An Interlacing Approach for Bounding the Sum of Laplacian - PowerPoint PPT Presentation

An Interlacing Approach for Bounding the Sum of Laplacian Eigenvalues of Graphs Aida Abiad Tilburg University joint work with M.A. Fiol, W.H. Haemers and G. Perarnau

Introduction Laplacian matrix A generalization of Grone’s result Eigenvalue interlacing Some applications Two cases of interlacing A variation of Grone-Merris’s result   1 − 1 0 0 − 1 3 − 1 − 1   L =   0 − 1 1 0   0 − 1 0 1 spectrum: { 4 1 , 1 2 , 0 1 } Aida Abiad

Introduction Laplacian matrix A generalization of Grone’s result Eigenvalue interlacing Some applications Two cases of interlacing A variation of Grone-Merris’s result m < n λ 1 ≥ λ 2 ≥ · · · ≥ λ n µ 1 ≥ µ 2 ≥ · · · ≥ µ m Aida Abiad

Introduction Laplacian matrix A generalization of Grone’s result Eigenvalue interlacing Some applications Two cases of interlacing A variation of Grone-Merris’s result m < n λ 1 ≥ λ 2 ≥ · · · ≥ λ n µ 1 ≥ µ 2 ≥ · · · ≥ µ m Interlacing: λ i ≥ µ i ≥ λ n − m + i 1 ≤ i ≤ m Aida Abiad

Introduction Laplacian matrix A generalization of Grone’s result Eigenvalue interlacing Some applications Two cases of interlacing A variation of Grone-Merris’s result m < n λ 1 ≥ λ 2 ≥ · · · ≥ λ n µ 1 ≥ µ 2 ≥ · · · ≥ µ m Interlacing: λ i ≥ µ i ≥ λ n − m + i 1 ≤ i ≤ m Aida Abiad

Introduction Laplacian matrix A generalization of Grone’s result Eigenvalue interlacing Some applications Two cases of interlacing A variation of Grone-Merris’s result m < n λ 1 ≥ λ 2 ≥ · · · ≥ λ n µ 1 ≥ µ 2 ≥ · · · ≥ µ m Interlacing: λ i ≥ µ i ≥ λ n − m + i 1 ≤ i ≤ m λ 1 , λ 2 , · · · , λ n eigenvalues of a matrix A µ 1 , µ 2 , · · · , µ m eigenvalues of a matrix B Aida Abiad

Introduction Laplacian matrix A generalization of Grone’s result Eigenvalue interlacing Some applications Two cases of interlacing A variation of Grone-Merris’s result B is a principal submatrix of A . 1 Aida Abiad

Introduction Laplacian matrix A generalization of Grone’s result Eigenvalue interlacing Some applications Two cases of interlacing A variation of Grone-Merris’s result B is a principal submatrix of A . 1 If P = { U 1 , . . . , U m } is a partition of { 1 , . . . , n } we can take for B 2 the so-called quotient matrix of A with respect to P . Aida Abiad

Introduction Laplacian matrix A generalization of Grone’s result Eigenvalue interlacing Some applications Two cases of interlacing A variation of Grone-Merris’s result [Schur 1923] Let G be a graph with vertex degrees d 1 ≥ d 2 ≥ · · · ≥ d n , and Laplacian matrix L with eigenvalues λ 1 ≥ λ 2 ≥ · · · ≥ λ n (= 0 ) . Then, m m � � λ i ≥ d i i = 1 i = 1 Aida Abiad

Introduction Laplacian matrix A generalization of Grone’s result Eigenvalue interlacing Some applications Two cases of interlacing A variation of Grone-Merris’s result [Schur 1923] Let G be a graph with vertex degrees d 1 ≥ d 2 ≥ · · · ≥ d n , and Laplacian matrix L with eigenvalues λ 1 ≥ λ 2 ≥ · · · ≥ λ n (= 0 ) . Then, m m � � λ i ≥ d i i = 1 i = 1 Proof: 1 Let B be a principal m × m submatrix of L indexed by the subindexes corresponding to the m largest degrees, with eigenvalues µ 1 ≥ µ 2 ≥ · · · ≥ µ m . Then, m m � � µ i = tr B = d i , i = 1 i = 1 and, by interlacing, µ i ≤ λ i . Aida Abiad

Introduction Laplacian matrix A generalization of Grone’s result Eigenvalue interlacing Some applications Two cases of interlacing A variation of Grone-Merris’s result Aida Abiad

Introduction Laplacian matrix A generalization of Grone’s result Eigenvalue interlacing Some applications Two cases of interlacing A variation of Grone-Merris’s result Aida Abiad

Introduction Laplacian matrix A generalization of Grone’s result Eigenvalue interlacing Some applications Two cases of interlacing A variation of Grone-Merris’s result Aida Abiad

Introduction Laplacian matrix A generalization of Grone’s result Eigenvalue interlacing Some applications Two cases of interlacing A variation of Grone-Merris’s result The isoperimetric number i of G is defined as � � i ( G ) = min | ∂ ( U , U ) | / | U | : 0 < | U | ≤ n / 2 . U ⊂ V [Mohar 1989] Let G be a graph with Laplacian eigenvalues λ 1 ≥ λ 2 ≥ · · · ≥ λ n (= 0 ) and isoperimetric number i ( G ) . Then, i ( G ) ≥ λ n − 1 / 2 . Aida Abiad

