Eigenvalues of Graphs Operator Theory and Krein Spaces (dedicated - - PowerPoint PPT Presentation

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Eigenvalues of Graphs Operator Theory and Krein Spaces (dedicated - - PowerPoint PPT Presentation

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Eigenvalues of Graphs Operator Theory and Krein Spaces (dedicated to the memory of Hagen Neidhardt) Samuel Mohr December 20, 2019 Institut fr Mathematik Technische Universitt Ilmenau Gefrdert durch die Deutsche Forschungsgemeinschaft

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Eigenvalues of Graphs

Operator Theory and Krein Spaces (dedicated to the memory of Hagen Neidhardt)

Samuel Mohr December 20, 2019

Institut für Mathematik Technische Universität Ilmenau Gefördert durch die Deutsche Forschungsgemeinschaft (DFG) – Projektnummer 327533333

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Introduction Eigenvalues Spectrum

??

What is a graph? And why am I speaking about graphs?

Samuel Mohr Eigenvalues of Graphs December 20, 2019 2

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Introduction Eigenvalues Spectrum

Graphs

A graph is a combinatorial object consisting of a fjnite set of vertices and links, which connect ver- tices and are called edges.

G V E

Samuel Mohr Eigenvalues of Graphs December 20, 2019 3

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Introduction Eigenvalues Spectrum

Graphs

A graph is a combinatorial object consisting of a fjnite set of vertices and links, which connect ver- tices and are called edges.

G V E

Samuel Mohr Eigenvalues of Graphs December 20, 2019 3

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Introduction Eigenvalues Spectrum

Graphs

A graph is a combinatorial object consisting of a fjnite set of vertices and links, which connect ver- tices and are called edges.

G V E

Samuel Mohr Eigenvalues of Graphs December 20, 2019 3

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Introduction Eigenvalues Spectrum

Graphs

A graph is a combinatorial object consisting of a fjnite set of vertices and links, which connect ver- tices and are called edges.

G V E

Samuel Mohr Eigenvalues of Graphs December 20, 2019 3

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Introduction Eigenvalues Spectrum

Graphs

A graph is a combinatorial object consisting of a fjnite set of vertices and links, which connect ver- tices and are called edges.

G = (V, E)

Samuel Mohr Eigenvalues of Graphs December 20, 2019 3

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Introduction Eigenvalues Spectrum

Graphs

A graph is a combinatorial object consisting of a fjnite set of vertices and links, which connect ver- tices and are called edges.

G = (V, E)

Samuel Mohr Eigenvalues of Graphs December 20, 2019 3

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Introduction Eigenvalues Spectrum

Applications

Algorithms.

Samuel Mohr Eigenvalues of Graphs December 20, 2019 4

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Introduction Eigenvalues Spectrum

Applications

Algorithms.

Samuel Mohr Eigenvalues of Graphs December 20, 2019 4

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Introduction Eigenvalues Spectrum

Applications

Algorithms.

Samuel Mohr Eigenvalues of Graphs December 20, 2019 4

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Introduction Eigenvalues Spectrum

Applications

− → Algorithms.

Samuel Mohr Eigenvalues of Graphs December 20, 2019 4

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Introduction Eigenvalues Spectrum

Theoretical Aspects

Structure, Connectedness, Eigenvalues

Samuel Mohr Eigenvalues of Graphs December 20, 2019 5

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Introduction Eigenvalues Spectrum

Theoretical Aspects

Structure, Connectedness, Eigenvalues

Samuel Mohr Eigenvalues of Graphs December 20, 2019 5

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Introduction Eigenvalues Spectrum

Theoretical Aspects

Structure, Connectedness, Eigenvalues

Samuel Mohr Eigenvalues of Graphs December 20, 2019 5

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Introduction Eigenvalues Spectrum

Defjnition

Let G = (V, E) be a graph, V = {1, 2, . . . , n}. Then AG ∈ Rn×n with AG(x, y) =    1 if {x, y} ∈ E,

  • therwise,

is the adjacency matrix of G. is eigenvalue of G if is eigenvalue of AG.

Samuel Mohr Eigenvalues of Graphs December 20, 2019 6

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Introduction Eigenvalues Spectrum

Defjnition

Let G = (V, E) be a graph, V = {1, 2, . . . , n}. Then AG ∈ Rn×n with AG(x, y) =    1 if {x, y} ∈ E,

  • therwise,

is the adjacency matrix of G. λ is eigenvalue of G if λ is eigenvalue of AG.

