On a lower bound on the Laplacian eigenvalues of a graph

Transcription

1 On a lower bound on the Laplacian eigenvalues of a graph Akihiro Munemasa (joint work with Gary Greaves and Anni Peng) Graduate School of Information Sciences Tohoku University May 22, 2016 JCCA 2016, Kyoto University A. Munemasa (Tohoku University) Laplacian Eigenvalue JCCA / 9

2 Laplacian Eigenvalue of a Graph Definition The Laplacian matrix L of Γ is a matrix indexed by X = V (Γ) with deg(x) if x = y, L xy = 1 if x y, 0 if x y. Laplacian eigenvalues of Γ are eigenvalues of the matrix L. A. Munemasa (Tohoku University) Laplacian Eigenvalue JCCA / 9

3 Laplacian Eigenvalue of a Graph Definition The Laplacian matrix L of Γ is a matrix indexed by X = V (Γ) with deg(x) if x = y, L xy = 1 if x y, 0 if x y. Laplacian eigenvalues of Γ are eigenvalues of the matrix L. If X = n, then the Laplacian eigenvalues are µ 1 µ 2 > 0 } = {{ = 0} = µ n, c where c is the number of connected components. A. Munemasa (Tohoku University) Laplacian Eigenvalue JCCA / 9

4 Laplacian Eigenvalue of a Graph Definition The Laplacian matrix L of Γ is a matrix indexed by X = V (Γ) with deg(x) if x = y, L xy = 1 if x y, 0 if x y. Laplacian eigenvalues of Γ are eigenvalues of the matrix L. If Γ is obtained from Γ by deleting an edge, then µ m (Γ) µ m (Γ ) for all m {1, 2,..., n}. A. Munemasa (Tohoku University) Laplacian Eigenvalue JCCA / 9

5 A Lower Bound Theorem (Brouwer and Haemers, 2008) Let Γ have vertex degrees and Laplacian eigenvalues d 1 d 2 d n, µ 1 µ 2 µ n = 0. Let m {1, 2,..., n}. If Γ K m (n m)k 1, then µ m d m m + 2. A. Munemasa (Tohoku University) Laplacian Eigenvalue JCCA / 9

6 A Lower Bound Theorem (Brouwer and Haemers, 2008) Let Γ have vertex degrees and Laplacian eigenvalues d 1 d 2 d n, µ 1 µ 2 µ n = 0. Let m {1, 2,..., n}. If Γ K m (n m)k 1, then µ m d m m + 2. m = 1: by Grone and Merris (1994). m = 2: by Li and Pan (2000). m = 3: by Guo (2007), and conjectured the above theorem. A. Munemasa (Tohoku University) Laplacian Eigenvalue JCCA / 9

7 A Lower Bound Theorem (Brouwer and Haemers, 2008) Let Γ have vertex degrees and Laplacian eigenvalues d 1 d 2 d n, µ 1 µ 2 µ n = 0. Let m {1, 2,..., n}. If Γ K m (n m)k 1, then µ m d m m + 2. µ m (K m (n m)k 1 ) = 0 = d m m + 1. A. Munemasa (Tohoku University) Laplacian Eigenvalue JCCA / 9

8 A Lower Bound Theorem (Brouwer and Haemers, 2008) Let Γ have vertex degrees and Laplacian eigenvalues d 1 d 2 d n, µ 1 µ 2 µ n = 0. Let m {1, 2,..., n}. If Γ K m (n m)k 1, then µ m d m m + 2. Problem. Characterize graphs Γ which achieve equality: µ m = d m m + 2. A. Munemasa (Tohoku University) Laplacian Eigenvalue JCCA / 9

9 Proof Technique Lemma (Interlacing) Let N be a real symmetric matrix of order n. λ 1 (N) λ n (N). If M is a principal submatrix of N, or a quotient matrix of N, with eigenvalues λ 1 (M) λ m (M), then the eigenvalues of M interlace those of N, that is λ i (N) λ i (M) λ n m+i (N) for i = 1,..., m. A. Munemasa (Tohoku University) Laplacian Eigenvalue JCCA / 9

