Some regular signed graphs with only two distinct eigenvalues
aa r X i v : . [ m a t h . C O ] S e p Some regular signed graphs with only two distincteigenvalues
Farzaneh Ramezani
Department of Mathematics,K.N.Toosi University of Technology, Tehran, IranP.O. Box 16315-1618
Abstract . We consider signed graphs, i.e, graphs with positive or negative signs on their edges.We determine the admissible parameters for the { , , . . . , } -regular signed graphs which have onlytwo distinct eigenvalues. For each obtained parameter we provide some examples of signed graphshaving two distinct eigenvalues. It turns out to construction of infinitely many signed graphs of eachmentioned valency with only two distinct eigenvalues. We prove that for any k ≥ k -regular graphs having maximum eigenvalue √ k . Moreover for each m ≥ m , − m ]. These yield infinite family of k -regularRamanujan graphs, for each k . Keywords:
Signed graphs, two distinct eigenvalues.
Mathematics Subject Classification:
We consider only simple graphs, i.e., graphs with out loops and multiple edges. A signature on agraph Γ = ( V Γ , E Γ ) is a function σ : E Γ → { , − } . A graph Γ provided with a signature σ is called a signed graph , and will be denoted by Σ = (Γ , σ ). We call the graph Γ the ground graph of the signedgraph Σ, and denote it by | Σ | . Two signed graphs Σ and Σ are isomorphic , denoted by Σ ∼ = Σ , ifthere is a graph isomorphism from | Σ | to | Σ | , which preserves the signs of edges. For a subset X ofvertices of a signed graph Σ = (Γ , σ ), the signed graph of Σ induced on X is a signed graph where theground is the subgraph of | Σ | induced on X where the corresponding sign function is the restriction of σ to E h X i . By Σ \ X we mean the signed graph induced on V | P | \ X . With a signed k -regular graph ,we mean a signed graph whose ground is k -regular. For a graph Γ, by Γ + (resp. Γ − ) we mean the allpositive (resp. the all negative) signed graph with ground Γ. For a signed graph Σ = (Γ , σ ), by − Σ,we mean the signed graph (Γ , − σ ).The adjacency matrix , A Σ of the signed graph Σ = (Γ , σ ) on the vertex set V = { v , v , . . . , v n } , isan n × n matrix whose entries are A Σ ( i, j ) = (cid:26) σ ( v i v j ) , if v i is adjacent to v j ;0 , otherwise. Email: [email protected]. ordinary adjacency matrix of the graph Γ, is denoted by A Γ . Whose entries are the absolutevalues of the entries of A Σ .The spectrum of a signed graph is the eigenvalues of its adjacency matrix and will be denoted by[ λ m , λ m , . . . , λ m s s ], where m i is the multiplicity of eigenvalue λ i , for i = 1 , , . . . , s . By m ( λ ) = m Σ ( λ ),we denote the multiplicity of λ as an eigenvalue of the signed graph Σ. By O n , J n and I n , we meanthe all zero matrix, all one matrix, and identity matrix of order n , respectively. If n is already known,then we simply use O, J, I . For a matrix X of order m × n , by X i we mean the i th row of X , for i = 1 , , . . . , m .An n × n matrix C , with (0 , ± weighing matrix , if CC t = C t C = αI n , where α is a positive integer, called the weight of the weighing matrix C . A signed graph is called weighing ifits adjacency matrix is a weighing matrix. Weighing matrices are well studied in the literature andseveral constructions and examples are known, see for instance [6, 11, 12].The following is definition of a FSRSG which is introduced in [15] and updated in [16]. A signedregular graph Σ is called FSRSG whenever it satisfies the following conditions:(i) | Σ | is k -regular, with n vertices,(ii) if Σ contains at least one positive edge, then there exist t ∈ Z such that t + xy − t − xy = t , for anyedges xy ,(iii) there exist ρ ∈ Z such that ρ + xy − ρ − xy = ρ , for all non-adjacent vertices x and y , where ρ + xy , ρ − xy are the numbers of positive and negative length two paths joining the vertices. We denote a FSRSGhaving just mentioned parameters with FSRSG( n, k, t, ρ ).We refer to a signed graph with only two distinct eigenvalues, by STE . The following is a generaldetermination of STE’s. The result is independently obtained in [15] and [18].
