Achievable multiplicity partitions in the inverse eigenvalue problem of a graph
Mohammad Adm, Shaun Fallat, Karen Meagher, Shahla Nasserasr, Sarah Plosker, Boting Yang
aa r X i v : . [ m a t h . SP ] D ec ACHIEVABLE MULTIPLICITY PARTITIONS IN THE INVERSEEIGENVALUE PROBLEM OF A GRAPH
MOHAMMAD ADM , SHAUN FALLAT , KAREN MEAGHER , SHAHLA NASSERASR ,SARAH PLOSKER , AND BOTING YANG Abstract.
Associated to a graph G is a set S ( G ) of all real-valued symmetricmatrices whose off-diagonal entries are nonzero precisely when the correspond-ing vertices of the graph are adjacent, and the diagonal entries are free to bechosen. If G has n vertices, then the multiplicities of the eigenvalues of anymatrix in S ( G ) partition n ; this is called a multiplicity partition.We study graphs for which a multiplicity partition with only two integersis possible. The graphs G for which there is a matrix in S ( G ) with partitions[ n − ,
2] have been characterized. We find families of graphs G for which thereis a matrix in S ( G ) with multiplicity partition [ n − k, k ] for k ≥
2. We focus ongeneralizations of the complete multipartite graphs. We provide some methodsto construct families of graphs with given multiplicity partitions starting fromsmaller such graphs. We also give constructions for graphs with matrix in S ( G )with multiplicity partition [ n − k, k ] to show the complexities of characterizingthese graphs. Introduction
Let G be a simple graph on n vertices and consider the set S ( G ) of all possiblereal-valued weighted symmetric adjacency matrices associated to G , where the di-agonal entries are free in that they may or may not be zero (the restriction hereinto simple graphs avoids some unnecessary confusion when stating and proving re-sults). The notation λ ( n i ) i is used to denote the eigenvalue λ i with multiplicity n i .In this work, we order the eigenvalues according to their corresponding multiplic-ities. That is, a matrix A ∈ S ( G ) has spectrum σ ( A ) = { λ ( n )1 , . . . , λ ( n ℓ ) ℓ } , where n ≥ · · · ≥ n ℓ and the eigenvalues λ i are distinct. The list of eigenvalues could alsobe ordered according to their values, this is distinctly different from the orderingconsidered in this work; see [10].If q ( A ) is the number of distinct eigenvalues of a symmetric matrix A , then for agiven graph G , we let q ( G ) = min { q ( A ) : A ∈ S ( G ) } , and refer to this parameter as Date : December 3, 2019.2010
Mathematics Subject Classification.
Key words and phrases. inverse eigenvalue problem, multiplicity partition, adjacency matrix,minimum rank, distinct eigenvalues, graphs. Department of Applied Mathematics and Physics, Palestine Polytechnic University, Hebron,Palestine. Department of Mathematics and Statistics, University of Konstanz, Konstanz, Germany. Department of Mathematics and Statistics, University of Regina, Regina, SK, S4S 0A2,Canada. Department of Mathematics and Computer Science, Brandon University, Brandon, MB R7A6A9, Canada. Department of Computer Science, University of Regina, Regina, SK, S4S 0A2, Canada. the minimum number of distinct eigenvalues of G . It is well-known that q ( G ) = 1if 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 − q ( G ) = c , for c ∈ { , . . . , n − } are given in [2]. For example, ifa connected graph G on at least three vertices has q ( G ) = 2, then G cannot havea cut edge (an edge whose deletion results in a disconnected graph), and any twonon-adjacent vertices of G must have at least two common neighbours (Lemma 4.2and Corollary 4.5 of [2], respectively). Moreover, Corollary 3.6 of the same papergives a construction for a graph on n vertices satisfying q ( G ) = c for any 1 ≤ c ≤ n .For positive integers n ≥ · · · ≥ n ℓ , a partition [ n , n , . . . , n ℓ ] of n is said tobe achievable by G if there exists an A ∈ S ( G ) such that the spectrum of A is { λ ( n )1 , . . . , λ ( n ℓ ) ℓ } for some set of distinct values λ i . If [ n , n , . . . , n ℓ ] is a partitionof n , we use the notation M P ([ n , n , . . . , n ℓ ]) to denote the set of all graphs on n vertices for which the partition [ n , . . . , n ℓ ] of n is achievable.Given a graph, some natural questions arise. For example, what multiplicitypartitions are achievable by a given graph or a family of graphs? This question isconsidered in [1, 3] where all achievable partitions are listed for all graphs on fewerthan 6 vertices. Another question is if a certain multiplicity partition is achievablefor a graph, is it possible to characterize any other multiplicity partitions that arealso achievable for the graph? In particular, for two partitions [ n , n , . . . , n ℓ ] and[˜ n , ˜ n , . . . , ˜ n m ] of n , when is M P ([ n , n , . . . , n ℓ ]) ⊆ M P ([˜ n , ˜ n , . . . , ˜ n m ])?Another natural approach is to characterize the graphs in M P ([ n , . . . , n ℓ ]) forsome partition [ n , . . . , n ℓ ]. This has been answered for a limited number of par-titions. For example, the main result in [14] is that every connected graph on n vertices is in M P ([1 , , . . . , M P ([ n ]) is the graph on n vertices with no edges. The set M P ([ n − , n vertices that have one connected component that is a complete graph and theremaining components are isolated vertices (this includes the graph on n verticeswith no edges). The graphs in M P ([ n − , 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 verticesachieves a bipartition [ n − k, k ] for some k = 1 , , . . . , ⌊ n ⌋ . The initial results thatwe 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 everyvertex of G incident to every vertex of H . The following theorem by Monfared andShader [14, Theorem 5.2] gives a sufficient condition for the minimum number ofdistinct eigenvalues to be 2. CHIEVABLE MULTIPLICITY PARTITIONS IN THE IEVP OF A GRAPH 3
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 , or in one oftwo families of trees. Further, Theorem 2.5 of the same paper proves that manybipartite graphs G have the property q ( G ) = 2. This gives another large and diversefamily of graphs with only 2 distinct eigenvalues.The results in [14] and [11] indicate that the collection of graphs with onlytwo distinct eigenvalues is very large and likely very difficult to characterize. Thusrather than trying to characterize all graphs G with q ( G ) = 2, we define the minimalmultiplicity bipartition M B ( G ) to be the least integer k ≤ ⌊ n ⌋ such that G achievesthe multiplicity bipartition [ n − k, k ] (hence if k = M B ( G ), then G ∈ M P ([ n − k, k ])but G / ∈ M P ([ n − m, m ]) for any m < k ). If M B ( G ) = k , then we say G has multiplicity bipartition k . The multiplicity bipartition is only defined for graphsthat admit only two distinct eigenvalues and if G has n vertices, then M B ( G ) ≤ n/ 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 itsspectrum 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 ( J n ) denotethe n × m ( n × n ) matrix with all entries equal to one, and we let 0 n denote the n × n zero matrix. Lemma 2.2.
