The largest H -eigenvalue and spectral radius of Laplacian tensor of non-odd-bipartite generalized power hypergraphs
aa r X i v : . [ m a t h . C O ] O c t The largest H -eigenvalue and spectral radius of Laplacian tensor ofnon-odd-bipartite generalized power hypergraphs ∗ Yi-Zheng Fan , † , Murad-ul-Islam Khan , Ying-Ying Tan , . School of Mathematical Sciences, Anhui University, Hefei 230601, P. R. China . Department of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, P. R. China Abstract : Let G be a simple graph or hypergraph, and let A ( G ) , L ( G ) , Q ( G ) be the adjacency, Laplacianand signless Laplacian tensors of G respectively. The largest H -eigenvalues (respectively, the spectralradii) of L ( G ) , Q ( G ) are denoted respectively by λ L max ( G ) , λ Q max ( G ) (respectively, ρ L ( G ) , ρ Q ( G )). It isknown that for a connected non-bipartite simple graph G , λ L max ( G ) = ρ L ( G ) < ρ Q ( G ). But this doesnot hold for non-odd-bipartite hypergraphs. We will investigate this problem by considering a class ofgeneralized power hypergraphs G k, k , which are constructed from simple connected graphs G by blowingup each vertex of G into a k -set and preserving the adjacency of vertices.Suppose that G is non-bipartite, or equivalently G k, k is non-odd-bipartite. We get the followingspectral properties: (1) ρ L ( G k, k ) = ρ Q ( G k, k ) if and only if k is a multiple of 4; in this case λ L max ( G k, k ) <ρ L ( G k, k ). (2) If k ≡
2( mod 4), then for sufficiently large k , λ L max ( G k, k ) < ρ L ( G k, k ). Motivated by thestudy of hypergraphs G k, k , for a connected non-odd-bipartite hypergraph G , we give a characterizationof L ( G ) and Q ( G ) having the same spectra or the spectrum of A ( G ) being symmetric with respect tothe origin, that is, L ( G ) and Q ( G ), or A ( G ) and −A ( G ) are similar via a complex (necessarily non-real)diagonal matrix with modular-1 diagonal entries. So we give an answer to a question raised by Shao etal., that is, for a non-odd-bipartite hypergraph G , that L ( G ) and Q ( G ) have the same spectra can notimply they have the same H -spectra. Keywords:
Non-odd-bipartite hypergraph; Laplacian tensor; largest H -eigenvalue; spectral radius; spec-trum; H -spectrum A hypergraph G = ( V ( G ) , E ( G )) consists of a set of vertices say V ( G ) = { v , v , . . . , v n } and aset of edges say E ( G ) = { e , e , . . . , e m } where e j ⊆ V ( G ). If | e j | = k for each j = 1 , , . . . , m ,then G is called a k -uniform hypergraph. In particular, the 2-uniform hypergraphs are exactlythe classical simple graphs. For a k -uniform hypergraph G , if we add to G some edges withcardinality less than k , the resulting hypergraph denoted by G o is one with loops; and those ∗ Supported by National Natural Science Foundation of China (11371028), Natural Science Research Foundationof Anhui Provincial Department of Education (KJ2015A322), Scientific Research Fund for Fostering DistinguishedYoung Scholars of Anhui University (KJJQ1001), Project of Academic Innovation Team of Anhui University(KJTD001B), Open Project of School of Mathematical Sciences of Anhui University (ADSY201501). † Corresponding author. E-mail addresses: [email protected](Y.-Z. Fan), [email protected] (M.Khan), [email protected] (Y.-Y. Tan) loops of G o . The degree d v ( G ) or simply d v of a vertex v ∈ V ( G ) isdefined as d v ( G ) = |{ e j : v ∈ e j ∈ E ( G ) }| . So, a loop contributes 1 to the degree of the vertexto which it is attached.An even uniform hypergraph G is called odd-bipartite if V ( G ) has a bipartition V ( G ) = V ∪ V such that each edge has an odd number of vertices in both V and V . Hu, Qi and Shao [4]introduced the cored hypergraphs and the power hypergraphs , where the cored hypergraph is onesuch that each edge contains at least one vertex of degree 1, and the k -th power of a simplegraph G , denoted by G k , is obtained from G by replacing each edge (a 2-set) with a k -set byadding ( k −