Theorem 1.1.
A signed graph is a STE if and only if it is a FSRSG with ρ = 0.Recently some problems on the spectrum of signed adjacency matrices have attracted many stud-ies. The authors of [3] has settled few open problems in the spectral theory of signed graphs. Theconstruction problem of STE’s is also mentioned there. There are some results on the problem inliterature, see [9, 14, 15, 18]. It is known that the only graphs with two distinct eigenvalues of theordinary adjacency matrix are the complete graphs, which is extensively far from the known results forthe signed graphs. We consider the same problem for signed graphs. We use the well known method,called the star complement technique , for construction of STE’s. For more details on the notions anddefinitions see [17, 19]. It is well adopted for the family of signed graphs as well as any other symmetricmatrices, see [2, 15]. During the preparation of this paper we encounter interesting results from othercombinatorial areas such as symmetric weighing matrices, Hadamard matrices, Conference matricesand Ramanujan graphs. The adjacency matrix of a FSRSG satisfies a quadratic equation of the form A − tA Σ − kI = ρA, (1)2here A is the complement of | Σ | , and t , k , ρ are as in the definition of FSRSG’s. From the equality(1), for any STE the corresponding ρ parameter leads to be zero, hence for the adjacency matrix of aSTE, the following equality must hold. A − tA Σ − kI = 0 . (2)Therefore the eigenvalues of a STE must be as follows. λ = t + √ t + 4 k , λ = t − √ t + 4 k . From now on, we denote t + 4 k , by b . It is well-known that if a monic integral polynomial admits anon-integral root α , then α is irrational and the algebraic conjugate of α is a root of the polynomialwith the same multiplicity. This leads to the following useful lemma. Lemma 2.1.
Let Σ = (Γ , σ ) be a STE, then one of the followings holds. • t = 0 and A Σ is a weighing matrix of weight k , or • t = 0, then b must be a perfect square. Proof.
The first part follows by putting t = 0, in (2). For the second part as contradiction supposethat b is not a perfect square, so − t + √ b is irrational. Therefore since the characteristic polynomial of Σis a monic integral polynomial, the algebraic conjugate of − t + √ b , that is, t −√ b must be an eigenvalueof Σ with the same multiplicity, say m . By the assumption Σ has the spectrum ( t + √ b ) m , ( t −√ b ) m .On the other hand trace of A Σ is zero, thus we have the following equality, m t + √ b m t − √ b , which implies mt = 0, so t = 0, a contradiction. Hence b must be a perfect square. (cid:3) Note that in a signed k -regular graphr, t may be an integer in the interval [ − k + 1 , k − t, λ , λ ) is called admissible for a STE, if for k = − λ λ , either b = t + 4 k is a perfect square or t is zero. With a simple calculation we observe the following. Theorem 2.2.
The following parameters are all possible admissible ( t, λ , λ )’s of the STE’s withvalencies 5 , , . . . , k = 5 (4 , , − , √ , −√ − , − k = 6 (5 , − , − , √ −√ − , − − , − k = 7 (6 , , − , √ , −√ − , , − k = 8 (7 , , − , , − , √ , −√ − , − , − , , − k = 9 (8 , , − − − , , − k = 10 (9 , , − , , − , √ , −√ − , , − − , , − Table 1.
The admissible parameters for the STE’s with { , , . . . , } -regular groundsThe notion of line of signed graphs is defined in [21]. The line graph of Σ is the signed graphΛ(Σ) = (Λ( G ) , σ Λ ), where Λ( G ) is the ordinary line graph of the underlying graph and σ Λ is a sign3unction such that every cycle in Σ becomes a cycle with the same sign in the line graph, and any threeedges incident with a common vertex become a negative triangle in the line graph. The adjacencymatrix is A Λ(Σ) = 2 I m − H (Σ) t H (Σ). Where H (Σ) is the incidence matrix of a signed graph Σ on n vertices with m edges, which is defined in [20], as follows. It is the n × m matrix H (Σ) = [ η i,j ], where η i,j = (cid:26) , if v i is not incident with e j ; ± , if v i is incident with e j .such that for an edge v i v j , the equality η ik η jk = σ ( v i v j ) holds. The incidence matrix is unique up tonegation of columns. The following theorem from [8, Theorem 3.10] will be useful. Theorem 2.3.
