Sharp upper and lower bounds for the spectral radius of a - PowerPoint PPT Presentation

Sharp upper and lower bounds for the spectral radius of a nonnegative weakly irreducible tensor and its application South China Normal University, Lihua YOU Joint work with Xiaohua HUANG, Xiying YUAN, Yujie SHU, Pingzhi YUAN July 7, 2018

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k -uniform hypergraph Outline of the Talk Part I : Matrix and its spectrum 1 Part II : Tensor and its spectrum 2 Part III : Applications to a k -uniform hypergraph 3 Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k -uniform hypergraph Part I : Matrix and its spectrum Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k -uniform hypergraph Definitions and notations M : a real matrix of order n . λ 1 , λ 2 , . . . , λ n : the eigenvalues of M with | λ 1 | ≥ | λ 2 | ≥ . . . ≥ | λ n | . ρ ( M ) : the spectral radius of M , ρ ( M ) = | λ 1 | . If M is a nonnegative matrix, it follows from the Perron-Frobenius theorem that the spectral radius ρ ( M ) is a eigenvalue of M . If M is a nonnegative irreducible matrix, it follows from the Perron-Frobenius theorem that ρ ( M ) = λ 1 is simple. Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k -uniform hypergraph Definitions and notations Let G be a graph. A ( G ) = ( a ij ) : the adjacency matrix of G, where a ij = 1 if v i and v j are adjacent and 0 otherwise. The spectral radius of A ( G ) : ρ ( G ) . diag ( G ) = diag ( d 1 , d 2 , . . . , d n ) : the diagonal matrix of vertex degrees of a graph G . The signless Laplacian matrix of G : Q ( G ) = diag ( G ) + A ( G ) . The spectral radius of Q ( G ) : q ( G ) . Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k -uniform hypergraph Definitions and notations D ( G ) = ( d ij ) : the distance adjacency matrix of G. The spectral radius of D ( G ) : ρ D ( G ) . Tr ( G ) = diag ( D 1 , D 2 , . . . , D n ) : the diagonal matrix of vertex transmission of G . The distance signless Laplacian matrix of G : Q ( G ) = Tr ( G ) + D ( G ) . The spectral radius of Q ( G ) : q D ( G ) . Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k -uniform hypergraph Definitions and notations → Let − G be a digraph. → → A ( − G ) = ( a ij ) : the adjacency matrix of − G , where a ij is equal to the number of arc ( v i , v j ) . → → The spectral radius of A ( − G ) : ρ ( − G ) . → diag ( − G ) = diag ( d + 1 , d + 2 , . . . , d + n ) : the diagonal matrix of → vertex out-degrees of − G . → The signless Laplacian matrix of − G : → → → Q ( − G ) = diag ( − G ) + A ( − G ) . → → The spectral radius of Q ( − G ) : q ( − G ) . Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k -uniform hypergraph Notation of a graph Let G = ( V, E ) be a simple (connected) graph with vertex set V = V ( G ) = { v 1 , v 2 , . . . , v n } and edge set E = E ( G ) . d i : the degree of vertex v i . i ∼ j : v i is adjacent to v j . � d j i ∼ j m i = d i : the average degree of the neighbors of v i in G . d uv : the distance between u and v , is the length of the shortest path connecting them in G . n � D i = d ij : the distance degree of vertex v i in G , or the j = 1 transmission of vertex v i in G . n � T i = d ij D j : the second distance degree of vertex v i in G . j = 1 Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k -uniform hypergraph Some known results Theorem 1 Let G = ( V, E ) be a simple connected graph on n vertices. Then � √ m i m j , i ∼ j � ρ ( G ) ≤ max . 1 ≤ i,j ≤ n Moreover, the equality holds if and only if one of the following two conditions holds: (1) m 1 = m 2 = . . . = m n ; (2) G is a bipartite graph and the vertices of same partition have the same average degree. K.C. Das, P. Kumar, Some new bounds on the spectral radius of graphs, Discrete Math. 281 (2004) 149–161. Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k -uniform hypergraph Some known results Theorem 2 Let G = ( V, E ) be a connected graph on n vertices, for any d i + d j + √ ( d i − d j ) 2 + 4m i m j 1 ≤ i, j ≤ n , g ( i, j ) = . Then we have 2 min { g ( i, j ) , i ∼ j } ≤ q ( G ) ≤ max { g ( i, j ) , i ∼ j } , and one of the equalities holds if and only if one of the following conditions holds: (1) G is a regular graph; (2) G is a bipartite semi-regular graph; A.D. Maden, K.C. Das, A.S. Cevik, Sharp upper bounds on the spectral radius of the signless Laplacian matrix of a graph, Appl Math Comput. 