Achievable multiplicity partitions in the inverse eigenvalue problem of a graph

Associated to a graph $G$ is a set $\mathcal{S}(G)$ of all real-valued symmetric matrices whose off-diagonal entries are non-zero precisely when the corresponding vertices of the graph are adjacent, and the diagonal entries are free to be chosen. If $G$ has $n$ vertices, then the multiplicities of the eigenvalues of any matrix in $\mathcal{S}(G)$ partition $n$; this is called a multiplicity partition. We study graphs for which a multiplicity partition with only two integers is possible. The graphs $G$ for which there is a matrix in $\mathcal{S}(G)$ with partitions $[n-2,2]$ have been characterized. We find families of graphs $G$ for which there is a matrix in $\mathcal{S}(G)$ with multiplicity partition $[n-k,k]$ for $k\geq 2$. We focus on generalizations of the complete multipartite graphs. We provide some methods to construct families of graphs with given multiplicity partitions starting from smaller such graphs. We also give constructions for graphs with matrix in $\mathcal{S}(G)$ with multiplicity partition $[n-k,k]$ to show the complexities of characterizing these graphs.


Introduction
Let G be a simple graph on n vertices and consider the set S(G) of all possible real-valued weighted symmetric adjacency matrices associated to G, where the diagonal entries are free in that they may or may not be zero (the restriction herein to simple graphs avoids some unnecessary confusion when stating and proving results). The notation λ (ni) i is used to denote the eigenvalue λ i with multiplicity n i . In this work, we order the eigenvalues according to their corresponding multiplicities. That is, a matrix A ∈ S(G) has spectrum σ(A) = (λ (n1) 1 , . . . , λ (n ℓ ) ℓ ), where n 1 ≥ · · · ≥ n ℓ and the eigenvalues λ i are distinct. The list of eigenvalues could also be ordered according to their values, this is distinctly different from the ordering considered in this work; see [10].
If q(A) is the number of distinct eigenvalues of a symmetric matrix A, then for a given graph G, we let q(G) = min{q(A) : A ∈ S(G)}, and refer to this parameter as the minimum number of distinct eigenvalues of G. It is well-known that q(G) = 1 if and only if G has no edges, and q(G) = n if and only if G is a path on n vertices; see [2]. Graphs with q(G) = n − 1 are characterized in [4]. Some conditions for graphs attaining q(G) = c, for c ∈ {2, . . . , n − 2} are given in [2]. For example, if a connected graph G on at least three vertices has q(G) = 2, then G cannot have a cut edge (an edge whose deletion results in a disconnected graph), and any two non-adjacent vertices of G must have at least two common neighbours (Lemma 4.2 and Corollary 4.5 of [2], respectively). Moreover, Corollary 3.6 of the same paper gives a construction for a graph on n vertices satisfying q(G) = c for any 1 ≤ c ≤ n.
For positive integers n 1 ≥ · · · ≥ n ℓ , a partition [n 1 , n 2 , . . . , n ℓ ] of n is said to be achievable by G if there exists an A ∈ S(G) such that the spectrum of A is λ (n1) 1 , . . . , λ (n ℓ ) ℓ for some set of distinct values λ i . If [n 1 , n 2 , . . . , n ℓ ] is a partition of n, we use the notation M P ([n 1 , n 2 , . . . , n ℓ ]) to denote the set of all graphs on n vertices for which the partition [n 1 , . . . , n ℓ ] of n is achievable.
Given a graph, some natural questions arise. For example, what multiplicity partitions are achievable by a given graph or a family of graphs? This question is considered in [1,3] where all achievable partitions are listed for all graphs on fewer than 6 vertices. Another question is if a certain multiplicity partition is achievable for a graph, is it possible to characterize any other multiplicity partitions that are also achievable for the graph? In particular, for two partitions [n 1 , n 2 , . . . , n ℓ ] and [ñ 1 ,ñ 2 , . . . ,ñ m ] of n, when is M P ([n 1 , n 2 , . . . , n ℓ ]) ⊆ M P ([ñ 1 ,ñ 2 , . . . ,ñ m ])?