2) new vertices. These two kinds of hypergraphs are both odd-bipartite.Recently spectral hypergraph theory has emerged as an important field in algebraic graphtheory. Let G be a k -uniform hypergraph on n vertices v , v , . . . , v n . The adjacency tensor A ( G ) of G is defined as A ( G ) = ( a i i ...i k ), a k th order n -dimensional symmetric tensor, where a i i ...i k = k − if { v i , v i , . . . , v i k } ∈ E ( G ) and a i i ...i k = 0 otherwise. Let D ( G ) be a k thorder n -dimensional diagonal tensor, where d i...i = d v i for all i ∈ [ n ] := { , , . . . , n } . Then L ( G ) = D ( G ) − A ( G ) is called the Laplacian tensor of G , and Q ( G ) = D ( G ) + A ( G ) is calledthe signless Laplacian tensor of G .For a hypergraph G o with loops, the adjacency tensor of G o is defined as the same as thatof G , i.e. A ( G o ) = A ( G ). The Laplacian tensor and the signless Laplacian tensor are defined by L ( G o ) = D ( G o ) − A ( G ) and Q ( G o ) = D ( G o ) + A ( G ), respectively. So, even if G o is not uniform,the adjacency, Laplacian and signless Laplacian tensor of G o are all k th order n -dimensionaltensors.In general, a real tensor (also called hypermatrix ) T = ( t i ...i k ) of order k and dimension n refers to a multidimensional array with entries t i ...i k such that t i ...i k ∈ R for all i j ∈ [ n ] and j ∈ [ k ]. The tensor T is called symmetric if its entries are invariant under any permutationof their indices. A subtensor of T is a multidimensional array with entries t i ...i k such that i j ∈ S j ⊆ [ n ] for some S j ’s and j ∈ [ k ], denoted by T [ S | S | · · · | S k ]. If S = S = · · · = S k =: S ,then we simply write T [ S | S | · · · | S k ] as T [ S ], which is called the principal subtensor of T . If k = 2, then T [ S ] is exactly the principal submatrix of T ; and if k = 1, then T [ S ] is the subvectorof T .Given a vector x ∈ R n , T x k is a real number, and T x k − is an n -dimensional vector, whichare defined as follows: T x k = X i ,i ,...,i k ∈ [ n ] t i i ...i k x i x i · · · x i k , ( T x k − ) i = X i ,...,i k ∈ [ n ] t ii ...i k x i · · · x i k for i ∈ [ n ] . Let I be the identity tensor of order k and dimension n , that is, i i i ...i k = 1 if and only if i = i = · · · = i k ∈ [ n ] and i i i ...i k = 0 otherwise. Definition 1.1 [9]
Let T be a k th order n -dimensional real tensor. For some λ ∈ C , if thepolynomial system ( λ I − T ) x k − = 0 , or equivalently T x k − = λx [ k − , has a solution x ∈ C n \{ } , then λ is called an eigenvalue of T and x is an eigenvector of T associated with λ ,where x [ k − := ( x k − , x k − , . . . , x k − n ) . x is a real eigenvector of T , surely the corresponding eigenvalue λ is real. In this case, x iscalled an H -eigenvector and λ is called an H -eigenvalue . The spectral radius of T is defined as ρ ( T ) = max {| λ | : λ is an eigenvalue of T } . Denote respectively the largest H -eigenvalues (re-spectively, the spectral radii) of A ( G ) , L ( G ) , Q ( G ) by λ A max ( G ) , λ L max ( G ) , λ Q max ( G ) (respectively, ρ A ( G ) , ρ L ( G ) , ρ Q ( G )). By Perron-Frobenius theorem of nonnegative tensors (see [1, 2, 12]), λ A max ( G ) = ρ A ( G ), λ Q max ( G ) = ρ Q ( G ). But this does not hold for the Laplacian tensors ingeneral.Qi [8] showed that ρ L ( G ) ≤ ρ Q ( G ), and posed a question of identifying the conditions underwhich the equality holds. So λ L max ( G ) ≤ ρ L ( G ) ≤ ρ Q ( G ) = λ Q max ( G ) . (1 . Theorem 1.2 [5]
Let G be a connected k -uniform hypergraph. Then λ L max ( G ) = λ Q max ( G ) if andonly if k is even and G is odd-bipartite. Denote by
Spec ( A ( G )), Spec ( L ( G )) and Spec ( Q ( G )) the spectra of A ( G ), L ( G ) and Q ( G )respectively, and by Hspec ( L ( G )), Hspec ( L ( G )) and Hspec ( Q ( G )) the sets of distinct H -eigenvalues of A ( G ), L ( G ) and Q ( G ) respectively. Shao et al. [10] gave some characterizationson these different types of spectra. Theorem 1.3 [10]