Let G be a graph on n vertices with q ( G ) = 2 . Then G ∈ M P ([ n − k, k ]) if and only if there is an A ∈ S ( G ) with A = u u T + u u T + · · · + u k u Tk , where k ≤ ⌊ n ⌋ and { u , u , . . . , 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 ) = (cid:8) ( n − k ) , ( k ) (cid:9) . Therefore, A can be written as A = V ( I k ⊕ O n − k ) V T = U U T , where V is a unitary matrix, U = V [1 , . . . , n | , . . . , k ] and O n − k is the ( n − k ) × ( n − k )all zeros matrix. Let U = ( u , u , . . . , u k ) where u i ∈ R n , i = 1 , . . . , k . Each u i is acolumn of V , so they are orthonormal. Moreover, A = u u T + u u T + · · · + u k u Tk .Conversely, assume there is a matrix A = u u T + u u T + · · · + u k u Tk , where { u , u , . . . , u k } is a set of orthonormal vectors in R n . Then the spectrum for A is (cid:8) ( n − k ) , ( k ) (cid:9) , so G ∈ M P ([ n − k, k ]). (cid:3) We summarize the known characterizations of graphs with given minimal mul-tiplicity bipartitions in the following lemma.
Lemma 2.3.
Let G be a connected graph on n vertices. Then(1) M B ( G ) = 1 if and only if G is the complete graph, K n .(2) M B ( G ) = 2 if and only if G = ( K p ∪ K q ) ∨ ( K p ∪ K q ) ∨ · · · ∨ ( K p k ∪ K q k ) , for non-negative integers p , . . . , p k , q , . . . , q k with k > , and G is notisomorphic to either one of a complete graph or ( K p ∪ K q ) ∨ K . M. ADM, S. FALLAT, K. MEAGHER, S. NASSERASR, S. PLOSKER, AND B. YANG (3) If
M B ( G ) = k , then G does not have an independent set (a set of verticesfor 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 weinclude a proof for completeness. From Lemma 2.2 there is a matrix A ∈ S ( G )with A = u u T + u u T + · · · + u k u Tk where u i , i = 1 , , . . . , k , are orthonormal vectors. Let U = [ u , u , . . . , u k ] where u i ∈ R n , i = 1 , . . . , k . If G has an independent set of size k + 1, then the rowsof U form k + 1 orthogonal vectors in R k , which is impossible. Hence there is noindependent set of size k + 1. (cid:3) The following lemma indicates that for a connected graph G with q ( G ) = 2, ifthe union of the pairwise common neighbourhood of an independent set of verticesis not empty, then it cannot be too small. Lemma 2.4 (Theorem 4.4, [2]) . Consider a connected graph G with q ( G ) = 2 , andlet S be an independent set of vertices. If ∪ v i ,v j ∈ S ( N ( v i ) ∩ N ( v j )) is not empty,then | S | ≤ | ∪ i,j ∈ S ( N ( v i ) ∩ N ( v j )) | . For a given graph G , the minimum rank among all matrices (positive semidefinitematrices) 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 ) withspectrum { ( n − k ) , ( k ) } . This implies the following lemma. Lemma 2.5. If G is any graph with q ( G ) = 2 , then mr( G ) ≤ mr + ( G ) ≤ M B ( G ) . The parameter mr( G ) has been extensively studied [9]; and any lower bound onthe minimum rank or the minimum positive semidefinite rank of a graph, is also alower bound on the minimal multiplicity bipartition of the graph. For example, wemay use the above bound on minimum rank together with [9, Obs. 1.6] to deducethe next result. Lemma 2.6. If G is any graph with M B ( G ) = k , then any induced path of G haslength no more than k . Complete Multipartite Graphs
As is standard, we use the notation K p ,p ,...,p ℓ for the complete ℓ -partite graph,where ℓ is a positive integer. The set of vertices is partitioned into ℓ disjoint parts V ∪ V ∪ · · · ∪ V ℓ ; part V i has p i vertices for i ∈ { , . . . , ℓ } ; no two vertices from apart are adjacent, while any two vertices from different parts are adjacent.In this section we show that the value of q ( K p ,p ,...,p ℓ ) is either 2 or 3, dependingon the size of its parts. We also provide an upper bound for M B ( K p ,p ,...,p ℓ ) inthe case of q ( K p ,p ,...,p ℓ ) = 2. The question of which other multiplicity partitionscan be achieved by a complete multipartite graph remains open.In an unpublished manuscript (see [5]), it is shown that any complete multipartitegraph K p ,p ,...,p ℓ satisfies q ( K p ,p ,...,p ℓ ) ≤
3. The basic idea employed in the proof
CHIEVABLE MULTIPLICITY PARTITIONS IN THE IEVP OF A GRAPH 5 of this inequality is to note that the matrix B = [ b u,v ] with entries defined as b u,v = u, v ∈ V i , √ p i p j if u ∈ V i , v ∈ V j , and i = j .satisfies B ∈ S ( K p ,p ,...,p ℓ ) and q ( B ) = 3. Furthermore, it can be easily verifiedthat the eigenvalues of B are {− , , ℓ − } with multiplicities ℓ − , P ℓi =1 ( p i − , K p ,p ,...,p ℓ ∈ M P ([ P ℓi =1 ( p i − , ℓ − , Lemma 3.1.