Assume Σ is a signed k -regular graph. Let the eigenvalues of Σ be λ , . . . , λ b ( − Σ) = − k , − k < λ b ( − Σ)+1 , . . . , λ n − b (Σ) < k , and λ n − b (Σ) , . . . λ n = k . The line graph Λ(Σ) has eigenvalues λ b ( − Σ)+1 − k + 2 , . . . , λ n − b (Σ) − k + 2 and eigenvalue 2 with multiplicity m − n + b (Σ).Note that b (Σ) is the number of balanced components of Σ. In the following we will summarize the method which is extensively surveyed in [17, 19].Let A be an n × n matrix, and µ be an eigenvalue of A having multiplicity k . A k -subset X ofthe vertex set is called a star set if the sub-matrix of A , obtained by removing rows and columnscorresponding to X does not have µ as an eigenvalue. In signed graph context, a star set for aneigenvalue µ in a signed graph Σ is a subset X of vertices such that µ is not an eigenvalue of Σ \ X .The signed graph Σ \ X is called a star complement for µ in Σ. The following theorem, known as the reconstruction theorem, will be useful later. Theorem 2.4. [17]
Let X be a set of k vertices in a signed graph Σ and suppose the matrix A Σ isof the following form A Σ = (cid:18) A X BB t C (cid:19) , where A X is the adjacency matrix of the signed graph of Σ induced on X . Then X is a star set for µ in Σ if and only if µ is not an eigenvalue of C and µI − A X = B ( µI − C ) − B t . The star complement method will also be used for the construction of STE’s, we state the followingimmediate consequence of Theorem 2.4 and the fact that any real symmetric matrix has a partitionto star complements , [5], with out proof.
Proposition 2.5.
A signed graph Σ on n vertices, have spectrum [ λ a , λ n − a ] if and only if the vertexset of Σ has a partition X, Y such that the matrix A Σ can be written as the following (cid:18) A X BB t A Y (cid:19) , such that A X ( A Y ) doesn’t have eigenvalue λ ( λ ) and the following equalities hold. λ I − A X = B ( λ I − A Y ) − B t , λ I − A Y = B t ( λ I − A X ) − B. Constructions
We have the following consequence of Theorem 2.3 for the line graph of the all positive completegraph.
Proposition 3.1.
Let Σ be the all positive signed graph on the complete graph K n , that is K + n . ThenΛ( K + n ) is a signed graph with the following spectrum.[(2 − n ) n − , n ) − n +1 ] . Moreover the spectrum of the signed graph − Λ( K + n ) is[( n − n − , ( − n ) − n +1 ] . Proof.
It follows by Theorem 2.3 and the fact that K + n has spectrum [( n − , − n − ]. (cid:3) Note that the admissible ( t, λ , λ ) parameters of STE’s on k -regular graphs, with 5 ≤ k ≤ • Type 1. ( n − , n − , − − n + 2 , , − n + 1), • Type 2. ( k − , k , − − k + 2 , , − k ), • Type 3. (0 , √ k, −√ k ).Where n , k respectively denote the number of vertices, and valency of the signed graph Σ which is anSTE with the given parameters.We recall that in a STE’s we have k = − λ λ and t = λ + λ . Also from the triples ( t, λ , λ )and ( − t, − λ , − λ ), we only need to treat one of them, as if Σ is a STE with parameters ( t, λ , λ ),then − Σ have parameters ( − t, − λ , − λ ). Note also the only signed graphs posing the parameters( n − , n − , −
1) are the all positive complete graph. Thus we only need to consider the two remainedcases. ( k2 − , k2 , − ) We will present two family of examples of STE’s posing the parameters ( k − , k , − k − , k , − A Σ . m ( k n k , m ( −
2) = nk k . Lemma 3.2.