Let G be a connected k -uniform hypergraph. Then ρ L ( G ) = ρ Q ( G ) if andonly if Spec ( L ( G )) = Spec ( Q ( G )) . Theorem 1.4 [10]
Let G be a connected k -uniform hypergraph. Then the following conditionsare equivalent. (1) k is even and G is odd-bipartite. (2) Spec ( L ( G )) = Spec ( Q ( G )) and Hspec ( L ( G )) = Hspec ( Q ( G )) . (3) Hspec ( L ( G )) = Hspec ( Q ( G )) . (4) Spec ( A ( G )) = − Spec ( A ( G )) and Hspec ( A ( G )) = − Hspec ( A ( G )) , i.e. both Spec ( A ( G )) and Hspec ( A ( G )) are symmetric with respect to the origin. (5) Hspec ( A ( G )) = − Hspec ( A ( G )) . Suppose that k is even and G is connected. If G is odd-bipartite, then λ L max ( G ) = λ Q max ( G ),which implies that λ L max ( G ) = ρ L ( G ). Suppose that G is non-odd-bipartite. Then λ L max ( G ) <λ Q max ( G ). From the inequalities in (1.1), we want to know under which condition ρ L ( G ) = ρ Q ( G )or λ L max ( G ) = ρ L ( G ). If ρ L ( G ) = ρ Q ( G ), then λ L max ( G ) < λ Q max ( G ) = ρ L ( G ). If λ L max ( G ) <ρ L ( G ), it may occur ρ L ( G ) = ρ Q ( G ), which implies that the spectral radius is attained forsome eigenvalue whose eigenvectors can not be scaled into H -eigenvectors, which are called N -eigenvectors of L ( G ).In this paper we will discuss the above problem for the non-odd-bipartite generalized powerhypergraphs G k, k constructed from non-bipartite simple graphs G , which will be introduced3ater. In Section 2, we first give a method to compute the spectrum and the H -spectrum of L ( G k, k ) by computing the spectrum of certain matrices associated with the modified inducedsubgraph of the simple graph G . In particular, we given two explicit formulas for λ L max ( G k, k )and ρ L ( G k, k ) respectively. By using those results, in Section 3 we give a characterization for theequality ρ L ( G k, k ) = ρ Q ( G k, k ), i.e. k is a multiple of 4;in this case λ L max ( G k, k ) < ρ L ( G k, k ). If k ≡
2( mod 4), then for sufficiently large k , λ L max ( G k, k ) < ρ L ( G k, k ). So, given a connected non-bipartite simple graph G , except a small number of k , we always have λ L max ( G k, k ) < ρ L ( G k, k ).Motivated by the study of hypergraphs G k, k , for a connected non-odd-bipartite hypergraph G ,we show that Spec ( L ( G )) = Spec ( Q ( G )) (respectively, Spec ( A ( G )) = − Spec ( A ( G ))) if andonly if L ( G ) and Q ( G ) (respectively, A ( G ) and −A ( G )) are similar via a complex (necessarilynon-real) diagonal matrix with modular-1 diagonal entries.In the paper [10], Shao et al. remarked that “if G is connected, then Hspec ( L ( G )) = Hspec ( Q ( G )) = ⇒ Spec ( L ( G )) = Spec ( Q ( G )) . (1 . Spec ( L ( G )) = Spec ( Q ( G )) is equivalent to that L ( G ) is similar to Q ( G ) via a complex diagonal matrix withmodular-1 diagonal entries. However, by the results in [10], that Hspec ( L ( G )) = Hspec ( Q ( G ))is equivalent to that L ( G ) is similar to Q ( G ) via a diagonal matrix with ± Spec ( L ( G )) = Spec ( Q ( G )) ⇒ Hspec ( L ( G )) = Hspec ( Q ( G )). But this happens only when G is odd-bipartite by Theorem1.4. Similar discussion can apply to Spec ( A ( G )) and Hspec ( A ( G )) for the spectral symmetricproperty. So, for a connected non-odd-bipartite hypergraph G , the reverse implication in (1.2)is not true.Finally we introduce the generalized power hypergraphs defined in [6]. Definition 1.5 [6]
Let G = ( V, E ) be a simple graph. For any k ≥ and ≤ s ≤ k/ , thegeneralized power of G , denoted by G k,s , is defined as the k -uniform hypergraph with the vertexset { v : v ∈ V } ∪ { e : e ∈ E } , and the edge set { u ∪ v ∪ e : e = { u, v } ∈ E } , where v is an s -setcontaining v and e is a ( k − s ) -set corresponding to e . Note that if 1 ≤ s < k/