10 Proof Technique Lemma (Interlacing) Let N be a real symmetric matrix of order n. λ 1 (N) λ n (N). If M is a principal submatrix of N, or a quotient matrix of N, with eigenvalues λ 1 (M) λ m (M), then the eigenvalues of M interlace those of N, that is λ i (N) λ i (M) λ n m+i (N) for i = 1,..., m. The quotient matrix of N with respect to a partition X 1,..., X m of {1,..., n}, have average row sums of N as entries. A. Munemasa (Tohoku University) Laplacian Eigenvalue JCCA / 9

11 Reduction Assume Γ: graph with n vertices, 1 m n. µ m = d m m + 2. Let S = {x 1,..., x m } be a set of vertices with largest degrees: deg x i = d i (1 i m), d 1 d m d n. A. Munemasa (Tohoku University) Laplacian Eigenvalue JCCA / 9

12 Reduction Assume Γ: graph with n vertices, 1 m n. µ m = d m m + 2. Let S = {x 1,..., x m } be a set of vertices with largest degrees: deg x i = d i (1 i m), d 1 d m d n. Deleting an edge outside S does not change d m, and µ m will not increase: A. Munemasa (Tohoku University) Laplacian Eigenvalue JCCA / 9

13 Reduction Assume Γ: graph with n vertices, 1 m n. µ m (Γ) = d m (Γ) m + 2. Let S = {x 1,..., x m } be a set of vertices with largest degrees: deg x i = d i (1 i m), d 1 d m d n. Deleting an edge outside S does not change d m, and µ m will not increase: µ m (Γ) µ m (Γ ) d m (Γ ) m + 2 = d m (Γ) m + 2. A. Munemasa (Tohoku University) Laplacian Eigenvalue JCCA / 9

14 Reduction Assume Γ: graph with n vertices, 1 m n. µ m (Γ) = d m (Γ) m + 2. Let S = {x 1,..., x m } be a set of vertices with largest degrees: deg x i = d i (1 i m), d 1 d m d n. Deleting an edge outside S does not change d m, and µ m will not increase: µ m (Γ) µ m (Γ ) d m (Γ ) m + 2 = d m (Γ) m + 2. provided Γ K m (n m)k 1. A. Munemasa (Tohoku University) Laplacian Eigenvalue JCCA / 9

15 Reduction Assume Γ: graph with n vertices, 1 m n. µ m (Γ) = d m (Γ) m + 2. Let S = {x 1,..., x m } be a set of vertices with largest degrees: deg x i = d i (1 i m), d 1 d m d n. Deleting an edge outside S does not change d m, and µ m will not increase: µ m (Γ) µ m (Γ ) d m (Γ ) m + 2 = d m (Γ) m + 2. provided Γ K m (n m)k 1. Unless S = K m is a connected component, this reduction works, eventually we reach a graph Γ in which there are no edges outside S. A. Munemasa (Tohoku University) Laplacian Eigenvalue JCCA / 9

16 Γ: graph with n vertices, 1 m n Proposition Assume Γ is edge-minimal subject to d m. If Γ satisfies µ m = d m m + 2 > 0, then one of the following holds: (i) µ m = 1, and Γ is K m with a pending edge attached at a vertex, (ii) µ m 2, and Γ is K m with µ m 1 pending edges attached at each vertex, (iii) m = 2 and Γ = K 2,dm. A. Munemasa (Tohoku University) Laplacian Eigenvalue JCCA / 9

17 Γ: graph with n vertices, 1 m n Proposition Assume Γ is edge-minimal subject to d m. If Γ satisfies µ m = d m m + 2 > 0, then one of the following holds: (i) µ m = 1, and Γ is K m with a pending edge attached at a vertex, (ii) µ m 2, and Γ is K m with µ m 1 pending edges attached at each vertex, (iii) m = 2 and Γ = K 2,dm. K m K m A. Munemasa (Tohoku University) Laplacian Eigenvalue JCCA / 9

18 Γ: graph with n vertices, 1 m n Proposition Assume Γ is edge-minimal subject to d m. If Γ satisfies µ m = d m m + 2 > 0, then one of the following holds: (i) µ m = 1, and Γ is K m with a pending edge attached at a vertex, (ii) µ m 2, and Γ is K m with µ m 1 pending edges attached at each vertex, (iii) m = 2 and Γ = K 2,dm. K m Our contribution for (i): If satisfies µ m = 1 = d m m + 2 > 0, reduces to the case (i) after deleting edges outside S, then is K m with pending edges attached at the same vertex. A. Munemasa (Tohoku University) Laplacian Eigenvalue JCCA / 9