The followings hold. • The signed graph Λ( K ) is 6-regular, having spectrum [3 , − ]. • The signed graph Λ( K ) is 8-regular, having spectrum [4 , − ].5 The signed graph Λ( K ) is 10-regular, having spectrum [5 , − ]. Proof.
The assertion follows by Proposition 3.1. (cid:3)
The following is a signed graph with spectrum [3 , − ]. Figure 1.
The signed graph − Λ( K ) -regular signed graphs with spectrum m , − m Using Proposition 3.1, we find out the signed graph Λ( K ) has spectrum 4 , − . At this part usingProposition 2.5, we construct STE’s with spectrum 4 m , − m . The following lemma has crucial rolein that regard. Lemma 3.3.
Let W , W be two weighing matrices of order m and weight 4. The following matrixhas spectrum [4 m , − m ]. A ( W , W ) = O m W W W t O m W t W W t W t W O m . Proof.
We prove that the subset { , , . . . , m } is a star set for 4, and the subset { m +1 , m +2 , . . . , m } is a star set for −
2. Let C , C be the submatrices corresponding to the mentioned vertex setsrespectively. That is, C = O m , and C = (cid:18) O m W t W W t W O m (cid:19) . We set B = ( W | W ), now by Theorem 2.4, it suffices to prove the following equalities.4 I m = B (4 I m − C ) − B t , − I m − C = B t ( − I m − C ) − B. By a simple calculation we obtain the following equality,(4 I m − C ) − = I m W t W W t W
24 13 I m ! . W W t = W W t = 4 I m . B (4 I m − C ) − B t = ( W | W ) I m W t W W t W
24 13 I m ! (cid:18) W t W t (cid:19) = ( W | W ) (cid:18) W t W t (cid:19) = 4 I m . Thus the first equality holds, now we prove the second equality. Note that ( − I n − C ) − = − I n ,therefore we need to prove the following so that the assertion follows. B t ( − I m − C ) − B = − (cid:18) W t W t (cid:19) ( W | W ) = − (cid:18) W t W W t W W t W W t W (cid:19) = − I m − C . Now the assertion follows by Theorem 2.4. (cid:3)
Corollary 3.4.
If the weighing matrices W and W are chosen so that the matrix W t W has 0 , ± W t W has 0 , ± A ( W , W ) is correspondingto a 8-regular graph on 3 m vertices, having spectrum [4 m , − m ].If two weighing matrices W and W of weight 4 are so that the matrix W t W has 0 , ± semi-orthogonal weighing matrices. In the following we provide examples of semi-orthogonal weighing matrices of any even order m ≥
6. We should mention that a family of weighingmatrices of even order is presented in [6], but our method provides a different matrix. We make useof the following two pattern matrices X and Y of order m . For i, j = 1 , , . . . , m , the ( i, j )’th entriesof X and Y follows. X ( i, j ) = (cid:26) , j − i ≡ , m ;0 , other wise. Y ( i, j ) = (cid:26) , j − i ≡ , m − m ;0 , other wise.Note that the matrices X, Y have two entry 1 at each row and each column. Moreover
X, Y do notshare a common entry 1. We define the matrix W , similarly W as the following. • The first m × m block of W , say F = F X , is filled as follows. In fact the matrix X is expandedto the matrix F of size m × m . For i, j = 1 , , . . . , m , if X ( i, j ) = 0, then replace the ( i, j )’thentry of X with [0 , j = 1 , , . . . , m/ j ’th column the one entries occur in thecoordinates ( i , j ), and ( i , j ), where i < i , then replace the ( i , j )’th entry with [1 , i , j )’th entries with [1 , − • The next m × m block of W , say R = R X , is filled as follows. The entries of R X in thecorresponding zero coordinates of F X are zero. For i = 1 , , . . . , m , if non-zero entries of F X occur in the coordinates ( i, j ), ( i, j ), ( i, j ), ( i, j ), where j < j < j < j , then we set R X ( i, j ) = F X ( i, j ) , R X ( i, j ) = F X ( i, j ) ,R X ( i, j ) = − F X ( i, j ) , R X ( i, j ) = − F X ( i, j ) . Now set W to be the m × m matrix, (cid:18) FR (cid:19) . The matrix W is constructed similarly by considering Y instead of X . See the following matrices for illustration. X = , Y = . X = − − − − − − − R X = − − − − − − − − − − − − − − − − − − − F Y = − − − − − − − , R Y = − − − − − − − − − − − − − − − .W = − − − − − − − − − − − − − − − − − − − − − − − − − − , W = − − − − − − − − − − − − − − − − − − − − − − . We now prove some properties of the obtained matrices.