2, then G k,s is a cored hypergraphs and hence is odd-bipartite. Inparticular, G k, is exactly the k -th power of G . If s = k/ k being even), then G k,s is obtainedfrom G by only blowing up its vertices, G , = G . In this case, { u, v } is an edge of G if and onlyif u ∪ v is an edge of G k, k , where we use the bold v to denote the blowing-up of the vertex v in G . For simplicity, we write uv rather than u ∪ v , and call u a half edge of G k, k .If G = G o , a simple graph with loops (i.e. edges containing only one vertex), then ( G o ) k,s will have loops containing k − s vertices. In particular, ( G o ) k, k will have loops containing k vertices. That is, if { u } is a loop of G o , then the half edge u is a loop of ( G o ) k, k ; see Fig. 1.1. Lemma 1.6 [6]
Let G be a simple graph. The hypergraph G k, k is non-odd-bipartite if and onlyif G is non-bipartite. emma 1.7 [6] Let G be a connected simple graph. Then ρ A ( G ) = ρ A ( G k, k ) and ρ Q ( G ) = ρ Q ( G k, k ) . In the following for a simple graph G and its generalized power hypergraph G k, k , each vertex u of G is corresponding to the half edge u of G k, k , and u is always assumed to be contained in u . Clearly, each vertex in u can be considered as u . In addition, all k -uniform hypergraphs areeven uniform, i.e. k is even. A simple graph G The power hypgergraph G The generalized power hypergraph G , The generalized power hypergraph G , u wu w Modified induced subgraph G o [ u, w ] G o [ u, w ] k, k Fig. 1.1 (c.f. [6]) Constructing power hypergraphs from a simple graph, where a closed green curverepresents an edge and a closed red curve represents a loop
In this section we will give a method to compute the spectra and the H -spectra of generalizedpower hypergraphs G k, k . The eigenvector equation L ( G ) x k − = λx [ k − could be interpreted as[ d v − λ ] x k − v = X { v,v ,v ,...,v k }∈ E ( G ) x v x v · · · x v k , for each v ∈ V ( G ) . (2 . G be a simple graph on n vertices possibly with loops. Let u be an arbitrary fixed vertexof G k, k . Define a vector x on G k, k such that x u = 1 and x v = 0 for any other vertices v = u .It is easy to verify by (2.1) that d u is an eigenvalue (also an H -eigenvalue) of L ( G k, k ).From the above fact, we find that the vertices in the same half edge of G k, k may havedifferent values given by eigenvectors of L ( G k, k ). However, if λ = d v for some vertex v , we willhave the following property on the eigenvectors associated with λ .5enote by d u the common degree of the vertices in u . For a nonempty subset S ⊆ V ( G k, k ),denote x S := Q v ∈ S x v , where x is a vector defined on the vertices of G k, k . Lemma 2.1
Let G be a simple graph possibly with loops. Let u and ¯ u be two vertices in the samehalf edge u of G k, k . If x is an eigenvector of L ( G k, k ) corresponding an eigenvalue λ = d u , then x ku = x k ¯ u . Proof:
By the eigenvector equation (2.1),( d u − λ ) x k − u = X uv ∈ E ( G k, k ) x u \{ u } x v , ( d u − λ ) x k − u = X uv ∈ E ( G k, k ) x u \{ ¯ u } x v . So we have ( d u − λ ) x ku = ( d u − λ ) x k ¯ u . The result follows as λ = d u . (cid:4) Let λ be an eigenvalue of L ( G k, k ) such that λ / ∈ { d u : u ∈ V ( G ) } . By Lemma 2.1, theeigenvectors x of λ have the common modulus on the vertices in each half edge u , which willbe denoted by | x u | . By Lemma 2.1, if x u = 0, then x v = 0 for each v ∈ u . Otherwise, for each v ∈ u , x v x u = e i πℓvuk =: E vu , (2 . ℓ uu = 0 and ℓ vu ∈ { , , . . . , k − } . Suppose that x contains no zero entries. Define E u := Y v ∈ u E vu = x u x k/ u = e i π P v ∈ u ℓvuk . (2 . E = diag {E u : u ∈ V ( G ) } , and L E ( G ) = D ( G ) − EA ( G ) E . (2 . u as u , then E ¯ u = ±E u as x u = x k/ u E u = x k/ u E ¯ u and x ku = x k ¯ u .Let ¯ E = diag {E ¯ u : u ∈ V ( G ) } . Then ¯ E = E S , where S is a diagonal matrix with ± L ¯ E ( G ) = D ( G ) − ¯ E A ( G ) ¯ E = S − L E ( G ) S , and hence L ¯ E ( G ) has the same spectrumas L E ( G ). Lemma 2.2
Let λ be an eigenvalue of L ( G k, k ) corresponding to an eigenvector x , where G is asimple graph possibly with loops. Suppose that x contains no zero entries. If λ / ∈ { d u : u ∈ V ( G ) } as an eigenvalue of L ( G k, k ) , then λ is an eigenvalue of L E ( G ) with an eigenvector x such that x u = x k/ u for each u ∈ V ( G ) . Proof:
For each vertex u ∈ u , ( d u − λ ) x k − u = P uw ∈ E ( G k, k ) x u \{ u } x w . So by (2.3)( d u − λ ) x ku = X uw ∈ E ( G k, k ) x u x w = X uw ∈ E ( G k, k ) E u x k/ u E w x k/ w . ( d u − λ ) x k/ u = X uw ∈ E ( G ) E u x k/ w E w . λ is an eigenvalue of the matrix L E ( G ) with the eigenvector x defined in the lemma. (cid:4) The modified induced subgraph of a simple graph G induced by the vertex subset U ⊆ V ( G ),denoted by G o [ U ], is the induced subgraph G [ U ] together with d v ( G ) − d v ( G [ U ]) loops oneach vertex v ∈ U ; see Fig. 1.1. The Laplacian matrix of G o [ U ] is exactly L ( G )[ U ], i.e. L ( G o [ U ]) = L ( G )[ U ], and L ( G o [ U ] k, k ) = L ( G k, k )[ U ], where U = ∪{ u : u ∈ U } . Similarly, Q ( G o [ U ]) = Q ( G )[ U ] and Q ( G o [ U ] k, k ) = Q ( G k, k )[ U ]. Theorem 2.3
Let λ be an eigenvalue of L ( G k, k ) corresponding to an eigenvector x , where G is a simple graph possibly with loops. Suppose that λ / ∈ { d u : u ∈ V ( G ) } as an eigenvalue of L ( G k, k ) . Let U = ∪{ u : | x u | > } and U = { u : u ⊆ U } . Let E = diag {E u : u ∈ U } be definedas in (2.3). Then the following results hold. (1) G k, k [ U ] contains no isolated half edges, and hence G [ U ] contains no isolated vertices. (2) λ is an eigenvalue of L ( G k, k )[ U ] with x [ U ] as an eigenvector. (3) λ is an eigenvalue of L E ( G o [ U ]) with an eigenvector x such that x u = x k/ u for u ∈ U . Proof:
By (2.1), it is easy to verify the assertions (1) and (2). Note that L ( G k, k )[ U ] = L ( G o [ U ] k, k ), the assertion (3) follows from Lemma 2.2 as x [ U ] contains no zero entries. (cid:4) Corollary 2.4
Each eigenvalue λ of L ( G k, k ) is an eigenvalue of L E ( G o [ U ]) for some con-nected modified induced subgraph G o [ U ] and some choice of E . Furthermore, if λ is an H -eigenvalue of L ( G k, k ) , then λ is an eigenvalue of L ( G o [ U ]) . Proof:
Let x be an eigenvector of L ( G k, k ) corresponding to λ . If λ = d u for some halfedge u , then λ is an H -eigenvalue of the Laplacian matrix L ( G o [ u ]). Otherwise, let U and U bedefined as in Theorem 2.3. Then λ is an eigenvalue of L ǫ ( G o [ U ]), where E = diag {E u : u ∈ U } .We may assume that G o [ U ] is connected, as otherwise λ must be a Laplacian eigenvalue of someconnected component of G o [ U ]. If x is real, so is x [ U ]. From the notations (2.2) and (2.3), foreach half edge u ∈ U and each vertex v ∈ u , by Lemma 2.1, E vu = ± E u = ±
1. So, E = E − , and L E ( G o [ U ]) = D ( G )[ U ] − EA ( G [ U ]) E = E − ( D ( G )[ U ] − A ( G [ U ])) E = E − L ( G o [ U ]) E , which implies that L E ( G o [ U ]) has the same spectrum as L ( G o [ U ]). Therefore λ is an eigenvalueof L ( G o [ U ]). (cid:4) Lemma 2.5
Let G be a simple graph possibly with loops. Let E = diag {E u : u ∈ V ( G ) } , where E u = e i πℓuk for some ℓ u ∈ { , , . . . , k − } . Then each eigenvalue of L E ( G ) is an eigenvalue of L ( G k, k ) . Proof:
Let λ be an eigenvalue of L ǫ ( G ) associated with the eigenvector x . For each halfedge u of G k, k , there exists a function f u : u → { , , . . . , k − } such that f u ( u ) = 0 and7 i π P v ∈ u f u ( v ) k = E u . Now define a vector x defined on G k, k such that for each half edge u andeach v ∈ u , x v = x /ku e i πf u ( v ) k , (2 . x /ku is a root of the equation α k/ = x u . By the eigenvector equation of L E ( G ), for eachvertex u , ( d u − λ ) x u = X uw ∈ E ( G ) E u x w E w . So ( d u − λ ) x k/ u = X uw ∈ E ( G ) E u x k/ w E w . ( d u − λ ) x k − u = X uw ∈ E ( G ) E u x k/ − u x k/ w E w = X uw ∈ E ( G k, k ) x u \{ u } x w . For any other vertex v ∈ u ,( d u − λ ) x k − v = ( d u − λ )( x u e i πf u ( v ) k ) k − = X uw ∈ E ( G k, k ) x u \{ u } x w e − i πf u ( v ) k = X uw ∈ E ( G k, k ) x u \{ v } x w . Therefore λ is an eigenvalue of L ( G k, k ) with the eigenvector x defined as in (2.5). (cid:4) By Lemma 2.5, if taking E = I , then each eigenvalue of L ( G ) is an eigenvalue of L ( G k, k ).We will show those eigenvalues of L ( G ) are really H -eigenvalue of L ( G k, k ). Lemma 2.6