219 (2013) 5025–5032. Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k -uniform hypergraph Some known results Theorem 3 Let G = ( V, E ) be a connected graph on n vertices. Then � � � � � T i T j ρ D ( G ) ≤ max � , D i D j 1 ≤ i,j ≤ n D 1 = T 2 T 1 D 2 = . . . = T n and the equality holds if and only if D n . C.X. He, Y. Liu, Z.H. Zhao, Some new sharp bounds on the distance spectral radius of graph, MATCH Commum. Math. Comput. Chem. 63 (2010) 783–788. Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k -uniform hypergraph Some known results Theorem 4 (Hong and Y., 2014, AMC) Let G = ( V, E ) be a connected graph on n vertices, for all � 4TiTj D i + D j + ( D i − D j ) 2 + DiDj 1 ≤ i, j ≤ n , h ( i, j ) = . Then 2 q D ( G ) ≤ max 1 ≤ i,j ≤ n { h ( i, j ) } , and the equality holds if and only if D 1 + T 1 D 1 = . . . = D n + T n D n . W.X. Hong, L.H. You, Futher results on the spectral radius of matrices and graphs, Applied Math Comput. 239 (2014) 326–332. Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k -uniform hypergraph Notation of a digraph → Let − G = ( V, E ) be a digraph on n vertices. i ∼ j : ( v i , v j ) ∈ E . → i : the out-degree of the vertex v i in − d + G . d + � j m + i ∼ j : the average out-degree of the out-neighbors of v i i = d + i → in − G . Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k -uniform hypergraph Some known results Theorem 5 → Let − G = ( V, E ) be a strong connected digraph on n vertices. Then � � � � → � m + i m + ≤ ρ ( − m + i m + min j , i ∼ j G ) ≤ max 1 ≤ i,j ≤ n { j , i ∼ j , 1 ≤ i,j ≤ n and one of the equalities holds if and only if one of the following two conditions holds: (1) m + 1 = m + 2 = . . . = m + n , → (2) − G is a bipartite graph and the vertices of same partition have the same average out-degree. G.H. Xu, C.Q. Xu, Sharp bounds for the spectral radius of digraphs, Linear Algebra Appl. 430 (2009) 1607–1612. Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k -uniform hypergraph Some known results Theorem 6 → Let − G = ( V, E ) be a strong connected digraph on n vertices, j + √ d + i + d + ( d + i − d + j ) 2 + 4m + i m + G ( i, j ) = j for any i, j ∈ { 1, 2, . . . , n } . Then 2 → 1 ≤ i,j ≤ n { G ( i, j ) , i ∼ j } ≤ q ( − min G ) ≤ max 1 ≤ i,j ≤ n { G ( i, j ) , i ∼ j } . S.B. Bozkurt, D. Bozkurt, On the signless Laplacian spectral radius of digraphs, Ars Combin. 108 (2013) 193–200. Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k -uniform hypergraph Our results Theorem 7 (Y., Shu and Yuan, 2017, LAMA) Let A = ( a ij ) be an n × n nonnegative irreducible matrix with a ii = 0 for i = 1, 2, . . . , n , and the row sum r 1 , r 2 , . . . , r n . Let B = A + M , where M = diag ( t 1 , t 2 , . . . , t n ) with t i ≥ 0 for any n � i ∈ { 1, 2, . . . , n } , s i = a ij r j , ρ ( B ) be the spectral radius of B . j = 1 � 4sisj t i + t j + ( t i − t j ) 2 + rirj Let f ( i, j ) = for any 1 ≤ i, j ≤ n . Then 2 1 ≤ i,j ≤ n { f ( i, j ) , a ij � = 0 } ≤ ρ ( B ) ≤ max min (1) 1 ≤ i,j ≤ n { f ( i, j ) , a ij � = 0 } . L.H. You, Y.J. Shu, P.Z. Yuan, Sharp upper and lower bounds for the spectral radius of a nonnegative irreducible matrix and its applications, Linear Multilinear Algebra, 65(1) (2017), Lihua You SCNU 113–128.

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k -uniform hypergraph (Cont.) Moreover, the equalities in (1) hold if and only if one of the two conditions holds: r i = t j + s j (i) t i + s i r j for any i, j ∈ { 1, 2, . . . , n } ; (ii) There exist nonempty proper subsets U and W of [ n ] such that 1) [ n ] = U ∪ W with U ∩ W = φ , 2) a ij � = 0 only when i ∈ U, j ∈ W or i ∈ W, j ∈ U, r i = t j + s j 3) there exists ℓ > 0 such that ρ ( B ) = t i + ℓs i ℓr j for all i ∈ U and j ∈ W. In fact, ℓ > 1 when the left equality holds and ℓ < 1 when the right equality holds. Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k -uniform hypergraph Part II : Tensor and its spectrum Lihua You SCNU