An Interlacing Approach for Bounding the Sum of Laplacian Eigenvalues of Graphs

Transcription

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

2 Laplacian matrix Eigenvalue interlacing Two cases of interlacing L = spectrum: {4 1, 1 2, 0 1 }

3 Laplacian matrix Eigenvalue interlacing Two cases of interlacing m < n λ 1 λ 2 λ n µ 1 µ 2 µ m

4 Laplacian matrix Eigenvalue interlacing Two cases of interlacing 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

5 Laplacian matrix Eigenvalue interlacing Two cases of interlacing 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

6 Laplacian matrix Eigenvalue interlacing Two cases of interlacing 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

7 Laplacian matrix Eigenvalue interlacing Two cases of interlacing 1 B is a principal submatrix of A.

8 Laplacian matrix Eigenvalue interlacing Two cases of interlacing 1 B is a principal submatrix of A. 2 If P = {U 1,..., U m } is a partition of {1,..., n} we can take for B the so-called quotient matrix of A with respect to P.

9 Laplacian matrix Eigenvalue interlacing Two cases of interlacing [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, λ i d i

10 Laplacian matrix Eigenvalue interlacing Two cases of interlacing [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, λ i d i 1 Proof: 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, µ i = tr B = d i, and, by interlacing, µ i λ i.

11 Laplacian matrix Eigenvalue interlacing Two cases of interlacing

12 Laplacian matrix Eigenvalue interlacing Two cases of interlacing

13 Laplacian matrix Eigenvalue interlacing Two cases of interlacing

14 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 λ 1 λ 2 λ n (= 0) and isoperimetric number i(g). Then, i(g) λ n 1 /2. }.

15 Laplacian matrix Eigenvalue interlacing Two cases of interlacing [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. 2 Proof: Set m = 2 and take a partition {V 1 = U, V 2 = U}. Then, B = (U,U) U (U,U) n U (U,U) U (U,U) n U spectrum B: µ 1 µ 2 = 0 and µ 1 = traceb = (U,U) U (1 + U n U ) By interlacing, λ 1 µ 1 λ n 2+1 = λ n 1. So λ n 1 (U,U) n U ( n U ). For U n 2, we have λ n 1 2 (U,U) U 2i(G).

16 [Grone 1995] For a connected graph and 0 < m < n, then λ i d 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, λ i d i.

17 [Grone 1995] For a connected graph and 0 < m < n, then λ i d 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, λ i d i + (U, U) n m.

18 2 Proof: Let U V be the set containing the m vertices with largest degree.

19 2 Proof: Let U V be the set containing the m vertices with largest degree. B has row sum 0, so µ m+1 = λ n = 0 m+1 µ i = µ i = tr B = b m+1,m+1 = (U,U) n m d i + b m+1,m+1

20 2 Proof: Let U V be the set containing the m vertices with largest degree. B has row sum 0, so µ m+1 = λ n = 0 m+1 µ i = µ i = tr B = b m+1,m+1 = (U,U) n m d i + b m+1,m+1 Interlacing µ i λ i and add for i = 1, 2,..., m

21 Example of equality: The graph join of the complete graph K p with the empty graph K q, n = p + q.

22 Example of equality: The graph join of the complete graph K p with the empty graph K q, n = p + q.

23 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 }

24 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 d i + b m+1,m+1 = m(n 1) + m = mn = λ i.

25 If U U and we delete the vertices (and corresponding edges) of U\ U, [Theorem] For a connected graph and 0 < m < n, then λ i d i + (U, U). U Since (U,U) U 1, our result implies Grone s theorem.

26 Idea: bounding (U, U) or optimizing b = (U, U) /(n m)

27 The edge-connectivity The isoperimetric number The edge-connectivity κ e (G) of a graph G is the minimum size of a cut (U, U), provided 0 < U < n.

28 The edge-connectivity The isoperimetric number The edge-connectivity κ e (G) of a graph G is the minimum size of a cut (U, U), provided 0 < U < n. [Proposition] κ e (G) min 0<m<n { (n m) } (λ i d i ).

29 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.

30 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] λ n 1 2 i(g) λ n 1 (2d 1 λ n 1 ).

31 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] λ n 1 2 i(g) λ n 1 (2d 1 λ n 1 ). [Proposition] i(g) min n 2 m<n (λ i d i ).

32 The edge-connectivity The isoperimetric number Example: The graph join of the complete graph K p with the empty graph K q.

33 The edge-connectivity The isoperimetric number Example: The graph join of the complete graph K p with the empty graph K q. Laplacian spectrum: {n p, p q 1, 0 1 } degree sequence: {(n 1) p, p q }

34 The edge-connectivity The isoperimetric number Example: The graph join of the complete graph K p with the empty graph K q. Laplacian spectrum: {n p, p q 1, 0 1 } degree sequence: {(n 1) p, p q } Our bound gives i(g) min{p, n 2 }, which is better than Mohar s upper bound for all 0 q < n.

35 [Grone-Merris 1994] Let G be a connected graph of order n > 2 with Laplacian eigenvalues λ 1 λ 2 λ n. If the induced subgraph of a subset U V with U = m consists of r pairwise disjoint edges and m 2r isolated vertices, then λ i d u + m r. u U

36 [Grone-Merris 1994] Let G be a connected graph of order n > 2 with Laplacian eigenvalues λ 1 λ 2 λ n. If the induced subgraph of a subset U V with U = m consists of r pairwise disjoint edges and m 2r isolated vertices, then λ i d u + m r. u U [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, λ i d u + h. u U

37 [Theorem] Let G be a connected graph of order n > 2 with Laplacian eigenvalues λ 1 λ 2 λ n. 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 ϑ 1 be the largest Laplacian eigenvalue of G[U], then m+1 λ i d u + m E[U] + ϑ 1. u U

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

39 Thanks for your attention