Another natural approach is to characterize the graphs in M P ([n 1 , . . . , n ℓ ]) for some partition [n 1 , . . . , n ℓ ]. This has been answered for a limited number of partitions. For example, the main result in [14] is that every connected graph on n vertices is in M P ([1, 1, . . . , 1]). The only graph that is in M P ([n]) is the graph on n vertices with no edges. The set M P ([n − 1, 1]) is exactly the set of graphs on n vertices that have one connected component that is a complete graph and the remaining components are isolated vertices (this includes the graph on n vertices with no edges). The graphs in M P ([n − 2, 2]) have also been exactly characterized; this characterization is given in Lemma 2.3 and see any of [7,12,13,15]) for a proof.
The questions and studies mentioned above all fit under the general umbrella of the inverse eigenvalue problem (IEVP) of a graph, specifically looking at the achievable multiplicities of the eigenvalues in the IEVP. Many authors have provided partial answers to these questions; see [2,3,4,8,10,13]. Here, we add to the growing body of work, focusing on joins, complete multipartite graphs and the sets M P ([n − k, k]) for some k with k ≤ ⌊ n 2 ⌋.

Graphs with two Distinct Eigenvalues
Our goal is to consider the graphs G with q(G) = 2; any such graph on n vertices achieves a bipartition [n − k, k] for some k = 1, 2, . . . , ⌊ n 2 ⌋. The initial results that we state show that this is a very large set of graphs.
The join of two graphs G and H is the graph G ∨ H on vertex set V (G) ∪ V (H), where all edges of G and H are preserved, and edges are added to make every vertex of G incident to every vertex of H. The following theorem by Monfared and Shader [14,Theorem 5.2] gives a sufficient condition for the minimum number of distinct eigenvalues to be 2.
Theorem 2.1. Let G and H be two connected graphs on n vertices. Then q(G ∨ H) = 2.
In [11] a large number of graphs are shown to admit only two distinct eigenvalues. Theorem 3.2 of [11] proves for a tree T that q(T ) = 2, unless T is P 4 , or in one of two families of trees. Further, Theorem 2.5 of the same paper proves that many bipartite graphs G have the property q(G) = 2. This gives another large and diverse family of graphs with only 2 distinct eigenvalues.
The results in [14] and [11] indicate that the collection of graphs with only two distinct eigenvalues is very large and likely very difficult to characterize. Thus rather than trying to characterize all graphs G with q(G) = 2, we define the minimal multiplicity bipartition M B(G) to be the least integer k ≤ ⌊ n 2 ⌋ such that G achieves the multiplicity bipartition for any m < k). If M B(G) = k, then we say G has multiplicity bipartition k. The multiplicity bipartition is only defined for graphs that admit only two distinct eigenvalues and if G has n vertices, then M B(G) ≤ n/2.
It is easy to note that if S(G) has a matrix with only two distinct eigenvalues, then for any two distinct real numbers there exists a matrix in S(G) such that its spectrum consists of these two real numbers.
For an m × n matrix A, the notation A[α | β] is used to denote the submatrix of A lying in rows indexed by α and columns indexed by β. We let J n×m denote the n × m matrix where all of its entries are one.
where k ≤ ⌊ n 2 ⌋ and {u 1 , u 2 , . . . , u k } is a set of orthonormal vectors in R n . Proof. Assume that q(G) = 2 and G ∈ M P ([n − k, k]), then there is a matrix A ∈ S(G) with σ(A) = 0 (n−k) , 1 (k) . Therefore, A can be written as where V is a unitary matrix, U = V [1, . . . , n|1, . . . , k] and O n−k is the (n−k)×(n−k) all zeros matrix. Let U = (u 1 , u 2 , . . . , u k ) where u i ∈ R n , i = 1, . . . , k. Each u i is a column of V , so they are orthonormal. Moreover, We summarize the known characterizations of graphs with given minimal multiplicity bipartitions in the following lemma.
for non-negative integers p 1 , . . . , p k , q 1 , . . . , q k with k > 1, and G is not isomorphic to either one of a complete graph or (K p1 ∪ K q1 ) ∨ K 1 .
(3) If M B(G) = k, then G does not have an independent set (a set of vertices for which no two are adjacent) of size k + 1 or more.