Samuel Mohr Eigenvalues of Graphs December 20, 2019 6

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Introduction Eigenvalues Spectrum

Theorem of Perron and Frobenius

Theorem (Perron-Frobenius) Let A ∈ Rn×n

≥0 , n ≥ 2 be symmetric and irreducible and let

λ1(A) ≥ λ2(A) ≥ · · · ≥ λn(A) eigenvalues of A. Then:

1 λ1(A) is positive, has algebraic multiplicity one, and λ1(A) ≥ |λk(A)| for

k ∈ {1, . . . , n}. Furthermore, it is the only eigenvalue that has a eigenvector with only positive entries.

2 λ1(A) = −λn(A) if and only if A looks like

0 B

BT 0

  • r can be transformed by

simultaneous row and column changes of A.

Samuel Mohr Eigenvalues of Graphs December 20, 2019 7

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Introduction Eigenvalues Spectrum

Consequences

Theorem (Perron-Frobenius)

1 λ1(A) is positive, has algebraic multiplicity one, and λ1(A) ≥ |λk(A)| for

k ∈ {1, . . . , n}. Furthermore, it is the only eigenvalue that has a eigenvector with only positive entries. Largest eigenvalue is: positive, multiplicity 1, smaller or equal maximum degree, is k if G is k-regular.

Samuel Mohr Eigenvalues of Graphs December 20, 2019 8

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Introduction Eigenvalues Spectrum

Consequences

Theorem (Perron-Frobenius)

1 λ1(A) is positive, has algebraic multiplicity one, and λ1(A) ≥ |λk(A)| for

k ∈ {1, . . . , n}. Furthermore, it is the only eigenvalue that has a eigenvector with only positive entries. Largest eigenvalue is: positive, multiplicity 1, smaller or equal maximum degree, is k if G is k-regular.

Samuel Mohr Eigenvalues of Graphs December 20, 2019 8

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Introduction Eigenvalues Spectrum

Consequences

Theorem (Perron-Frobenius)

1 λ1(A) is positive, has algebraic multiplicity one, and λ1(A) ≥ |λk(A)| for

k ∈ {1, . . . , n}. Furthermore, it is the only eigenvalue that has a eigenvector with only positive entries. Largest eigenvalue is: positive, multiplicity 1, smaller or equal maximum degree, is k if G is k-regular.

Samuel Mohr Eigenvalues of Graphs December 20, 2019 8

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Introduction Eigenvalues Spectrum

Consequences

Theorem (Perron-Frobenius)

1 λ1(A) is positive, has algebraic multiplicity one, and λ1(A) ≥ |λk(A)| for

k ∈ {1, . . . , n}. Furthermore, it is the only eigenvalue that has a eigenvector with only positive entries. Largest eigenvalue is: positive, multiplicity 1, smaller or equal maximum degree, is k if G is k-regular.

Samuel Mohr Eigenvalues of Graphs December 20, 2019 8

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Introduction Eigenvalues Spectrum

Consequences

Theorem (Perron-Frobenius)

2 λ1(A) = −λn(A) if and only if A looks like

0 B

BT 0

  • r can be transformed by

simultaneous row and column changes of A. If λ1(A) = −λn(A), then: V = A ˙ ∪ B, no edge between any two vertices from A (respectively B), G is bipartite.

Samuel Mohr Eigenvalues of Graphs December 20, 2019 9

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Introduction Eigenvalues Spectrum

Consequences

Theorem (Perron-Frobenius)

2 λ1(A) = −λn(A) if and only if A looks like

0 B

BT 0

  • r can be transformed by

simultaneous row and column changes of A. If λ1(A) = −λn(A), then: V = A ˙ ∪ B, no edge between any two vertices from A (respectively B), G is bipartite.

Samuel Mohr Eigenvalues of Graphs December 20, 2019 9

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Introduction Eigenvalues Spectrum

Consequences

Theorem (Perron-Frobenius)

2 λ1(A) = −λn(A) if and only if A looks like

0 B

BT 0

  • r can be transformed by

simultaneous row and column changes of A. If λ1(A) = −λn(A), then: V = A ˙ ∪ B, no edge between any two vertices from A (respectively B), G is bipartite.

Samuel Mohr Eigenvalues of Graphs December 20, 2019 9

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Introduction Eigenvalues Spectrum

Independence Number

Petersen-Graph: Independence Number G 4. For regular graphs:

  • A. J. Hofgman, unpublished.

G

n 1 n

n and many other bounds … Eigenvalues: 3, 1, and -2.

