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

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

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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 Grones result

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An Interlacing Approach for Bounding the Sum

  • f Laplacian Eigenvalues of Graphs

Aida Abiad Tilburg University

joint work with M.A. Fiol, W.H. Haemers and G. Perarnau

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

L =     1 −1 −1 3 −1 −1 −1 1 −1 1     spectrum: {41, 12, 01}

Aida Abiad

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

m < n λ1 ≥ λ2 ≥ · · · ≥ λn µ1 ≥ µ2 ≥ · · · ≥ µm

Aida Abiad

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

m < n λ1 ≥ λ2 ≥ · · · ≥ λn µ1 ≥ µ2 ≥ · · · ≥ µm

Interlacing:

λi ≥ µi ≥ λn−m+i 1 ≤ i ≤ m

Aida Abiad

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

m < n λ1 ≥ λ2 ≥ · · · ≥ λn µ1 ≥ µ2 ≥ · · · ≥ µm

Interlacing:

λi ≥ µi ≥ λn−m+i 1 ≤ i ≤ m

Aida Abiad

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

m < n λ1 ≥ λ2 ≥ · · · ≥ λn µ1 ≥ µ2 ≥ · · · ≥ µm

Interlacing:

λi ≥ µi ≥ λn−m+i 1 ≤ i ≤ m λ1, λ2, · · · , λn eigenvalues of a matrix A µ1, µ2, · · · , µm eigenvalues of a matrix B

Aida Abiad

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Introduction A generalization of Grone’s result Some applications A variation of Grone-Merris’s result Laplacian matrix Eigenvalue interlacing Two cases of interlacing 1

B is a principal submatrix of A.

Aida Abiad

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Introduction A generalization of Grone’s result Some applications A variation of Grone-Merris’s result Laplacian matrix Eigenvalue interlacing Two cases of interlacing 1

B is a principal submatrix of A.

2

If P = {U1, . . . , Um} is a partition of {1, . . . , n} we can take for B the so-called quotient matrix of A with respect to P.

Aida Abiad

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

[Schur 1923] Let G be a graph with vertex degrees d1 ≥ d2 ≥ · · · ≥ dn, and Laplacian matrix L with eigenvalues λ1 ≥ λ2 ≥ · · · ≥ λn(= 0). Then,

m

  • i=1

λi ≥

m

  • i=1

di

Aida Abiad

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

[Schur 1923] Let G be a graph with vertex degrees d1 ≥ d2 ≥ · · · ≥ dn, and Laplacian matrix L with eigenvalues λ1 ≥ λ2 ≥ · · · ≥ λn(= 0). Then,

m

  • i=1

λi ≥

m

  • i=1

di

1

Proof: Let B be a principal m × m submatrix of L indexed by the subindexes corresponding to the m largest degrees, with eigenvalues µ1 ≥ µ2 ≥ · · · ≥ µm. Then,

m

  • i=1

µi = tr B =

m

  • i=1

di, and, by interlacing, µi ≤ λi.

Aida Abiad

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

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

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

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

The isoperimetric number i of G is defined as i(G) = min

U⊂V

  • |∂(U, U)|/|U| : 0 < |U| ≤ n/2
  • .

[Mohar 1989] Let G be a graph with Laplacian eigenvalues λ1 ≥ λ2 ≥ · · · ≥ λn(= 0) and isoperimetric number i(G). Then, i(G) ≥ λn−1/2 .

Aida Abiad

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

[Mohar 1989] Let G be a graph with Laplacian eigenvalues λ1 ≥ λ2 ≥ · · · ≥ λn(= 0) and isoperimetric number i. Then, i(G) ≥ λn−1/2 .

2

Proof: Set m = 2 and take a partition {V1 = U, V2 = U}. Then, B =  

|∂(U,U)| |U|

− |∂(U,U)|

|U|

− |∂(U,U)|

n−|U| |∂(U,U)| n−|U|

  spectrum B: µ1 ≥ µ2 = 0 and µ1 = traceB = |∂(U,U)|

|U|

(1 +

|U| n−|U|)

By interlacing, λ1 ≥ µ1 ≥ λn−2+1 = λn−1. So λn−1 ≤ |∂(U,U)|

|U|

(

n n−|U|).