Proof. The first statement is trivial. The second statement has appeared in [7,12,13,15]. The third statement is known (for example, there is a proof in [13]), but we include a proof for completeness. From Lemma 2.2 there is a matrix A ∈ S(G) with where u i , i = 1, 2, . . . , k, are orthonormal vectors. Let U = (u 1 , u 2 , . . . , u k ) where u i ∈ R n , i = 1, . . . , k. If G has an independent set of size k + 1, then the rows of U form k + 1 orthogonal vectors in R k , which is impossible. Hence there is no independent set of size k + 1.
The following lemma indicates that for a connected graph G with q(G) = 2, if the union of the pairwise common neighbourhood of an independent set of vertices is not empty, then it cannot be too small. ). Consider a connected graph G with q(G) = 2, and let S be an independent set of vertices.
For a given graph G, the minimum rank among all matrices (positive semidefinite matrices) in S(G) is denoted by mr(G) (mr + (G)). If a graph G on n vertices has q(G) = 2 and M B(G) = k, from Lemma 2.2 then there is a matrix A ∈ S(G) with spectrum {0 (n−k) , 1 (k) }. This implies the following lemma. The parameter mr(G) has been extensively studied [9]; and any lower bound on the minimum rank or the minimum positive semidefinite rank of a graph, is also a lower bound on the minimal multiplicity bipartition of the graph. For example, we may use the above bound on minimum rank together with [9, Obs. 1.6] to deduce the next result. Lemma 2.6. If G is any graph with M B(G) = k, then any induced path of G has length no more than k.

Complete Multipartite Graphs
As is standard, we use the notation K p1,p2,...,p ℓ for the complete ℓ-partite graph, where ℓ is a positive integer. The set of vertices is partitioned into ℓ parts V 1 ∪ V 2 ∪ · · · ∪ V ℓ ; part V i has p i vertices for i ∈ {1, . . . , ℓ}; no two vertices from a part are adjacent, while any two vertices from different parts are adjacent.
In this section we show that the value of q(K p1,p2,...,p ℓ ) is either 2 or 3, depending on the size of its parts. We also provide an upper bound for M B(K p1,p2,...,p ℓ ) in the case of q(K p1,p2,...,p ℓ ) = 2. The question of which other multiplicity partitions can be achieved by a complete multipartite graph remains open.
In an unpublished manuscript (see [5]), it is shown that any complete multipartite graph K p1,p2,...,p ℓ satisfies q(K p1,p2,...,p ℓ ) ≤ 3. The basic idea employed in the proof of this inequality is to note that the matrix B = [b uv ] with entries defined as satisfies B ∈ S(K p1,p2,...,p ℓ ) and q(B) = 3. Furthermore, it can be easily verified that the eigenvalues of There are several known results for the partitions that are achievable for the complete multipartite graphs; we list them in the following lemma.
Proof. The first statement is from [2]. The second statement follows from Theorem 2.1.
To see that the third statement holds, assume that p 2 + · · · + p ℓ < p 1 , and Finally, the last statement follows from the above comments on the unpublished manuscript [5].
Continuing with the complete multipartite graph, we consider the particular case p 1 ≤ p 2 + · · · + p ℓ . Along these lines, we will make use of the following lemmas to demonstrate that in this case q(K p1,p2,...,p ℓ ) = 2. We begin by stating a technical result that is a special case of [6, Lemma 10] (in the notation of [6], we are setting q = 0 and p = k ≥ 2). Lemma 3.2. Let k ≥ 2, and M 1 and M 2 be matrices that have k rows and no zero columns. Then there exists a k × k matrix R such that R T R = I k and M T 1 RM 2 has no zero entries.
Proof. Since for the given parameter d we have mr + (G) ≤ d ≤ |V |, it follows that there is an A ∈ S + (G)-the set of positive semidefinite matrices in S(G)-with rank(A) = d. Since G has no isolated vertices, A has no zero rows or columns. Since A is positive semidefinite, A can be written as where u 1 , u 2 , . . . , u d are the columns of U , then we may assume that u 1 , u 2 , . . . , u d are mutually orthogonal vectors in R n .
