , λ , . . . , λn be distinct real numbers. Then there exists a real symmetric matrix A ∈ S(G) with eigenvalues λ , λ , . . . , λn such that none of the eigenvectors of A has a zero entry. Lemma 3.12. Let G be a connected graph on n ≥ vertices. Then q(G ∨ Kn) = and MB(G ∨ Kn) = n.
Proof. Since G is a connected graph, by Theorem 3.11 there exists a matrix A ∈ S(G) with positive distinct eigenvalues λ > λ > · · · > λn and corresponding entry-wise nonzero unit eigenvectors v , . . . , vn such that where the scalars a i (i = , , . . . , n) are to be determined. Since U is an entry-wise nonzero matrix, if each a i is also nonzero, then Further, the rows of C are orthogonal and so where α i = a i + λ i , i = , . . . , n. Therefore, the eigenvalues of CC T are α i , i = , . . . , n. The values a i can be set so that they are all strictly positive, and α i for all i = , . . . , n are equal to some λ > λ . Then the spectrum of C T C is with multiplicity n, and λ also with multiplicity n. This implies that q(G ∨ Kn) = and MB(G ∨ Kn) ≤ n. Finally, the vertices in Kn form an independent set of size n, and so by Statement (3) of Lemma 2.3, it follows that MB(G ∨ Kn) = n.
The same proof can be used to prove the following result.  where α i = a i + λ i , i = , . . . , n − . Similar to the proof of the previous lemma, the spectrum of C T C is with multiplicity n, and λ with multiplicity n − . This implies that q(G ∨ Kn) = and MB(G ∨ K n− ) = n − .