For positive integers ℓ, p i , q i with i = 1 , , . . . , ℓ (1) q ( K p ,p ,...,p ℓ ) ∈ { , } .(2) q ( K p ,q ) = ( if p = q , otherwise.(3) If p + p + · · · + p ℓ = q + q + · · · + q ℓ ′ for ℓ, ℓ ′ ≥ , then q ( K p ,p ,...,p ℓ ,q ,q ,...,q ℓ ′ ) = 2 . (4) If p + · · · + p ℓ < p , then q ( K p ,p ,...,p ℓ ) = 3 .Proof. The first statement is from [2]. The second statement follows from Theo-rem 2.1.To observe that the third statement holds, note that p + p + · · · + p ℓ = q + q + · · · + q ℓ ′ for ℓ, ℓ ′ ≥ K p ,p ,...,p ℓ ,q ,q ,...,q ℓ ′ is isomorphic to the joinof K p ,p ,...,p ℓ and K q ,q ,...,q ℓ ′ , and hence satisfies q ( K p ,p ,...,p ℓ ,q ,q ,...,q ℓ ′ ) = 2 byTheorem 2.1.To see that the last statement holds, assume that p + · · · + p ℓ < p , and set n = p + p + · · · + p ℓ ; this implies n − p < p . If q ( K p ,p ,...,p ℓ ) = 2, then K p ,p ,...,p ℓ ∈ M P ([ n − k, k ]) with p ≤ k ≤ n − k (by Statement (3) of Lemma 2.3).Hence p ≤ k ≤ n − k ≤ n − p , implying 2 p ≤ n , which is a contradiction. Hence q ( K p ,p ,...,p ℓ ) ≥
3. Finally, the equality follows from the work in the unpublishedmanuscript [5]. (cid:3)
Continuing with the complete multipartite graph, we consider the particular case p ≤ p + · · · + p ℓ . Along these lines, we will make use of the following lemmas todemonstrate that in this case q ( K p ,p ,...,p ℓ ) = 2. We begin by stating a technicalresult that is a special case of [6, Lemma 10] (in the notation of [6], we are setting q = 0 and p = k ≥ Lemma 3.2.
Let k ≥ , and M and M be matrices that have k rows and no zerocolumns. Then there exists a k × k matrix R such that R T R = I k and M T RM has no zero entries. Lemma 3.3.
For any graph G with no isolated vertices and for any number d suchthat mr + ( G ) ≤ d ≤ | V ( G ) | , there exist vectors q , . . . , q d such that P i q i q Ti ∈ S ( G ) and each q i , i = 1 , . . . , d , is entry-wise nonzero.Proof. Since for the given parameter d we have mr + ( G ) ≤ d ≤ | V | , it follows thatthere is an A ∈ S + ( G )—the set of positive semidefinite matrices in S ( G )—withrank( 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 A = U U T , for some n × d M. ADM, S. FALLAT, K. MEAGHER, S. NASSERASR, S. PLOSKER, AND B. YANG matrix U . If U = [ u , u , . . . , u d ], where u , u , . . . , u d are the columns of U , thenwe may assume that u , u , . . . , u d are mutually orthogonal vectors in R n .Set M = U T , and M = I d ; each of these matrices have d rows and do not haveany zero columns (this follows since A has no zero rows). Hence, by Lemma 3.2,there exists an orthogonal d × d matrix R such that M T RM = U R has no zeroentries. Let Q = U R = [ q , q , . . . , q d ] then QQ T = U RR T U T = U U T = A ∈ S ( G ) . as needed. (cid:3) The next result, in which a slightly more general version originally appearedin [5], is a technical result concerning a bound on the minimum semidefinite rankof joins of graphs. This result is needed to complete our study on the minimumnumber of distinct eigenvalues of the complete multipartite graph and establish abound on the corresponding minimal multiplicity bipartition. We provide a proofhere for completeness of exposition. In this proof s is used to denote the vectorin 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 clearfrom context. Lemma 3.4.