Set M 1 = U T , and M 2 = I d ; each of these matrices have d rows and do not have any zero columns (this follows since A has no zero rows). Hence, by Lemma 3.2, as needed.
The next result, in which a slightly more general version originally appeared in [5], is a technical result concerning a bound on the minimum semidefinite rank of joins of graphs. This result is needed to complete our study on the minimum number of distinct eigenvalues of the complete multipartite graph and establish a bound on the corresponding minimal multiplicity bipartition. We provide a proof here for completeness of exposition. In this proof 1 s is used to denote the vector in R s with all entries equal to one. Similarly, 0 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.
i has the following form: Note that, in Lemma 3.4, if G has ℓ isolated vertices, then these vertices form an independent set. By Lemma 2.4, in order for q(G) = 2, the union of the mutual common neighbours of an independent set cannot have more than ℓ elements; therefore, ℓ ≤ d i s i . Moreover, if d = s 1 = 1 and G has isolated vertices, then there is a unique path from a vertex of G to an isolated vertex of G using the vertex of K s1 , which implies the graph cannot have only two distinct eigenvalues. It is unclear if the statement of Lemma 3.4 holds in other cases when G has ℓ isolated vertices.
The fact that the minimum number of distinct eigenvalues of complete multipartite graphs is at most three is a special case of Lemma 3.4. The next result shows Statement 4 of Lemma 3.1 gives the only family of complete multipartite graphs that do not have only two distinct eigenvalues.
Corollary 3.5. Any complete multipartite graph H = K p1,p2,...,p ℓ with p 1 ≥ p 2 ≥ · · · ≥ p ℓ , p 1 ≤ p 2 + · · · + p ℓ , and ℓ ≥ 3 achieves two distinct eigenvalues, and  For a vertex v in a graph G, a new graph G ′ can be constructed by cloning 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 M B(G) = k results in a graph G ′ with M B(G ′ ) ≤ k. The following proposition is proved in Theorem 6.3 of [12], it is also implied by Corollary 4 of [1]. In [12], this is used to characterize graphs G with M B(G) = 2 by constructing minimal such graphs (these are K 1 , K 1 ∪ K 1 , K 2,1 , K 2,2,...,2 and K 2,2,...,2,1 ) and constructing all the other such graphs by cloning vertices in the minimal graphs.   Proof. Assume M B(G) = M B(H) = k. If k = 1, there is nothing to prove, so assume k ≥ 2. Let n 1 be the number of vertices in G and n 2 the number of vertices in H. Let A ∈ S(G) be such that σ(A) = 0 (n1−k) , 1 (k) and let B ∈ S(H) be such that σ(B) = 0 (n2−k) , 1 (k) . By Schur's Theorem, there exists orthogonal matrices Q 1 and Q 2 such that By Lemma 3.2, there exists a k × k matrix R such that R T R = I k and M T 1 RM 2 has no zero entries. Define C as follows: Hence C is positive semidefinite and C ∈ S(G ∨ H) since M T 1 RM 2 is an entrywise nonzero matrix. It is easy to note that C has rank k since [M 1 RM 2 ] has a full-row rank. Therefore, null(C) = n 1 + n 2 − k. Moreover, C 2 = 2C which implies σ(C) = 0 (n1+n2−k) , 2 (k) . Hence q(G ∨ H) = 2 since C ∈ S(G ∨ H) and q(G ∨ H) > 1. In fact, this idea can be easily generalized as follows: Suppose G is a graph with q(G) = 2 that contains an independent set of vertices S = {v 1 , v 2 , . . . , v k } in which ∪ vi,vj ∈S (N (v i ) ∩ N (v j )) = ∅. Then for any graph H with q(H) = 2 and |H| < k, we have q(G ∨ H) > 2. To see this, it is enough to observe that in the graph G ∨ H we have | ∪ vi,vj ∈S (N (v i ) ∩ N (v j ))| = |H| < |S|, and hence the condition of Lemma 2.4 fails to hold.
We also note that the assumption of no isolated vertices in Theorem 3.9 is possibly a stronger condition than is in fact necessary; this assumption is used to ensure that the matrix M 2 in the proof has no zero columns. For instance, in the next result, which is a weaker version of Lemma 3.4, all the vertices of the second graph are isolated vertices. The proof of Lemma 3.10 is the same as the proof of Theorem 3.9, except that the matrix B is replaced with the identity matrix. We denote the graph on k vertices with no edges by K k .