For |U| ≤ n

2, we have λn−1 ≤ 2 |∂(U,U)| |U|

≤ 2i(G).

Aida Abiad

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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

  • i=1

λi ≥

m

  • i=1

di + 1. [Theorem] Let G be a connected graph on n = |V| vertices, having Laplacian matrix L with eigenvalues λ1 ≥ λ2 ≥ · · · ≥ λn(= 0). Let U be the vertex subset which contains the m largest degrees, with 0 < m < n. Then,

m

  • i=1

λi ≥

m

  • i=1

di .

Aida Abiad

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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

  • i=1

λi ≥

m

  • i=1

di + 1. [Theorem] Let G be a connected graph on n = |V| vertices, having Laplacian matrix L with eigenvalues λ1 ≥ λ2 ≥ · · · ≥ λn(= 0). Let U be the vertex subset which contains the m largest degrees, with 0 < m < n. Then,

m

  • i=1

λi ≥

m

  • i=1

di + |∂(U, U)| n − m .

Aida Abiad

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Introduction A generalization of Grone’s result Some applications A variation of Grone-Merris’s result 2

Proof: Let U ⊆ V be the set containing the m vertices with largest degree.

Aida Abiad

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Introduction A generalization of Grone’s result Some applications A variation of Grone-Merris’s result 2

Proof: Let U ⊆ V be the set containing the m vertices with largest degree. B has row sum 0, so µm+1 = λn = 0

m

  • i=1

µi =

m+1

  • i=1

µi = tr B =

m

  • i=1

di + bm+1,m+1 bm+1,m+1 = |∂(U,U)|

n−m

Aida Abiad

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Introduction A generalization of Grone’s result Some applications A variation of Grone-Merris’s result 2

Proof: Let U ⊆ V be the set containing the m vertices with largest degree. B has row sum 0, so µm+1 = λn = 0

m

  • i=1

µi =

m+1

  • i=1

µi = tr B =

m

  • i=1

di + bm+1,m+1 bm+1,m+1 = |∂(U,U)|

n−m

Interlacing µi ≤ λi and add for i = 1, 2, . . . , m

Aida Abiad

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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 Kp with the empty graph Kq, n = p + q.

Aida Abiad

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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 Kp with the empty graph Kq, n = p + q.

Aida Abiad

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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 Kp with the empty graph Kq, n = p + q. Laplacian spectrum: {np, pq−1, 01} degree sequence: {(n − 1)p, pq}

Aida Abiad

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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 Kp with the empty graph Kq, n = p + q. Laplacian spectrum: {np, pq−1, 01} degree sequence: {(n − 1)p, pq} U = {v1, . . . , vm} bm+1,m+1 = m

m

  • i=1

di + bm+1,m+1 = m(n − 1) + m = mn =

m

  • i=1

λi.

Aida Abiad

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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

  • i=1

λi ≥

m

  • i=1

di + |∂(U, U)| |∂U| . Since |∂(U,U)|

|∂U|

≥ 1, our result implies Grone’s theorem.

Aida Abiad

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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

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Introduction A generalization of Grone’s result Some applications A variation of Grone-Merris’s result The edge-connectivity The isoperimetric number

The edge-connectivity κe(G) of a graph G is the minimum size of a cut ∂(U, U), provided 0 < |U| < n.

Aida Abiad

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Introduction A generalization of Grone’s result Some applications A variation of Grone-Merris’s result The edge-connectivity The isoperimetric number

The edge-connectivity κe(G) of a graph G is the minimum size of a cut ∂(U, U), provided 0 < |U| < n. [Proposition] κe(G) ≤ min

0<m<n

  • (n − m)

m

  • i=1

(λi − di)

  • .

Aida Abiad

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Introduction A generalization of Grone’s result Some applications A variation of Grone-Merris’s result The edge-connectivity The isoperimetric number

Recall that the isoperimetric number is defined as i(G) = min

U⊂V

  • |∂(U, U)|/|U| : 0 < |U| ≤ n/2
  • .