Consider the graph G with no isolated vertices and positive integers d, s , s , . . . , s d . If < mr + ( G ) ≤ d ≤ | V ( G ) | , then q ( G ∨ ( K s ∪ K s ∪ · · · ∪ K s d )) =2 ; moreover, M B ( G ∨ ( K s ∪ K s ∪ · · · ∪ K s d )) = d .Proof. Let H = G ∨ ( K s ∪ K s ∪ · · · ∪ K s d ), and s = P di =1 s i . We constructa matrix B ∈ S ( H ) with σ ( B ) = { , β } , where the eigenvalue 0 has multiplicity | V ( H ) | − d and the eigenvalue β has multiplicity d . Using Lemma 3.3, there areentry-wise nonzero vectors q , . . . , q d such that A = P di =1 q i q Ti ∈ S ( G ). Constructvectors v , . . . , v d as follows v = q α s s − s , v = q s α s s − s − s , . . . , v d = q d s − s d α d s d the value of α i will be determined. Let q Ti = (cid:2) q ,i , q ,i , . . . , q n,i (cid:3) for i = 1 , . . . , d where | V ( G ) | = n .For j = 2 , . . . , d + 1, let B ,j = [ q j − , q j − , . . . , q j − ] be the n × s j − matrixwith all columns equal to q j − . The matrix B = P di =1 v i v Ti has the following form: B = A α B , . . . α d B ,d +1 α B T , α J s × s O . . . OO α J s × s O ... ...... . . . Oα d B T ,d +1 O . . . O α d J s d × s d . CHIEVABLE MULTIPLICITY PARTITIONS IN THE IEVP OF A GRAPH 7
Now, for a positive number β > max {|| q i || , i = 1 , . . . , d } , set α i = s β − || q i || s i .Then B ∈ S ( H ) and rank( B ) = d , which implies the eigenvalue zero has multiplicity | V ( H ) | − d . On the other hand, Bv i = βv i for each i = 1 , , . . . , d , and thevectors v , . . . , v d are linearly independent. Thus σ ( B ) = { , β } with the desiredmultiplicities, and since H has at least d independent vertices, d ≤ M B ( H ). (cid:3) Note that, in Lemma 3.4, if G has ℓ isolated vertices, then these vertices forman independent set. By Lemma 2.4, in order for q ( G ) = 2, the union of the mu-tual common neighbours of an independent set cannot have more than ℓ elements;therefore, ℓ ≤ P di s i . Moreover, if d = s = 1 and G has isolated vertices, thenthere is a unique path from a vertex of G to an isolated vertex of G using the vertexof K s , which implies the graph cannot have only two distinct eigenvalues. It isunclear if the statement of Lemma 3.4 holds in other cases when G has ℓ isolatedvertices.The fact that the minimum number of distinct eigenvalues of complete multipar-tite graphs is at most three is a special case of Lemma 3.4. The next result showsStatement 4 of Lemma 3.1 gives the only family of complete multipartite graphsthat do not have only two distinct eigenvalues. Corollary 3.5.
Any complete multipartite graph H = K p ,p ,...,p ℓ with p ≥ p ≥· · · ≥ p ℓ , p ≤ p + · · · + p ℓ , and ℓ ≥ achieves two distinct eigenvalues, and M B ( H ) = p .Proof. Note that H = ( K ∪ K ∪ · · · ∪ K ) | {z } p times ∨ K p ,...,p ℓ . Let G = K p ,...,p ℓ , d = p and s = s = · · · = s d = 1. Then mr + ( G ) = p ≤ p = d (see [17, Prop. 3.4]). Using Lemma 3.4, q ( H ) = 2 with M B ( H ) = p . (cid:3) A specific case of Corollary 3.5 is when all the parts have the same sizes.
Corollary 3.6.
Let G = K k,k,...,k , k ≥ be the complete multipartite graph on n vertices. Then q ( G ) = 2 and M B ( G ) = k . For a vertex v in a graph G , a new graph G ′ can be constructed by cloning (or duplicating ) v . The graph G ′ has vertex set V ( G ′ ) = V ( G ) ∪ { v ′ } and edgeset E ( G ′ ) = E ( G ) ∪ { v ′ u ; u ∈ N [ v ] } , where N [ v ] is the closed neighbourhood of v (that is, a neighbourhood of v containing v ). It turns out that cloning a vertex of agraph G with M B ( G ) = k results in a graph G ′ with M B ( G ′ ) ≤ k . The followingproposition 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 constructingminimal such graphs (these are K , K ∪ K , K , , K , ,..., and K , ,..., , ) andconstructing all the other such graphs by cloning vertices in the minimal graphs. Proposition 3.7.
Let G be a graph with G ∈ M P ([ n − k, k ]) . If H is obtainedfrom G by cloning a vertex in G , then H ∈ M P ([ n − k + 1 , k ]) . Suppose G is a graph with q ( G ) = 2. If H is obtained from G by cloning avertex, then M B ( H ) ≤ M B ( G ). It is not clear if this inequality is ever strict.By cloning vertices in K k,k,...,k , where k ≥
2, we have the following consequence,reminiscent of Lemma 2.3.
M. ADM, S. FALLAT, K. MEAGHER, S. NASSERASR, S. PLOSKER, AND B. YANG
Corollary 3.8. If G = W li =1 ( ∪ kj =1 K a i,j ) where k ≥ , i = 1 , , . . . , l, j = 1 , , . . . , k ,and a i,j are positive integers, then q ( G ) = 2 and M B ( G ) = k . (cid:3) Unlike the case where k = 2, for general k the previous corollary does not charac-terize all the graphs in M P ([ n − k, k ]). In Section 4, several graphs in M P ([ n − k, k ])that are not included in Corollary 3.8 are given. Theorem 3.9.