Lemma 3.5.
The matrix W , ( W ) obtained above, is a weighing matrix of weight four. Proof.
We first prove that W is orthogonal. Equivalently we prove that any two distinct rows of W are orthogonal. We first treat rows of R and F separately. Let F i , F j be the i th and j th rows of F respectively ( i < j ). Then one of the followings hold. • The 1 entries of X i and X j occur on different column. In this case, F i , F j wouldn’t have commoncolumns of non-zero entries, which means they are orthogonal. • X i ( k ) = X j ( k ) = 1 for only one column number, say 1 ≤ k ≤ m . In this case, by the definitionwe have F ( i, k −
1) = F ( i, k ) = 1, while F ( j, k −
1) = 1, F ( j, k ) = −
1. But for the otherentries, l = 2 k − , k either of F i ( l ) or F j ( l ) are zero, hence we have F i .F j = F ( i, k − F ( j, k −
1) + F ( i, k ) F ( j, k ) = 0 . Thus F i and F j are orthogonal.The orthogonality of row vectors of R can be seen similarly. Now we prove that for any i, j =1 , . . . , m/
2, the vectors F i and R j , are orthogonal. The following two cases may occur. • i = j ; In this case, by the definition, we have R i = [0 , . . . , , F ( i, j − , F ( i, j ) , , . . . , , − F ( i, j − , − F ( i, j ) , , . . . , . Where j , j are the numbers of non-zero columns of X i . Hence F i .R i = F ( i, j − + F ( i, j ) − F ( i, j − − F ( i, j ) = 1 + 1 − − . i = j ; In this case either X i , X j , and then R i , F j have no common column having non-zerocoordinates or X i , X j share only one common column of non-zero elements. In the formercase R i , F j have zero inner product, indeed. In the case that X i , X j share only one commoncolumn of non-zero elements, let k be the index of the common column of non-zero entry. Bythe definition we will have X i ( k ) = X j ( k ) = 1, so at first we have F j (2 k −
1) = F i (2 k − ,F j (2 k ) = − F i (2 k − , now depending on the occurrence of k , one of the followings may occur: R j (2 k −
1) = F i (2 k − R j (2 k ) = − F i (2 k − , or R j (2 k −
1) = − F i (2 k − R j (2 k ) = F i (2 k − . In both cases the orthogonality of F i and R j follows.Note that the matrix X has two 1 entries at each row and column, and in W , any entry 1 of X isreplaced with [1 ,
1] or [1 , − W has four entries equal to ±
1, then W has weight four, as desired. (cid:3) Now we prove the semi-orthogonality of W , and W . Proposition 3.6.
The matrices W and W are semi-orthogonal. Proof.
It suffices to prove for any appropriate i, j the inner product of the i ’th row of W with the j ’th row of W is equal to 0 , ±
2. Note that the matrices X and Y is chosen so that they never haveequal rows. This implies that X i and Y j have at most one common 1-entry. Hence the i ’th row of W and the j ’th row of W has either no common columns having non-zero coordinate or exactly twocommon columns having non-zero coordinates. In both cases depending on the corresponding entries,the inner product may be 0 , or ± (cid:3) The above statements result in the following theorem.
Theorem 3.7.
Let W , and W be two semi-orthogonal weighing matrices of order m . Then thesigned graph with adjacency matrix A ( W , W ) has spectrum [4 m , − m ]. Remark 3.8.
Using Lemma 3.3 and the mentioned construction of matrices W and W we mayconstruct signed 8-regular graphs on 3 m nods, ( m even) having spectrum [4 m , − m ], for m ≥ Figure 2.
