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

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

?? Introduction Eigenvalues Spectrum What is a graph? And why am I speaking about graphs? Samuel Mohr Eigenvalues of Graphs December 20, 2019 2

G V E 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. Samuel Mohr Eigenvalues of Graphs December 20, 2019 3

G V E 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. Samuel Mohr Eigenvalues of Graphs December 20, 2019 3

G V E 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. Samuel Mohr Eigenvalues of Graphs December 20, 2019 3

G V E 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. Samuel Mohr Eigenvalues of Graphs December 20, 2019 3

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

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

Algorithms . Introduction Eigenvalues Spectrum Applications Samuel Mohr Eigenvalues of Graphs December 20, 2019 4

Algorithms . Introduction Eigenvalues Spectrum Applications Samuel Mohr Eigenvalues of Graphs December 20, 2019 4

Algorithms . Introduction Eigenvalues Spectrum Applications Samuel Mohr Eigenvalues of Graphs December 20, 2019 4

Introduction Eigenvalues Spectrum Applications → Algorithms . − Samuel Mohr Eigenvalues of Graphs December 20, 2019 4

Connectedness, Eigenvalues Introduction Eigenvalues Spectrum Theoretical Aspects Structure, Samuel Mohr Eigenvalues of Graphs December 20, 2019 5

Eigenvalues Introduction Eigenvalues Spectrum Theoretical Aspects Structure, Connectedness, Samuel Mohr Eigenvalues of Graphs December 20, 2019 5

Introduction Eigenvalues Spectrum Theoretical Aspects Structure, Connectedness, Eigenvalues Samuel Mohr Eigenvalues of Graphs December 20, 2019 5

is eigenvalue of G if is eigenvalue of A G . Introduction Eigenvalues Spectrum Defjnition Let G = ( V , E ) be a graph, V = { 1 , 2 , . . . , n } . Then A G ∈ R n × n with  1 if { x , y } ∈ E ,  A G ( x , y ) = 0 otherwise ,  is the adjacency matrix of G . Samuel Mohr Eigenvalues of Graphs December 20, 2019 6

Introduction Eigenvalues Spectrum Defjnition Let G = ( V , E ) be a graph, V = { 1 , 2 , . . . , n } . Then A G ∈ R n × n with  1 if { x , y } ∈ E ,  A G ( x , y ) = 0 otherwise ,  is the adjacency matrix of G . λ is eigenvalue of G if λ is eigenvalue of A G . Samuel Mohr Eigenvalues of Graphs December 20, 2019 6

Introduction Eigenvalues Spectrum Theorem of Perron and Frobenius Theorem (Perron-Frobenius) Let A ∈ R n × n ≥ 0 , n ≥ 2 be symmetric and irreducible and let λ 1 ( A ) ≥ λ 2 ( A ) ≥ · · · ≥ λ n ( A ) eigenvalues of A. Then: 1 λ 1 ( A ) is positive, has algebraic multiplicity one, and λ 1 ( A ) ≥ | λ k ( A ) | for k ∈ { 1 , . . . , n } . Furthermore, it is the only eigenvalue that has a eigenvector with only positive entries. � 0 B � 2 λ 1 ( A ) = − λ n ( A ) if and only if A looks like or can be transformed by B T 0 simultaneous row and column changes of A. Samuel Mohr Eigenvalues of Graphs December 20, 2019 7

multiplicity 1, smaller or equal maximum degree, is k if G is k -regular. Introduction Eigenvalues Spectrum Consequences Theorem (Perron-Frobenius) 1 λ 1 ( A ) is positive, has algebraic multiplicity one, and λ 1 ( A ) ≥ | λ k ( A ) | for k ∈ { 1 , . . . , n } . Furthermore, it is the only eigenvalue that has a eigenvector with only positive entries. Largest eigenvalue is: positive, Samuel Mohr Eigenvalues of Graphs December 20, 2019 8

smaller or equal maximum degree, is k if G is k -regular. Introduction Eigenvalues Spectrum Consequences Theorem (Perron-Frobenius) 1 λ 1 ( A ) is positive, has algebraic multiplicity one, and λ 1 ( A ) ≥ | λ k ( 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, Samuel Mohr Eigenvalues of Graphs December 20, 2019 8

is k if G is k -regular. Introduction Eigenvalues Spectrum Consequences Theorem (Perron-Frobenius) 1 λ 1 ( A ) is positive, has algebraic multiplicity one, and λ 1 ( A ) ≥ | λ k ( 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, Samuel Mohr Eigenvalues of Graphs December 20, 2019 8