Aida Abiad

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Introduction A generalization of Grone’s result Some applications A variation of Grone-Merris’s result The edge-connectivity The isoperimetric number

Recall that the isoperimetric number is defined as i(G) = min

U⊂V

  • |∂(U, U)|/|U| : 0 < |U| ≤ n/2
  • .

[Mohar 1989] λn−1 2 ≤ i(G) ≤

  • λn−1(2d1 − λn−1).

Aida Abiad

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Introduction A generalization of Grone’s result Some applications A variation of Grone-Merris’s result The edge-connectivity The isoperimetric number

Recall that the isoperimetric number is defined as i(G) = min

U⊂V

  • |∂(U, U)|/|U| : 0 < |U| ≤ n/2
  • .

[Mohar 1989] λn−1 2 ≤ i(G) ≤

  • λn−1(2d1 − λn−1).

[Proposition] i(G) ≤ min

n 2 ≤m<n

m

  • i=1

(λi − di).

Aida Abiad

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Introduction A generalization of Grone’s result Some applications A variation of Grone-Merris’s result The edge-connectivity The isoperimetric number

Example: The graph join of the complete graph Kp with the empty graph Kq.

Aida Abiad

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Introduction A generalization of Grone’s result Some applications A variation of Grone-Merris’s result The edge-connectivity The isoperimetric number

Example: The graph join of the complete graph Kp with the empty graph Kq. Laplacian spectrum: {np, pq−1, 01} degree sequence: {(n − 1)p, pq}

Aida Abiad

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Introduction A generalization of Grone’s result Some applications A variation of Grone-Merris’s result The edge-connectivity The isoperimetric number

Example: The graph join of the complete graph Kp with the empty graph Kq. Laplacian spectrum: {np, pq−1, 01} degree sequence: {(n − 1)p, pq} Our bound gives i(G) ≤ min{p, ⌈ n

2⌉}, which is better than Mohar’s

upper bound for all 0 ≤ q < n.

Aida Abiad

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Introduction A generalization of Grone’s result Some applications A variation of Grone-Merris’s result

[Grone-Merris 1994] Let G be a connected graph of order n > 2 with Laplacian eigenvalues λ1 ≥ λ2 ≥ · · · ≥ λn. If the induced subgraph of a subset U ⊂ V with |U| = m consists of r pairwise disjoint edges and m − 2r isolated vertices, then

m

  • i=1

λi ≥

  • u∈U

du + m − r.

Aida Abiad

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Introduction A generalization of Grone’s result Some applications A variation of Grone-Merris’s result

[Grone-Merris 1994] Let G be a connected graph of order n > 2 with Laplacian eigenvalues λ1 ≥ λ2 ≥ · · · ≥ λn. If the induced subgraph of a subset U ⊂ V with |U| = m consists of r pairwise disjoint edges and m − 2r isolated vertices, then

m

  • i=1

λi ≥

  • u∈U

du + m − r. [Brouwer-Haemers 2012] Let G be a (not necessarily connected) graph with a vertex subset U, with m = |U|, and let h be the number of connected components of G[U] that are not connected components of G. Then,

m

  • i=1

λi ≥

  • u∈U

du + h.

Aida Abiad

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Introduction A generalization of Grone’s result Some applications A variation of Grone-Merris’s result

[Theorem] Let G be a connected graph of order n > 2 with Laplacian eigenvalues λ1 ≥ λ2 ≥ · · · ≥ λn. Given a vertex subset U ⊂ V, with m = |U| < n, let G[U] = (U, E[U]) and G[U] be the corresponding induced subgraphs. Let ϑ1 be the largest Laplacian eigenvalue of G[U], then

m+1

  • i=1

λi ≥

  • u∈U

du + m − |E[U]| + ϑ1.

Aida Abiad

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Introduction A generalization of Grone’s result Some applications A variation of Grone-Merris’s result

The coauthors ... M.A. Fiol W.H. Haemers G.Perarnau

Aida Abiad

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Introduction A generalization of Grone’s result Some applications A variation of Grone-Merris’s result

Thanks for your attention

Aida Abiad