Samuel Mohr Eigenvalues of Graphs December 20, 2019 10

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Introduction Eigenvalues Spectrum

Independence Number

Petersen-Graph: Independence Number G 4. For regular graphs:

  • A. J. Hofgman, unpublished.

G

n 1 n

n and many other bounds … Eigenvalues: 3, 1, and -2.

Samuel Mohr Eigenvalues of Graphs December 20, 2019 10

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Introduction Eigenvalues Spectrum

Independence Number

Petersen-Graph: Independence Number G 4. For regular graphs:

  • A. J. Hofgman, unpublished.

G

n 1 n

n and many other bounds … Eigenvalues: 3, 1, and -2.

Samuel Mohr Eigenvalues of Graphs December 20, 2019 10

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Introduction Eigenvalues Spectrum

Independence Number

Petersen-Graph: Independence Number α(G) = 4. For regular graphs:

  • A. J. Hofgman, unpublished.

G

n 1 n

n and many other bounds … Eigenvalues: 3, 1, and -2.

Samuel Mohr Eigenvalues of Graphs December 20, 2019 10

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Introduction Eigenvalues Spectrum

Independence Number

Petersen-Graph: Independence Number α(G) = 4. For regular graphs:

  • A. J. Hofgman, unpublished.

α(G) ≤ −λn λ1 − λn · n. and many other bounds … Eigenvalues: 3, 1, and -2.

Samuel Mohr Eigenvalues of Graphs December 20, 2019 10

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Introduction Eigenvalues Spectrum

Independence Number

Petersen-Graph: Independence Number α(G) = 4. For regular graphs:

  • A. J. Hofgman, unpublished.

α(G) ≤ −λn λ1 − λn · n. and many other bounds … Eigenvalues: 3, 1, and -2.

Samuel Mohr Eigenvalues of Graphs December 20, 2019 10

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Introduction Eigenvalues Spectrum

Eigenvalues of Graphs

Observation Eigenvalues of graphs describe its structure: Maximum degree, bipartite?, independence number, many more … Note The choice of 0 (representing equality), 1 (if there is an edge), and 0 (no edge) in AG was rather arbitrary.

  • ther representing matrices of graphs.

Samuel Mohr Eigenvalues of Graphs December 20, 2019 11

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Introduction Eigenvalues Spectrum

Eigenvalues of Graphs

Observation Eigenvalues of graphs describe its structure: Maximum degree, bipartite?, independence number, many more … Note The choice of 0 (representing equality), 1 (if there is an edge), and 0 (no edge) in AG was rather arbitrary. − → other representing matrices of graphs.

Samuel Mohr Eigenvalues of Graphs December 20, 2019 11

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Introduction Eigenvalues Spectrum

Defjnition

The spectrum of a graph G is the spectrum of its adjacency matrix AG, that is, its set of eigenvalues together with their multiplicities.

Samuel Mohr Eigenvalues of Graphs December 20, 2019 12

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Introduction Eigenvalues Spectrum

Examples

Petersen-Graph:

0 1 0 0 1 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 1 0 1 1 0 0

Eigenvalues: 3 1 1 1 1 1

5 times

2 2 2 2

4 times

Samuel Mohr Eigenvalues of Graphs December 20, 2019 13

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Introduction Eigenvalues Spectrum

Examples

Petersen-Graph:

     

0 1 0 0 1 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 1 0 1 1 0 0

     

Eigenvalues: 3 1 1 1 1 1

5 times

2 2 2 2

4 times

Samuel Mohr Eigenvalues of Graphs December 20, 2019 13

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Introduction Eigenvalues Spectrum

Examples

Petersen-Graph:

     

0 1 0 0 1 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 1 0 1 1 0 0

     

Eigenvalues: 3, 1, 1, 1, 1, 1

  • 5 times

, −2, −2, −2, −2

  • 4 times

Samuel Mohr Eigenvalues of Graphs December 20, 2019 13

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Introduction Eigenvalues Spectrum

Examples

Petersen-Graph:

     

0 1 0 0 1 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 1 0 1 1 0 0

     

Eigenvalues: 3, 1, 1, 1, 1, 1

  • 5 times

, −2, −2, −2, −2

  • 4 times

Samuel Mohr Eigenvalues of Graphs December 20, 2019 13

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Introduction Eigenvalues Spectrum

Examples

Petersen-Graph:

     

0 1 0 0 1 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 1 0 1 1 0 0

     