Let G and H be two graphs with no isolated vertices. Furtherassume that q ( G ) = q ( H ) = 2 with M B ( G ) = M B ( H ) . Then q ( G ∨ H ) = 2 and M B ( G ∨ H ) ≤ M B ( G ) (= M B ( H )) .Proof. Assume
M B ( G ) = M B ( H ) = k . If k = 1, there is nothing to prove, soassume k ≥
2. Let n be the number of vertices in G and n the number of verticesin H . Let A ∈ S ( G ) be such that σ ( A ) = (cid:8) ( n − k ) , ( k ) (cid:9) and let B ∈ S ( H ) be suchthat σ ( B ) = (cid:8) ( n − k ) , ( k ) (cid:9) . By Schur’s Theorem, there exists orthogonal matrices Q and Q such that A = Q T ( I k ⊕ O n − k ) Q and B = Q T ( I k ⊕ O n − k ) Q . Let M = Q [1 , . . . , k | , . . . , n ] and M = Q [1 , . . . , k | , . . . , n ] , so A = M T M and B = M T M . This also implies that M has no zero columns,since otherwise A would have a row and column of zeros which would imply that G would have an isolated vertex. Similarly, M has no zero columns.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. Define C as follows: C = (cid:20) M T M T R T (cid:21) (cid:2) M RM (cid:3) = (cid:20) M T M M T RM M T R T M M T R T RM (cid:21) = (cid:20) A M T RM M T R T M B (cid:21) . Hence C is positive semidefinite 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 ] hasa full-row rank. Therefore, null( C ) = n + n − k . Moreover, C = 2 C whichimplies σ ( C ) = (cid:8) ( n + n − k ) , ( k ) (cid:9) . Hence q ( G ∨ H ) = 2 since C ∈ S ( G ∨ H ) and q ( G ∨ H ) > (cid:3) In Theorem 3.9, where k = M B ( G ) = M B ( H ), if k = 1, then the inequality M B ( G ∨ H ) ≤ M B ( G ∨ H ) ≥ k = 2, then the inequality M B ( G ∨ H ) ≤ M B ( G ∨ H ) = 1 implies that G ∨ H is a complete graph with possiblyisolated vertices, which is a contradiction. Moreover, if k = 3, then the inequality M B ( G ∨ H ) ≤ M B ( G ∨ H ) = 1 , G ∨ H is either a complete graph with possibly isolated vertices, or a graph characterizedin Lemma 2.3 (2), either case is a contradiction.For cases of k ≥ M B ( G ∨ H ) ≤ k . We suspect, that in fact, equality among M B ( G ∨ H ) = M B ( G ) = M B ( H ) holds under the hypothesis of Theorem 3.9. A related matteris to determine if a version of Theorem 3.9 still holds in the case when M B ( G ) = M B ( H ). It turns out that the requirement of M B ( G ) = M B ( H ) is essential inconcluding that q ( G ∨ H ) = 2 as in Theorem 3.9. Consider the following example.Let G = Q (the 6-dimensional hypercube), and let S = { x, y, z } be the independent CHIEVABLE MULTIPLICITY PARTITIONS IN THE IEVP OF A GRAPH 9 set of vertices in G consisting of x = (000000) , y = (010101) , and z = (111111).Also observe that there are no common neighbours among any pair of vertices from S in G . Let H = P . Then we have q ( G ) = q ( H ) = 2, and M B ( G ) ≥ M B ( H ) = 1. However, in the graph G ∨ H , using the independent set S , it is easyto deduce that the condition of Lemma 2.4 fails. Hence q ( G ∨ H ) > G is a graph with q ( G ) = 2 that contains an independent set of vertices S = { v , v , . . . , v k } in which ∪ v i ,v j ∈ 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 | ∪ v i ,v j ∈ 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 possi-bly a stronger condition than is in fact necessary; this assumption is used to ensurethat the matrix M in the proof has no zero columns. For instance, in the nextresult, which is a weaker version of Lemma 3.4, all the vertices of the second graphare isolated vertices. The proof of Lemma 3.10 is the same as the proof of Theo-rem 3.9, except that the matrix B is replaced with the identity matrix. We denotethe graph on k vertices with no edges by K k . Lemma 3.10.
Let G be a graph with no isolated vertices and q ( G ) = 2 with G ∈ M P ([ n − k, k ]) for some k ≥ . Then q ( G ∨ K k ) = 2 with M B ( G ∨ K k ) ≤ k . Note that the multiplicity bipartition [ n − k, k ] for regular complete multipartitegraphs 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 ofthe join of two graphs can be large. Lemma 3.11.
For any graph G , q ( G ∨ K ) ≥ ⌈ q ( G )+12 ⌉ .Proof. The eigenvalues for any matrix in S ( G ) interlace the eigenvalues any matrix S ( G ∨ K ). (cid:3) 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 andlet λ , λ , . . . , λ n be distinct real numbers. Then there exists a real symmetric matrix A ∈ S ( G ) with eigenvalues λ , λ , . . . , 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 ≥ 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 > λ > · · · > λ n and corresponding entry-wise nonzero unit eigenvectors v , . . . , v n such that A = V T Λ V = U T U, where U = Λ V , Λ = diag( λ , . . . , λ n ), and V T = [ v , . . . , v n ]. The rows of U areorthogonal since V is unitary. Let C be the n × n matrix C = (cid:2) D U (cid:3) , where D = a . . . a . . . . . . a n , where the scalars a i ( i = 1 , , . . . , n ) are to be determined. Since U is an entry-wisenonzero matrix, if each a i is also nonzero, then C T C = (cid:20) D DUU T D U T U (cid:21) ∈ S ( G ∨ K n ) . Further, the rows of C are orthogonal and so CC T = α . . . α . . . . . . α n , where α i = a i + λ i , i = 1 , . . . , 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 > λ . Then the spectrum of C T C is 0 withmultiplicity n , and λ also with multiplicity n . This implies that q ( G ∨ K n ) = 2and 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 . (cid:3) The same proof can be used to prove the following result.