Introduction Laplacian matrix A generalization of Grone’s result Eigenvalue interlacing Some applications Two cases of interlacing A variation of Grone-Merris’s result [Mohar 1989] Let G be a graph with Laplacian eigenvalues λ 1 ≥ λ 2 ≥ · · · ≥ λ n (= 0 ) and isoperimetric number i . Then, i ( G ) ≥ λ n − 1 / 2 . Proof: 2 Set m = 2 and take a partition { V 1 = U , V 2 = U } . Then,   | ∂ ( U , U ) | − | ∂ ( U , U ) | | U | | U | B =   − | ∂ ( U , U ) | | ∂ ( U , U ) | n −| U | n −| U | spectrum B : µ 1 ≥ µ 2 = 0 and µ 1 = trace B = | ∂ ( U , U ) | | U | ( 1 + n −| U | ) | U | By interlacing, λ 1 ≥ µ 1 ≥ λ n − 2 + 1 = λ n − 1 . So λ n − 1 ≤ | ∂ ( U , U ) | n ( n −| U | ) . | U | 2 , we have λ n − 1 ≤ 2 | ∂ ( U , U ) | For | U | ≤ n ≤ 2 i ( G ) . | U | Aida Abiad

Introduction A generalization of Grone’s result Some applications A variation of Grone-Merris’s result [Grone 1995] For a connected graph and 0 < m < n , then m m � � λ i ≥ d i + 1 . i = 1 i = 1 [Theorem] Let G be a connected graph on n = | V | vertices, having Laplacian matrix L with eigenvalues λ 1 ≥ λ 2 ≥ · · · ≥ λ n (= 0 ) . Let U be the vertex subset which contains the m largest degrees, with 0 < m < n . Then, m m � � λ i ≥ d i . i = 1 i = 1 Aida Abiad

Introduction A generalization of Grone’s result Some applications A variation of Grone-Merris’s result [Grone 1995] For a connected graph and 0 < m < n , then m m � � λ i ≥ d i + 1 . i = 1 i = 1 [Theorem] Let G be a connected graph on n = | V | vertices, having Laplacian matrix L with eigenvalues λ 1 ≥ λ 2 ≥ · · · ≥ λ n (= 0 ) . Let U be the vertex subset which contains the m largest degrees, with 0 < m < n . Then, m m d i + | ∂ ( U , U ) | � � λ i ≥ n − m . i = 1 i = 1 Aida Abiad

Introduction A generalization of Grone’s result Some applications A variation of Grone-Merris’s result Proof: Let U ⊆ V be the set containing the m vertices with 2 largest degree. Aida Abiad

Introduction A generalization of Grone’s result Some applications A variation of Grone-Merris’s result Proof: Let U ⊆ V be the set containing the m vertices with 2 largest degree. B has row sum 0, so µ m + 1 = λ n = 0 m m + 1 m � � � µ i = µ i = tr B = d i + b m + 1 , m + 1 i = 1 i = 1 i = 1 b m + 1 , m + 1 = | ∂ ( U , U ) | n − m Aida Abiad

Introduction A generalization of Grone’s result Some applications A variation of Grone-Merris’s result Proof: Let U ⊆ V be the set containing the m vertices with 2 largest degree. B has row sum 0, so µ m + 1 = λ n = 0 m m + 1 m � � � µ i = µ i = tr B = d i + b m + 1 , m + 1 i = 1 i = 1 i = 1 b m + 1 , m + 1 = | ∂ ( U , U ) | n − m Interlacing µ i ≤ λ i and add for i = 1 , 2 , . . . , m Aida Abiad

Introduction A generalization of Grone’s result Some applications A variation of Grone-Merris’s result Example of equality: The graph join of the complete graph K p with the empty graph K q , n = p + q . Aida Abiad

Introduction A generalization of Grone’s result Some applications A variation of Grone-Merris’s result Example of equality: The graph join of the complete graph K p with the empty graph K q , n = p + q . Aida Abiad

Introduction A generalization of Grone’s result Some applications A variation of Grone-Merris’s result Example of equality: The graph join of the complete graph K p with the empty graph K q , n = p + q . Laplacian spectrum: { n p , p q − 1 , 0 1 } degree sequence: { ( n − 1 ) p , p q } Aida Abiad

Introduction A generalization of Grone’s result Some applications A variation of Grone-Merris’s result Example of equality: The graph join of the complete graph K p with the empty graph K q , n = p + q . Laplacian spectrum: { n p , p q − 1 , 0 1 } degree sequence: { ( n − 1 ) p , p q } U = { v 1 , . . . , v m } b m + 1 , m + 1 = m m m � � d i + b m + 1 , m + 1 = m ( n − 1 ) + m = mn = λ i . i = 1 i = 1 Aida Abiad

Introduction A generalization of Grone’s result Some applications A variation of Grone-Merris’s result If U � = ∂ U and we delete the vertices (and corresponding edges) of U \ ∂ U , [Theorem] For a connected graph and 0 < m < n , then m m d i + | ∂ ( U , U ) | � � λ i ≥ . | ∂ U | i = 1 i = 1 Since | ∂ ( U , U ) | ≥ 1, our result implies Grone’s theorem. | ∂ U | Aida Abiad

Introduction A generalization of Grone’s result Some applications A variation of Grone-Merris’s result Idea: bounding | ∂ ( U , U ) | or optimizing b = | ∂ ( U , U ) | / ( n − m ) Aida Abiad