A signed graph with spectrum [4 , − ] , .3 STE’s with parameters ( , √ k , −√ k ) Signed graphs with spectrum √ k n , −√ k n are in correspondence with symmetric weighing matrices.Symmetric weighing matrices of small weights have been studied extensively and some constructionsare announced, for more details see [6, 11, 12]. Recently Huang [10] has proved the SensitivityConjecture from theoretical computer science, by presenting a special signing of hypercubes. Theobtained signed hypercubes admit spectrum ±√ k . His method is improved in [3] to a construction ofsymmetric weighing matrices. The following is their construction. Theorem 3.9. [3]
If Σ is a signed k -regular graph of order n with spectrum [ √ k n , −√ k n ], then thefollowing is the adjacency matrix of a signed ( k + 1)-regular graph with spectrum √ k + 1 n , −√ k + 1 n . Ac (Σ) = (cid:18) A Σ I n I n − A Σ (cid:19) . In [9] the authors have provided a family containing infinitely many signed 4-regular graphs havingjust two distinct eigenvalues 2 , −
2. More precisely for each m ≥
3, they construct a signed 4-regulargraph with spectrum [2 m , − m ]. We call their examples, the STE ’s. This yields the following result.
Theorem 3.10.
For any k ≥ k -regular graphs withdistinct eigenvalues ±√ k . Proof.
Note the graph K has spectrum ±
1. Now repeatedly applying Theorem 3.9, for any positive k we find a signed k -regular graph with distinct eigenvalues ±√ k . It is easily seen that the grounds of thesigned graphs which are just constructed, are connected. As already discussed for any m ≥
3, a signed4-regular graph with spectrum [2 m , − m ] exists. Let Σ be a signed graph with spectrum [2 m , − m ].Then Ac (Σ) would give a signed graph with spectrum [ √ m , −√ m ]. By repeating the procedure weget a signed graph with spectrum [ √ m , −√ m ], [ √ m , −√ m ], . . . , [ √ k k − m , −√ k k − m ]. Since m belongs to an infinite set of integer numbers hence the assertion follows. (cid:3) The above theorem provides an infinite family of connected signed k -regular graphs having eigenvalues ±√ k . The ground of the mentioned graphs are bipartite. By the similar construction, starting with asymmetric conference matrix of order 6, we provide other examples of signed k -regular graphs havingeigenvalues ±√ k . In this case the ground is not bipartite. The following is the corresponding signed5-regular graph. Figure 3.
The Pentagon, its spectrum is [ √ , −√ ]Therefore we have the following result. 10 heorem 3.11. For any k ≥ k -regular graph on 3 . k − vertices with distincteigenvalues ±√ k , in which the ground is not bipartite. Proof.
By applying the method stated in Theorem 3.9, on the signed graph of Figure 3, the assertionfollows. (cid:3)
Concluding Remarks
The vertex number of STE’s which are provided at this paper are summarized at the following table.We proposed two infinite families of connected STE’s.( t, λ , λ ) number of vertices of instances(1 , , −
2) 10(2 , , −
2) 15, 6 q , q ≥ , , −
2) 21(0 , √ k, −√ k ) 3 . k − , 2 k − m , m = 3 , , . . . In [4] the authors have conjectured that every connected k -regular graph Γ has a sign function σ such that the maximum spectrum of the corresponding signed graph is at most 2 √ k −
1. Then theypropose to construct new families of Ramanujan graphs by using lift of graphs. Our instance of signed8-regular graphs have maximum eigenvalue 4, which is smaller than 2 √
7. Hence an infinite familyof 8-regular Ramanujan graphs can be constructed. Also by Theorem 3.10 for any k ≥ √ k is constructed, which again yields an infinitefamily of k -regular Ramanujan graphs.Moreover it is well-known that the Kronecker product of two weighing matrices of orders m , n ,and weights α, β, respectively, yields an orthogonal matrix of order mn , with weight αβ . Hence bythe Kronecker product of the known weighing matrices with the above obtained matrices some newinstances will be constructed. References [1]
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