Lemma 3.14.
Let G be a connected graph on n ≥ vertices. Then q ( G ∨ K n − ) = 2 and M B ( G ∨ K n − ) = n − .Proof. As in the proof of the previous lemma, there exists a matrix A ∈ S ( G ) witheigenvalues λ > λ > · · · > λ n − > λ n = 0and corresponding entry-wise nonzero unit eigenvectors v , . . . , v n such that A = V T Λ V = U T U, where V T = [ v , . . . , v n ], Λ = diag( λ , . . . , λ n − , U = DV [1 , . . . , n − | , . . . , n ],and D = diag( √ λ , . . . , p λ n − ). Hence the rows of U are orthogonal since V isunitary. Let C be the ( n − × (2 n −
1) matrix C = (cid:2) D U (cid:3) . Since U is anentry-wise nonzero matrix, as long as each a i is also nonzero, the matrix C T C = (cid:20) D DUU T D U T U (cid:21) ∈ S ( G ∨ K n − ) . Further, the rows of C are orthogonal and so CC T = α . . . α . . . . . . α n − , CHIEVABLE MULTIPLICITY PARTITIONS IN THE IEVP OF A GRAPH 11 where α i = a i + λ i , i = 1 , . . . , n −
1. Similar to the proof of the previous lemma,the spectrum of C T C is 0 with multiplicity n , and λ with multiplicity n −
1. Thisimplies that q ( G ∨ K n ) = 2 and M B ( G ∨ K n − ) = n − (cid:3) Constructions
Corollary 3.8 provides an infinite family of graphs in
M P ([ n − k, k ]) for variousvalues 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, weconsider 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 graphsin Corollary 3.8. Example 4.1.
Let G = ( K a ∪ K b ∪ K c ) ∨ ( K a ∪ K b ∪ K c ) with a i , b i , c i > , so that q ( G ) = 2 and M B ( G ) = k by Corollary 3.8. For i = 1 , and t i ∈ R to be determined, set v ,i = (cid:2) t i √ a i , . . . , t i √ a i | {z } a i times , √ b i , . . . , √ b i | {z } b i times , − t i √ c i ( t i + 1) , . . . , − t i √ c i ( t i + 1) | {z } c i times (cid:3) T v ,i = (cid:2) − t i √ a i ( t i + 1) , . . . , − t i √ a i ( t i + 1) | {z } a i times , t i √ b i , . . . , t i √ b i | {z } b i times , √ c i , . . . , √ c i | {z } c i times (cid:3) T v ,i = (cid:2) √ a i , . . . , √ a i | {z } a i times , − t i √ b i ( t i + 1) , . . . , − t i √ b i ( t i + 1) | {z } b i times , t i √ c i , . . . , t i √ c i | {z } c i times (cid:3) T . Clearly k v ,i k = k v ,i k = k v ,i k = t i + 1 + t i ( t i + 1) and the three vectors are pairwise orthogonal for each i .Now, form three overall vectors by concatenation, so v = (cid:20) v , v , (cid:21) , v = (cid:20) v , v , (cid:21) , v = (cid:20) v , v , (cid:21) . These three vectors all have the same norm and are pairwise orthogonal. Let A = I − v v T + v v T + v v T ) .Let x, y be two vertices in G , then the ( x, y ) -entry of A is given by [ A ] x,y = a i (cid:16) t i + 1 + t i ( t i +1) (cid:17) if x ∈ V ( K a i ) , and y ∈ V ( K a i ); b i (cid:16) t i + 1 + t i ( t i +1) (cid:17) if x ∈ V ( K b i ) , and y ∈ V ( K b i ); c i (cid:16) t i + 1 + t i ( t i +1) (cid:17) if x ∈ V ( K c i ) , and y ∈ V ( K c i ) . If x and y are both in K a i ∪ K b i ∪ K c i , but not both in the same clique (inducedcomplete graph), then [ A ] x,y = 0 . Assume that x ∈ K a , then [ A ] x,y = √ a √ a (cid:16) t t + t ( t +1) t ( t +1) + 1 (cid:17) if y ∈ V ( K a ); √ a √ b (cid:16) t + − t t ( t +1) + − t t +1 (cid:17) if y ∈ V ( K b ); √ a √ c (cid:16) − t t t +1 + − t ( t +1) + t (cid:17) if y ∈ V ( K c ) . If t and t are distinct and positive, these are all nonzero. Similarly we can showthat for any x ∈ K a ∪ K b ∪ K c and any y ∈ K a ∪ K b ∪ K c that [ A ] x,y is notequal to zero. Therefore, A ∈ S ( G ) . Theorem 4.2.
If there are ℓ matrices M i of order k with the following properties(1) the rows of M i are orthogonal, and(2) for i = j all the entries of M Ti M j are nonzero,then the graph G = ℓ _ j =1 ( ∪ ki =1 K a i,j ) has multiplicity bipartition [ n − k, k ] provided all a i,j ≥ .Proof. For h = 1 , , . . . , k , construct vectors v h as follows: v h = [ v h, , v h, , . . . , v h,ℓ ] T ,where each vector v h,j represents the vertices in a ∪ ki =1 K a i,j . The vertices in K a i,j all receive the same value, namely √ a i,j [ M j ] h,i . Following the approach inLemma 4.1, set V = [ v , . . . , v k ], then M = V T V ∈ S ( G ) and the spectrum of M is { ( n − k ) , k v i k ( k ) } . (cid:3) In the following, we give examples of matrices that satisfy the conditions ofTheorem 4.2 with order less than or equal to five, where t >
1. Such a matrix oforder 2 is M t = (cid:20) t − t (cid:21) . Similarly, for orders 3, 4, and 5 we have, M t = t − tt + 1 − tt + 1 t − tt + 1 t ,M t = t − t t − t t t − − t t ,M s = p − s r p − s rr p − ss r p − − s r p , CHIEVABLE MULTIPLICITY PARTITIONS IN THE IEVP OF A GRAPH 13 with p = − s + s − s +2 s and r = ss +1 .Note that in the construction in Lemma 4.1, the numerator in the entries in thevectors come from the entries of the 3 × M t . This method can easily begeneralized.The following Lemma provides graphs with M B ( G ) = 3 that are not listed inCorollary 3.8. Lemma 4.3.