Eigenvalues: 3.2, 1.3, 1, 1, 1, 0.4

  • 1’s

, −1.6, −2, −2, −2.4

  • 2’s

Samuel Mohr Eigenvalues of Graphs December 20, 2019 13

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Introduction Eigenvalues Spectrum

Examples

Petersen-Graph:

     

0 1 0 0 1 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0 1 1 1 0 0 1 0 0 1 1 0 0 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 1 0 1 1 0 0

     

Eigenvalues: 3.2, 1.3, 1, 1, 1, 0.4

  • 1’s

, −1.6, −2, −2, −2.4

  • 2’s

Samuel Mohr Eigenvalues of Graphs December 20, 2019 13

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Introduction Eigenvalues Spectrum

Examples

Petersen-Graph:

     

0 1 0 0 1 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0 1 1 1 0 0 1 0 0 1 1 0 0 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 1 0 1 1 0 0

     

Eigenvalues: 3.6, 1.4, 1, 1, 1, 0

  • 1’s

, −1.7, −2, −2, −2.3

  • 2’s

Samuel Mohr Eigenvalues of Graphs December 20, 2019 13

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Introduction Eigenvalues Spectrum

Interlacing

Two sequences of real numbers λ1 ≥ λ2 ≥ · · · ≥ λn and η1 ≥ η2 ≥ · · · ≥ ηm with n ≥ m interlace if λi ≥ ηi ≥ λn−m+i. Theorem (see Haemers – 1995) Let G be a graph and H be obtained from G by removing

  • vertices. Then the spectra of AG and AH interlace.

Samuel Mohr Eigenvalues of Graphs December 20, 2019 14

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Introduction Eigenvalues Spectrum

Interlacing

Two sequences of real numbers λ1 ≥ λ2 ≥ · · · ≥ λn and η1 ≥ η2 ≥ · · · ≥ ηm with n ≥ m interlace if λi ≥ ηi ≥ λn−m+i. Theorem (see Haemers – 1995) Let G be a graph and H be obtained from G by removing

  • vertices. Then the spectra of AG and AH interlace.

Samuel Mohr Eigenvalues of Graphs December 20, 2019 14

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Introduction Eigenvalues Spectrum

??

Is a graph uniquely determined by its spectrum?

Samuel Mohr Eigenvalues of Graphs December 20, 2019 15

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Introduction Eigenvalues Spectrum

Cospectral Graphs

Eigenvalues: −2, 0, 0, 0, 2. Eigenvalues: 2 0 0 0 2.

Samuel Mohr Eigenvalues of Graphs December 20, 2019 16

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Introduction Eigenvalues Spectrum

Cospectral Graphs

Eigenvalues: −2, 0, 0, 0, 2. Eigenvalues: −2, 0, 0, 0, 2.

Samuel Mohr Eigenvalues of Graphs December 20, 2019 16

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Introduction Eigenvalues Spectrum

Connected Cospectral Graphs

Eigenvalues: ∼2.7, 1, ∼0.2, −1, −1, ∼−1.9. Eigenvalues: ∼2.7, 1, ∼0.2, −1, −1, ∼−1.9.

Samuel Mohr Eigenvalues of Graphs December 20, 2019 17

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Introduction Eigenvalues Spectrum

Problem

Observation Graphs are not uniquely determined by their spectra. Conjecture (Van Dam, Haemers – 2009) Almost every graph is determined by its spectrum. n 1-4 5 6 7 8 9 10 11 12 ratio 0.0 0.095 0.064 0.105 0.139 0.186 0.213 0.211 0.188

Samuel Mohr Eigenvalues of Graphs December 20, 2019 18

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Introduction Eigenvalues Spectrum

Problem

Observation Graphs are not uniquely determined by their spectra. Conjecture (Van Dam, Haemers – 2009) Almost every graph is determined by its spectrum. n 1-4 5 6 7 8 9 10 11 12 ratio 0.0 0.095 0.064 0.105 0.139 0.186 0.213 0.211 0.188

Samuel Mohr Eigenvalues of Graphs December 20, 2019 18

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Introduction Eigenvalues Spectrum

Problem

Observation Graphs are not uniquely determined by their spectra. Conjecture (Van Dam, Haemers – 2009) Almost every graph is determined by its spectrum. n 1-4 5 6 7 8 9 10 11 12 ratio 0.0 0.095 0.064 0.105 0.139 0.186 0.213 0.211 0.188

Samuel Mohr Eigenvalues of Graphs December 20, 2019 18

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Introduction Eigenvalues Spectrum

!!

Many thanks for listening!

Samuel Mohr Eigenvalues of Graphs December 20, 2019 19