19 Γ: graph with n vertices, 1 m n Proposition Assume Γ is edge-minimal subject to d m. If Γ satisfies µ m = d m m + 2 > 0, then one of the following holds: (i) µ m = 1, and Γ is K m with a pending edge attached at a vertex, (ii) µ m 2, and Γ is K m with µ m 1 pending edges attached at each vertex, (iii) m = 2 and Γ = K 2,dm. K m Our contribution for (i): If satisfies µ m = 1 = d m m + 2 > 0, reduces to the case (i) after deleting edges outside S, then is K m with pending edges attached at the same vertex. Similar result is false for (ii) and (iii). A. Munemasa (Tohoku University) Laplacian Eigenvalue JCCA / 9

20 Γ: graph with n vertices, 1 m n Proposition Assume Γ is edge-minimal subject to d m. If Γ satisfies µ m = d m m + 2 > 0, then one of the following holds: (i) µ m = 1, and Γ is K m with a pending edge attached at a vertex, (ii) µ m 2, and Γ is K m with µ m 1 pending edges attached at each vertex, (iii) m = 2 and Γ = K 2,dm. K m As for the case (ii).... A. Munemasa (Tohoku University) Laplacian Eigenvalue JCCA / 9

21 Case (ii) Γ is K m with µ m 1 pending edges at each vertex Γ K m [ ] (m + µm 1)I J 1 L( ) = I m 1 I m M [ ] (m + µm 1)I J (µ m 1)I m I m M [ ] (µm 1) (µ m 1) 1 1 (2m 2m) (2 2) A. Munemasa (Tohoku University) Laplacian Eigenvalue JCCA / 9

22 Case (ii) Γ is K m with µ m 1 pending edges at each vertex Γ K m [ ] (m + µm 1)I J 1 L( ) = I m 1 I m M [ ] (m + µm 1)I J (µ m 1)I m I m M [ ] (µm 1) (µ m 1) 1 1 (2m 2m) (2 2) A. Munemasa (Tohoku University) Laplacian Eigenvalue JCCA / 9

23 Case (ii) Γ is K m with µ m 1 pending edges at each vertex Γ K m [ ] (m + µm 1)I J 1 L( ) = I m 1 I m M [ ] (m + µm 1)I J (µ m 1)I m I m M [ ] (µm 1) (µ m 1) 1 1 (2m 2m) (2 2) achieves equality = λ 1 (M ) m(µm 1) m 1. A. Munemasa (Tohoku University) Laplacian Eigenvalue JCCA / 9

24 Γ: graph with n vertices, 1 m n Proposition Assume Γ is edge-minimal subject to d m. If Γ satisfies µ m = d m m + 2 > 0, then one of the following holds: (i) µ m = 1, and Γ is K m with a pending edge attached at a vertex, (ii) µ m 2, and Γ is K m with µ m 1 pending edges attached at each vertex, (iii) m = 2 and Γ = K 2,dm. What if µ m = 0? A. Munemasa (Tohoku University) Laplacian Eigenvalue JCCA / 9

25 Γ: graph with n vertices, 1 m n Proposition If µ m = d m m + 2 = 0, then one of the following holds: (i) m = 2 and Γ = nk 1. (ii) m = 3 and Γ = 2K 2 (n 4)K 1. (iii) Γ = (K m tk 2 ) (n m)k 1 for some 0 < t m. 2 A. Munemasa (Tohoku University) Laplacian Eigenvalue JCCA / 9

26 Γ: graph with n vertices, 1 m n Proposition If µ m = d m m + 2 = 0, then one of the following holds: (i) m = 2 and Γ = nk 1. (ii) m = 3 and Γ = 2K 2 (n 4)K 1. (iii) Γ = (K m tk 2 ) (n m)k 1 for some 0 < t m. 2 Thank you for your attention! A. Munemasa (Tohoku University) Laplacian Eigenvalue JCCA / 9