Let H = K α,α be a complete bipartite graph with α ≤ a . Suppose G = (( K a \ H ) ∪ K b ) ∨ ( K a ∪ K b ) ∨ · · · ∨ ( K a ℓ ∪ K b ℓ ) , with a i , b i > and ℓ ≥ . Then q ( G ) = 2 and M B ( G ) = 3 .Proof. Let v = [ v , , v , , . . . , v ,l ] T , v = [ v , , v , , . . . , v ,l ] T , and v = [ v , , v , , . . . , v ,l ] T ,where v ,i = h , . . . , , | {z } a i times − r a i b i w i , . . . , − r a i b i w i | {z } b i times i T v ,i = h w i , . . . , w i , | {z } a i times r a i b i , . . . , r a i b i | {z } b i times i T v , = (cid:2) β, . . . , β, | {z } α times − β, . . . , − β, | {z } α times , . . . , | {z } a − α times (cid:3) T v , = (cid:2) a, − a, , . . . , (cid:3) T , where β = p w and all vectors v ,i , i ≥ i >
2, if wechoose the value of w i large enough so that P ℓi =2 a i (1 + w i ) − α (1 + w ) >
0, thensetting a so that a = P ℓi =1 a i (1 + w i ) − α (1 + w )2results in a vector v that has the same norm as vectors v and v . The vectors v T , v T , and v T form orthogonal rows of a 3 × | V ( H ) | matrix U , where the or-thogonality of the columns of U represents the edges and non-edges of H . Thus, U T U ∈ S ( G ), which completes the proof. (cid:3) This method can be extended to the graphs covered by Theorem 4.2.Using a similar approach, we can also remove edges across the join operation.Define an operation ( K a ∪ K b ) ˙ ∨ ( K a ∪ K b ) that is the join of K a ∪ K b and K a ∪ K b with two disjoint edges removed between K a and K a . Lemma 4.4.
Suppose G = ( K a ∪ K b ) ˙ ∨ ( K a ∪ K b ) ∨· · ·∨ ( K a ℓ ∪ K b ℓ ) , with a i ≥ for i ≥ , b ≥ , and ℓ ≥ . Then q ( G ) = 2 and M B ( G ) = 3 .Proof. Use the same vectors v ,i and v ,i as in the proof of Lemma 4.3. Set v , = (cid:2) β, − β, , . . . , , , . . . , (cid:3) T , where there are a − b zeros, and v , = h (1+ w w ) β , − (1+ w w ) β , , . . . , , (1+ w w ) β , − (1+ w w ) β , , . . . , , i T , where there are a − b − v ,i = for i = 3 , . . . , ℓ . Let v be the vector formed by concatenating v ,i for i = 1 , . . . , ℓ .The norm of v is 2 β + 4 (1+ w w ) β . This is a continuous function in β and ittakes values in the interval ((4 + 2 √ √ w w , ∞ ). The norm of v is at least4 + 2 w + 2 w . Since it is possible to choose w and w so that the norm of v islarger than (4 + 2 √ √ w w , it is also possible to choose β so that the normof v equals k v k . (cid:3) 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 samelower bound. Let P n be the graph on n vertices labeled by 1 , , . . . , n, ′ , ′ , . . . , n ′ .Vertices i and i ′ are adjacent for all i ∈ { , . . . , n } . If i ∈ , . . . , n −
1, then i and i ′ are adjacent to vertices i − , ( i − ′ , i + 1 , ( i + 1) ′ . Vertices 1 and 1 ′ are adjacentto vertices 2 and 2 ′ . Vertices n and n ′ are adjacent to n − n − ′ . Note that P n is the strong product of P n and P . Lemma 4.5.
For any n , q ( P n ) = 2 and M B ( P n ) = n − .Proof. Order the vertices of P n by (1 , ′ , , ′ , . . . , n, n ′ ). For i ∈ { , . . . , n − } let u i be the vector with the (2 i + 1) and (2 i + 2)-entries equal to 1, the 2 i + 3 entry equalto 2 and the 2 i + 4 entry equal to − q ( P n ) = 2 and M B ( P n ) ≤ n − P n has an induced path of length n − (cid:3) 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 n by cloning vertices. The next result follows from Proposition 3.7. Corollary 4.6. If G is a path of cliques of size at least 4 with k the longest inducedpath in G , then q ( G ) = 2 and M B ( G ) = k − . 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 eigenval-ues, 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 ) = M B ( P ) = 3. Question 5.1.