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 3.9 and induction. We skip the details since a more general result is shown in Corollary 3.5.
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 1 ).
The next theorem is the main result of [14].
Theorem 3.12 (Theorem 4.3 [14]). Let G be a connected graph on n vertices and let λ 1 , λ 2 , . . . , λ n be distinct real numbers. Then there exists a real symmetric matrix A ∈ S(G) with eigenvalues λ 1 , λ 2 , . . . , λ n such that none of the eigenvectors of A has a zero entry. Lemma 3.13. Let G be a connected graph on n ≥ 2 vertices. Then q(G ∨ K n ) = 2 and M B(G ∨ K n ) = n.
Proof. Since G is a connected graph, by Theorem 3.12 there exists a matrix A ∈ S(G) with positive distinct eigenvalues λ 1 > λ 2 > · · · > λ n and corresponding entry-wise nonzero unit eigenvectors v 1 , . . . , v n such that Since U is an entry-wise nonzero matrix, if each a i is also non-zero, then Further, the rows of C are orthogonal and so . . , n. Therefore, the eigenvalues of CC T are α i , i = 1, . . . , n. The values a i can be set so that they are all strictly positive, and α i for all i = 1, . . . , n are equal to some λ 0 > λ 2 1 . Then the spectrum of C T C is 0 with multiplicity n, and λ 0 also with multiplicity n. This implies that q(G ∨ K n ) = 2 and M B(G ∨ K n ) ≤ n. Finally, the vertices in K n form an independent set of size n, and so by Statement 3 of Lemma 2.3, it follows that M B(G ∨ K n ) = n.
The same proof can be used to prove the following result.

Constructions
Corollary 3.8 provides an infinite family of graphs in M P ([n − k, k)] for various values of n and k. Corollary 3.8 gives a complete characterization of graphs with M P ([n − k, k)] for k = 2, but not for any larger value of k. In this section, we consider graphs that are in M P ([n − k, k)] but not covered in Corollary 3.8. First, we consider a direct construction of matrices corresponding to some of the graphs in Corollary 3.8. Clearly (t i + 1) 2 and the three vectors are pairwise orthogonal for each i. Now, form three overall vectors by concatenation, so These three vectors all have the same norm and are pairwise orthogonal.
. Let x, y be two vertices in G, then the (x, y)-entry of A is given by .
If x and y are both in K ai ∪ K bi ∪ K ci , but not both in the same clique (induced complete graph), then [A] x,y = 0.
Assume that x ∈ K a1 , then If t 1 and t 2 are distinct and positive, these are all non-zero. Similarly we can show that for any x ∈ K a1 ∪ K b1 ∪ K c1 and any y ∈ K a2 ∪ K b2 ∪ K c2 that [A] x,y is not equal to zero. Therefore, A ∈ S(G).
The matrix M in Theorem 4.2 of order at most 5 can be given as follows. These matrices satisfy the conditions of Theorem 4.2 if t > 1 and all values of t are distinct. If k = 2, then If k = 3, then If k = 4, then and r = s s+1 . Note that in the construction in Lemma 4.1, the numerator in the entries in the vectors come from the entries of the 3 × 3 matrix M t . This method can be generalized.
The following Lemma provides graphs with M B(G) = 3 that are not listed in Corollary 3.8. Lemma 4.3. Let H = K α,α be a complete bipartite graph with 2α ≤ a 1 . Suppose with a i , b i > 2 and ℓ ≥ 2. Then q(G) = 2 and M B(G) = 3.
where β = 1 + w 2 1 and all vectors v 3i , i ≥ 3 are zero vectors. For i > 2, if we choose the value of w i large enough so that ℓ i=2 a i (1 + w 2 i ) − α(1 + w 2 1 ) > 0, then setting a so that results in a vector v 3 that has the same norm as vectors v 1 and v 2 . The vectors v T 1 , v T 2 , and v T 3 form orthogonal rows of a 3 × |V (H)| matrix U , where the orthogonality of the columns of U represents the edges and non-edges of H. Thus, U T U ∈ S(G), which completes the proof.