Let G be a simple graph possibly with loops. Each eigenvalue of L ( G ) is an H -eigenvalue of L ( G k, k ) . Proof:
Let x be an eigenvector of L ( G ) corresponding to an eigenvalue λ . Let x be a vectordefined on G k, k as follows. For each u ∈ V ( G ), x u = sgn( x u ) | x u | /k , x v = | x u | /k , for each vertex v ∈ u \{ u } . (2 . x u = x u , for each u ∈ V ( G ) . Also, since k is even, x k − u = (sgn( x u ) | x u | /k ) k − = sgn( x u ) | x u | ( | x u | /k ) k/ − = x u x u \{ u } . By the eigenvector equation of L ( G ), ( d u − λ ) x u = P uw ∈ E ( G ) x w , so we have( d u − λ ) x k − u = ( d u − λ ) x u x u \{ u } = X uw ∈ E ( G ) x w x u \{ u } = X uw ∈ E ( G k, k ) x u \{ u } x w . For any other vertex v ∈ u ,( d u − λ ) x k − v = ( d u − λ )(sgn( x u ) x u ) k − = sgn( x u )( d u − λ ) x k − u = X uw ∈ E ( G k, k ) sgn( x u ) x u \{ u } x w = X uw ∈ E ( G k, k ) x u \{ v } x w . So λ is an H -eigenvalue of L ( G k, k ). (cid:4) orollary 2.7 Let G be a simple graph, and let G o [ U ] be a connected modified induced sub-graphs of G . Let E = diag {E u : u ∈ U } , where E u = e i πℓuk for some ℓ u ∈ { , , . . . , k − } .Then each eigenvalue of L E ( G o [ U ]) is an eigenvalue of L ( G k, k ) . In particular, each eigenvalueof L ( G o [ U ]) is an H -eigenvalue of L ( G k, k ) . Proof:
By Lemma 2.5, if λ is an eigenvalue of L E ( G o [ U ]) with an eigenvector x , then λ isan eigenvalue of L ( G o [ U ] k, k ) = L ( G k, k )[ U ] with an eigenvector x whose entries are defined asin (2.5), where U = ∪{ u : u ∈ U } . Extending the eigenvector x defined on G o [ U ] k, k to G k, k by assigning zeros to the vertices outside U , we will get a vector y . It is easy to verify by (2.1)that y is an eigenvector of L ( G k, k ) corresponding the eigenvalue λ .If E = I , L E ( G o [ U ]) = L ( G o [ U ]) and x could be taken real. In this case, by Lemma 2.6 wetake the real eigenvector x whose entries are defined as in (2.6). Then by a similar discussion, λ is an H -eigenvalue of L ( G k, k ). (cid:4) By Corollary 2.4 and Corollary 2.7, we get the following main result.
Theorem 2.8
Let G be a simple graph. Then, regardless of multiplicities, the spectrum of L ( G k, k ) consists of all eigenvalues of L E ( G o [ U ]) for all choices of E as defined in Corollary 2.7and all connected modified induced subgraphs G o [ U ] of G .Furthermore, regardless of multiplicities, the H -spectrum of L ( G k, k ) consists of all eigenval-ues of L ( G o [ U ]) for all connected modified induced subgraphs G o [ U ] of G . Corollary 2.9
Let G be a simple graph. Then λ L max ( G k, k ) = λ L max ( G ) , ρ L ( G k, k ) = max { ρ ( L E ( G o [ U ])) } ,where the maximum is taken over all all choices of E as defined in Corollary 2.7 and all connectedmodified induced subgraphs G o [ U ] of G . Proof:
By the interlacing theorem of the eigenvalues of real symmetric matrices (see [3]), λ L max ( G ) is the maximum of all largest eigenvalues of the principal submatrices of L ( G ). Thefirst equality follows from Theorem 2.8. The second equality is easily seen also by Theorem 2.8. (cid:4) Along the line of discussion in this section, one can easily get the spectrum of the adjacencytensor or the signless Laplacian tensor, where the H -spectra of these tensors are discussed in[7]. Theorem 2.10