What is the multiplicity bipartition for the complement of a pathon at least vertices? The graphs in
M P ([ n − , T with M B ( T ) = 2.One of the types of trees considered in [11] are denoted by S rm,n (these are called type-one trees ). The graph S rm,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, thesetrees are formed by taking K ,k ∪ K ,ℓ and added one additional edge to make thegraph connected. If the edge is added between two leaves in K ,k ∪ K ,ℓ , then the CHIEVABLE MULTIPLICITY PARTITIONS IN THE IEVP OF A GRAPH 15 resulting graph is S k − ,ℓ − ; if the edge is added between a leaf and a non-leaf thenthe resulting graph is either S k,ℓ − or S k − ,ℓ ; finally, if the edge is added betweentwo non-leaves the resulting graph is S k,ℓ .Note that if T = S km,n with k = 2 , ,
4, then T is formed by taking ( K ∪ K m ′ ) ∨ ( K ∪ K n ′ ) and removing a single edge across the join. In Lemma 4.4, it is shownthat in some cases two edges can be removed across the join. We conjecture thatit is also possible to remove a single edge across the join in many cases and achievethe multiplicity bipartition [ n − , Conjecture 5.1.
Let T = S km,n with k = 2 , , and m, n > . Then q ( T ) = 2 and M B ( T ) = 3 . Corollary 3.8 gives many graphs in
M P ([ n − k, k ]), but it is not a characterization.Is it possible to determining the minimal (in terms of the vertex cloning) graphs in M P ([ n − k, k ]), and then develop a characterization of the graphs in M P ([ n − k, k ])?Lemmas 4.2 and 4.3 of [16] show that graphs K , ,..., and K , ,..., , can achieveany multiplicity bipartition [ n − k, k ] for k = 2 , . . . , ⌊ n ⌋ . Since all graphs G with M B ( G ) = 2 can be obtained from K , ,..., and K , ,..., , by cloning vertices, andsince cloning can preserve the multiplicity bipartition, this implies that if a graphachieves the multiplicity bipartition [ n − , n − k, k ] for k >
2. Therefore, we have the following result.
Lemma 5.1.
For any n , M P ([ n − , ⊂ M P ([ n − k, k ]) for k ∈ { , . . . , ⌊ n/ ⌋} . This raises the open question of whether or not
M P ([ n − k, k ]) ⊂ M P ([ n − k − , k + 1]) for larger values of k . Characterizing graphs with M P ([ n − k, k ]) for k > G with M B ( G ) = 3) would answer this question partiallybut this is likely a harder question. We also suspect that it is possible to generalizeLemma 3.4 to show that a graph G ∨ ( K s ∪ K s ∪ · · · ∪ K s d ) (with the conditionsstated in the Lemma) is in M P ([ n − k, k ]) for all k ≥ d . This leads to our nextconjecture. Conjecture 5.2.
The complete multipartite graph K k,k,...,k can achieve all multi-plicity partitions except [ n − i, i ] for i < k . Theorem 3.9 proves for two graphs G and H with q ( G ) = q ( H ) = 2 and M B ( G ) = M B ( H ), that M B ( G ∨ H ) ≤ M B ( G ). We conjecture that M B ( G ∨ H ) = M B ( G ) and that this holds in a more general setting. Conjecture 5.3.
Let G and H be two graphs such that q ( G ) = q ( H ) = 2 with M B ( G ) = k and M B ( H ) = k , and mr + ( G ) = k and mr + ( H ) = k . Then q ( G ∨ H ) = 2 with M B ( G ∨ H ) = k , where k = max { k , k } . Lemma 2.5 states that for any graph G with q ( G ) = 2, we have mr + ( G ) ≤ M B ( G ), and currently we are not aware of any examples of such graphs in whichthis inequality is strict. However, we expect that these two graph parameters maydiffer in general for graphs that can achieve two distinct eigenvalues. Question 5.2.
Does there exist a graph G with M B ( G ) > mr + ( G ) ? Finally we would like to consider the family of strongly regular graphs. Anystrongly regular graph G has only three distinct eigenvalues, so q ( G ) ≤
3. Our finalquestion is the following.
Question 5.3.
Which strongly regular graphs (other than K n , K n,n , and theircomplements) admit only two distinct eigenvalues? Among the strongly regulargraphs G that admit only two distinct eigenvalues, determine M B ( G ) . A strongly regular graph has parameters ( n, k ; a, c ) where n is the number ofvertices in the graph, and each vertex has degree k . The number of common verticesin the neighbourhoods of two adjacent vertices is a , and non-adjacent vertices is c .If a strongly regular graph G has c = 1, then by Lemma 2.4, q ( G ) = 3. Acknowledgments
The work in this paper was a joint project of the Discrete Mathematics ResearchGroup at the University of Regina, attended by all of the authors. Dr. Adm’sresearch was supported by the German Academic Exchange Service (DAAD) withfunds from the German Federal Ministry of Education and Research (BMBF) andthe People Programme (Marie Curie Actions) of the European Union’s SeventhFrame-work Programme (FP7/2007-2013) under REA grant agreement No.605728(P.R.I.M.E. - Postdoctoral Researchers International Mobility Experience) duringhis delegation to the University of Regina and work at University of Konstanz andrevised during his work at Palestine Polytechnic University. Dr. Fallat’s researchwas supported in part by NSERC Discovery Research Grants, Application Nos.:RGPIN-2014-06036 and RGPIN-2019-03934. Dr. Meagher’s research was supportedin part by an NSERC Discovery Research Grant, Application No.: RGPIN-03952-2018. Dr. Nasserasr’s research was supported in part by an NSERC DiscoveryResearch Grant, Application No.: RGPIN-2019-05275. Dr. Plosker’s research wassupported by NSERC Discovery Grant number 1174582, the Canada Foundationfor Innovation (CFI) grant number 35711, and the Canada Research Chairs (CRC)Program grant number 231250. Dr. Yang’s research was supported in part by anNSERC Discovery Research Grant, Application No.: RGPIN-2018-06800.
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