Introduction Eigenvalues Spectrum Consequences Theorem (Perron-Frobenius) 1 λ 1 ( A ) is positive, has algebraic multiplicity one, and λ 1 ( A ) ≥ | λ k ( 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

no edge between any two vertices from A (respectively B ), G is bipartite. Introduction Eigenvalues Spectrum Consequences Theorem (Perron-Frobenius) � 0 B � 2 λ 1 ( A ) = − λ n ( A ) if and only if A looks like or can be transformed by B T 0 simultaneous row and column changes of A. If λ 1 ( A ) = − λ n ( A ) , then: V = A ˙ ∪ B , Samuel Mohr Eigenvalues of Graphs December 20, 2019 9

G is bipartite. Introduction Eigenvalues Spectrum Consequences Theorem (Perron-Frobenius) � 0 B � 2 λ 1 ( A ) = − λ n ( A ) if and only if A looks like or can be transformed by B T 0 simultaneous row and column changes of A. If λ 1 ( A ) = − λ n ( A ) , then: V = A ˙ ∪ B , no edge between any two vertices from A (respectively B ), Samuel Mohr Eigenvalues of Graphs December 20, 2019 9

Introduction Eigenvalues Spectrum Consequences Theorem (Perron-Frobenius) � 0 B � 2 λ 1 ( A ) = − λ n ( A ) if and only if A looks like or can be transformed by B T 0 simultaneous row and column changes of A. If λ 1 ( A ) = − λ n ( 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

and many other bounds … For regular graphs: A. J. Hofgman, unpublished. n G n 1 n Independence Number G 4. Eigenvalues: 3, 1, and -2. Introduction Eigenvalues Spectrum Independence Number Petersen-Graph: Samuel Mohr Eigenvalues of Graphs December 20, 2019 10

and many other bounds … For regular graphs: A. J. Hofgman, unpublished. n G n 1 n Independence Number G 4. Introduction Eigenvalues Spectrum Independence Number Petersen-Graph: Eigenvalues: 3, 1, and -2. Samuel Mohr Eigenvalues of Graphs December 20, 2019 10

and many other bounds … For regular graphs: A. J. Hofgman, unpublished. n G n 1 n Independence Number G 4. Introduction Eigenvalues Spectrum Independence Number Petersen-Graph: Eigenvalues: 3, 1, and -2. Samuel Mohr Eigenvalues of Graphs December 20, 2019 10

and many other bounds … For regular graphs: A. J. Hofgman, unpublished. n G n 1 n Introduction Eigenvalues Spectrum Independence Number Petersen-Graph: Independence Number α ( G ) = 4. Eigenvalues: 3, 1, and -2. Samuel Mohr Eigenvalues of Graphs December 20, 2019 10

and many other bounds … Introduction Eigenvalues Spectrum Independence Number Petersen-Graph: Independence Number α ( G ) = 4. For regular graphs: A. J. Hofgman, unpublished. − λ n α ( G ) ≤ · n . λ 1 − λ n Eigenvalues: 3, 1, and -2. Samuel Mohr Eigenvalues of Graphs December 20, 2019 10

Introduction Eigenvalues Spectrum Independence Number Petersen-Graph: Independence Number α ( G ) = 4. For regular graphs: A. J. Hofgman, unpublished. − λ n α ( G ) ≤ · n . λ 1 − λ n and many other bounds … Eigenvalues: 3, 1, and -2. Samuel Mohr Eigenvalues of Graphs December 20, 2019 10

Note The choice of 0 (representing equality), 1 (if there is an edge), and 0 (no edge) in A G was rather arbitrary. other representing matrices of graphs. Introduction Eigenvalues Spectrum Eigenvalues of Graphs Observation Eigenvalues of graphs describe its structure: Maximum degree, bipartite?, independence number, many more … Samuel Mohr Eigenvalues of Graphs December 20, 2019 11