Let G be a simple graph. Then, regardless of multiplicities, the spectrum of A ( G k, k ) (respectively, Q ( G k, k ) ) consists of all eigenvalues of A E ( G [ U ]) (respectively, Q E ( G o [ U ]) )for all choices of E as defined in Corollary 2.7 and all connected induced subgraphs G [ U ] (re-spectively, all connected modified induced subgraphs G o [ U ] ) of G , where A E ( G [ U ]) = EA ( G [ U ]) E and Q E ( G o [ U ]) = D ( G )[ U ] + EA ( G [ U ]) E .Furthermore, regardless of multiplicities, the H -spectrum of A ( G k, k ) (respectively, Q ( G k, k ) )consists of all eigenvalues of A ( G [ U ]) (respectively, Q ( G o [ U ]) ) for all connected induced sub-graphs G [ U ] (respectively, all connected modified induced subgraphs G o [ U ] ) of G . The largest H -eigenvalue and spectral radius of Laplacian ten-sor Let G be a connected simple graph. If G is bipartite, then G k, k is odd-bipartite by Lemma 1.6.So by Theorem 1.2, λ L max ( G k, k ) = λ Q max ( G k, k ), which implies that ρ L ( G k, k ) = λ L max ( G k, k ) = λ Q max ( G k, k ) = ρ Q ( G k, k ) . If G is non-bipartite, then G k, k is non-odd-bipartite also by Lemma 1.6. By Theorem 1.2, λ L max ( G k, k ) < λ Q max ( G k, k ) = ρ Q ( G k, k ) . However, it may occur that ρ L ( G k, k ) = ρ Q ( G k, k ). Lemma 3.1
Let G be a connected non-bipartite graph. If k is a multiple of , then ρ L ( G k, k ) = ρ Q ( G k, k ) , or equivalently Spec ( L ( G k, k )) = Spec ( Q ( G k, k )) . Proof:
It suffices to prove that ρ Q ( G k, k ) is an eigenvalue of L ( G k, k ) as ρ L ( G k, k ) ≤ ρ Q ( G k, k ). Let x be an eigenvector Q ( G k, k ) corresponding to ρ Q ( G k, k ) =: ρ . By the eigen-vector equation of Q ( G k, k ), for each vertex u ∈ u ,( ρ − d u ) x k − u = X uw ∈ E ( G k, k ) x u \{ u } x w . (3 . y such that for each half edge u , y u = ix u , y v = x v for any other v ∈ u \{ u } . (3 . k is a multiple of 4, by (3.1) it is easy to verify( d u − ρ ) y k − u = X uw ∈ E ( G k, k ) y u \{ u } y w , and for any other vertex v ∈ u ,( d u − ρ ) y k − v = X uw ∈ E ( G k, k ) y u \{ v } y w . So ρ Q ( G k, k ) is an eigenvalue of L ( G k, k ) with y as an eigenvector. (cid:4) We give some remarks for Lemma 3.1. For each half edge u of G k, k , defineΓ u = i , Γ v = 1 , for any other vertex v ∈ u . Then we get a diagonal matrix Γ = diag { Γ v : v ∈ V ( G k, k ) } . From the proof of Lemma 3.1,if x is an eigenvector of Q ( G k, k ) corresponding to an eigenvalue λ , then Γ x is an eigenvectorof L ( G k, k ) also corresponding to the eigenvalue λ . Furthermore, according the tensor productintroduced in [11], L ( G k, k ) = Γ − ( k − Q ( G k, k )Γ , (3 . L ( G k, k ) and Q ( G k, k ) are diagonal similar, and hence Spec ( L ( G k, k )) = Spec ( Q ( G k, k )) by [11, Theorem 2.3] though Hspec ( L ( G k, k )) = Hspec ( Q ( G k, k )) by Theorem1.4 as G k, k is not odd-bipartite. From (3.3), one can get −A ( G k, k ) = Γ − ( k − A ( G k, k )Γ , (3 . Spec ( A ( G k, k )) = − Spec ( A ( G k, k )), i.e. the spectrum is symmetric with respect to the origin,though Hspec ( A ( G k, k )) = − Hspec ( A ( G k, k )) by Theorem 1.4 as G k, k is not odd-bipartite.Secondly, the eigenvector y in the proof of Lemma 3.1 can also be defined in a way differentfrom (3.2). For each half edge u , arbitrarily choose k -subset U from u , and define y u = e i πk if u ∈ U , and y v = x v if v ∈ u \ U . One can also find a diagonal matrix Γ based on this definitionof y to make (3.3) and (3.4) hold.Motivated by the above discussion, we get a result complementary to Theorems 1.3 and 1.4. Theorem 3.2
Let G be a connected non-odd-bipartite even uniform hypergraph. Then the fol-lowing are equivalent. (1) ρ L ( G ) = ρ Q ( G ) . (2) L ( G ) and Q ( G ) are similar via a complex (necessarily non-real) diagonal matrix withmodular- diagonal entries. (3) Spec ( L ( G )) = Spec ( Q ( G )) . (4) A ( G ) and −A ( G ) are similar via a complex (necessarily non-real) diagonal matrix withmodular- diagonal entries. (5) Spec ( A ( G )) = − Spec ( A ( G )) . (6) − ρ A ( G ) ∈ Spec ( A ( G )) . Proof:
It is clear that (2) ⇒ (3) ⇒ (1) and (4) ⇒ (5) ⇒ (6) by [11, Theorem 2.3]. We willtake the proof technique from [10]. If ρ L ( G ) = ρ Q ( G ), taking λ = ρ Q ( G ) e i φ as an eigenvalue of L ( G ), by Perron-Frobenius Theorem for nonnegative weakly irreducible tensors (see [13]), thereexists a nonsingular diagonal matrix Γ with | Γ | = I such that L ( G ) = e i φ Γ − ( k − Q ( G )Γ . (3 . e i φ = 1 by comparing the diagonal entries of both sides of (3.5), and L ( G ) = Γ − ( k − Q ( G )Γ , (3 . −A ( G ) = Γ − ( k − A ( G )Γ . So, if (1) holds, we can get (2) and (4). Note that the matrix Γ can not be taken as real;otherwise, Γ would have both 1 and − G is odd-bipartite by [10,Theorem 2.1]; a contradiction.Now suppose (6) holds, i.e. − ρ A ( G ) ∈ Spec ( A ( G )). By Perron-Frobenius Theorem, therealso exists a nonsingular diagonal matrix ¯Γ with | ¯Γ | = I such that A ( G ) = − ¯Γ − ( k − A ( G )¯Γ , (3 . L ( G ) = ¯Γ − ( k − Q ( G )¯Γ , which implies that (2) holds. (cid:4) From the proof of Theorem 3.2, that
Spec ( L ( G )) = Spec ( Q ( G )) is equivalent to that L ( G )is similar to Q ( G ) via a complex diagonal matrix with modular-1 diagonal entries. However,by the results in [10], that Hspec ( L ( G )) = Hspec ( Q ( G )) is equivalent to that L ( G ) is similarto Q ( G ) via a diagonal matrix with ± Spec ( L ( G )) = Spec ( Q ( G )) ⇒ Hspec ( L ( G )) = Hspec ( Q ( G )). Butthis happens only when G is odd-bipartite by Theorem 1.4. Similar discussion can apply to Spec ( A ( G )) and Hspec ( A ( G )) for the spectral symmetric property. Theorem 3.3
Let G be a connected non-bipartite graph. Then ρ L ( G k, k ) = ρ Q ( G k, k ) if andonly if k is a multiple of . In this case, λ L max ( G k, k ) < ρ L ( G k, k ) . Proof:
The sufficiency follows by Lemma 3.1. By Corollary 2.9, suppose that ρ L ( G k, k ) = ρ ( L E ( G o [ U ])) for some connected modified induced subgraphs G o [ U ] of G and some E . As |L E ( G o [ U ]) | = Q ( G o [ U ]), by Perron-Frobenius Theorem for nonnegative weakly irreducible ten-sors (see [13]) or for nonnegative irreducible matrices (see [3]) and Lemma 1.7, ρ L ( G k, k ) = ρ ( L E ( G o [ U ])) ≤ ρ ( |L E ( G o [ U ]) | ) = ρ Q ( G o [ U ]) ≤ ρ Q ( G ) = ρ Q ( G k, k ) . If ρ L ( G k, k ) = ρ Q ( G k, k ), then ρ Q ( G o [ U ]) = ρ Q ( G ), which implies that U = V ( G ) as G isconnected. So ρ ( L E ( G )) = ρ Q ( G ). Assume that λ = e i φ ρ Q ( G ) is an eigenvalue of L E ( G ). ByPerron-Fronenius Theorem, there exists a diagonal matrix Γ = diag { e i θ u : u ∈ V ( G ) } such that L E ( G ) = e i φ Γ − Q ( G )Γ . (3 . e i φ Γ − D ( G )Γ = D ( G ) , e i φ Γ − EA ( G ) E Γ = −A ( G ) . (3 . e i φ = 1. As G is non-bipartite, letting C m +1 be an odd cycle of G with edges v i v i +1 for i = 1 , , . . . , m +1, where v m +2 = v . Using the second equality of (3.9), for i = 1 , , . . . , m +1, e − i θ vi E v i E v i +1 e i θ vi +1 = − . Thus m +1 Y i =1 (cid:16) e − i θ vi E v i E v i +1 e i θ vi +1 (cid:17) = − , and hence m +1 Y i =1 E v i = − . E v = e i πℓuk for some ℓ u ∈ { , , . . . , k − } , e i π P m +1 i =1 ℓvik = − , which implies that k is a multiple of 4. (cid:4) Next we discuss the case of k ≡
2( mod 4). In this case, ρ L ( G k, k ) < ρ Q ( G k, k ) by Theorem3.3. But, can we have λ L max ( G k, k ) = ρ L ( G k, k )? Theorem 3.4
Let G be a connected non-bipartite graph. Suppose that k ≡
2( mod 4) . Then forsufficiently large k , λ L max ( G k, k ) < ρ L ( G k, k ) . Proof:
Let k = 4 l + 2, and let ˜ E = e i πlk I . Then L ˜ E ( G ) = D ( G ) − ˜ EA ( G ) ˜ E = D ( G ) − e i πl l +1 A ( G ) . If k → ∞ (i.e. l → ∞ ), then L ˜ E ( G ) → D ( G ) + A ( G ) = Q ( G ). As ρ ( L ˜ E ( G )) is continuous in theentries of L ˜ E ( G ), if k → ∞ , ρ ( L ˜ E ( G )) → ρ ( Q ( G )) = ρ Q ( G k, k ) . By Corollary 2.9, ρ L ( G k, k ) = max { ρ ( L E ( G o [ U ])) } ≥ ρ ( L ˜ E ( G )) . Note that ρ L ( G k, k ) < ρ Q ( G k, k ) by Theorem 3.3. So, ρ L ( G k, k ) → ρ Q ( G k, k ) = ρ ( Q ( G )) . (3 . G is non-bipartite, by Corollary 2.9, λ L max ( G k, k ) = λ L max ( G ) = ρ ( L ( G )) < ρ ( Q ( G )) . (3 . k , λ L max ( G k, k ) < ρ L ( G k, k ). (cid:4) By Theorem 3.3 and Theorem 3.4, we pose the following conjecture.
Conjecture 3.5
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