This method can be extended to the graphs covered by Theorem 4.2.
The norm of v 3 is 2β 2 + 4 (1+w1w2) 2 . This is a continuous function in β 2 and it takes values in the interval ((4 + 2 √ 2) √ 1 + w 1 w 2 , ∞). The norm of v 1 is at least 4 + 2w 2 1 + 2w 2 2 . Since it is possible to choose w 1 and w 2 so that the norm of v 1 is larger than (4 + 2 √ 2) √ 1 + w 1 w 2 , it is also possible to choose β so that the norm of v 3 equals v 1 .
This method can also be extended to the graph in Theorem 4.2. From Theorem 2.1 we know that q(P n ∨ P n ) = 2 and Lemma 2.6 implies that M B(P n ∨ P n ) ≥ n − 1. We consider a related graph that achieves this same lower bound. Let P 2 n be the graph on n vertices labeled by 1, 2, . . . , n, 1 ′ , 2 ′ , . . . , n ′ . Vertices i and i ′ are adjacent for all i ∈ {1, . . . , n}. If i ∈ 2, . . . , n − 1, then i and i ′ are adjacent to vertices i − 1, (i − 1) ′ , i + 1, (i + 1) ′ . Vertices 1 and 1 ′ are adjacent to vertices 2 and 2 ′ . Vertices n and n ′ are adjacent to n − 1 and (n − 1) ′ .
Lemma 4.5. For any n, q(P 2 n ) = 2 and M B(P 2 n ) = n − 1. Proof. For i ∈ {0, . . . , n−2} let U i be the vector with the (2i+1) and (2i+2)-entries equal to 1, the 2i + 3 entry equal to 2 and the 2i + 4 entry equal to −2. These vectors satisfy the conditions of Lemma 2.2, so q(P 2 n ) = 2 and M B(P 2 n ) < n − 1. The result follows since P 2 n has an induced path of length n and Lemma 2.6. A graph is a path of cliques if its set of vertices can be partitioned into clusters, such that each cluster is a clique of size at least two, and the cliques form a path. A path of cliques whose clusters have at least two vertices can be obtained from P 2 n by cloning vertices. The next result follow from Proposition 3.7.
Corollary 4.6. If G is a path of cliques of size at least 4 with k the longest induced path in G, then q(G) = 2 and M B(G) = k − 1.

Open Problems
In [11] a large number of graphs are shown to admit only two distinct eigenvalues. In fact, they prove that many bipartite graphs G have the property that q(G) = 2. This gives another large and diverse family of graphs with only 2 distinct eigenvalues, and for all of these graph it is interesting to consider the multiplicity bipartition. This family includes the complements of many trees, in particular they show that q(P n ) = 2 if n ≥ 6. The only results we have are that M B(P 6 ) = M B(P 7 ) = 3.
Question 5.1. What is the multiplicity bipartition for the complement of a path on at least 8 vertices?
The graphs in M P ([n−2, 2]) have been exactly characterized (see [7,12,13,15]), from this characterization it can be seen that there are no trees T with M B(T ) = 2.
One of the types of trees considered in [11] are denoted by S r m,n (these are called type-one trees). The graph S r m,n is formed by taking a path on r vertices and adding m leaves to one end point and n leaves to the other end point. Alternately, these trees are formed by taking K 1,k ∪ K 1,ℓ and added one additional edge to make the graph connected. If the edge is added between two leaves in K 1,k ∪ K 1,ℓ , then the resulting graph is S 4 k−1,ℓ−1 ; if the edge is added between a leaf and a non-leaf then the resulting graph is either S 3 k,ℓ−1 or S 3 k−1,ℓ ; finally, if the edge is added between two non-leaves the resulting graph is S 2 k,ℓ . Note that if T = S k m,n with k = 2, 3, 4, then T is formed by taking (K 1 ∪ K m ′ ) ∨ (K 1 ∪ K n ′ ) and removing a single edge across the join. In Lemma 4.4, it is shown that in some cases two edges can be removed across the join. We conjecture that it is also possible to remove a single edge across the join in many cases and achieve the multiplicity bipartition [n − 3, 3].