Spectral Characterizations of Anti-Regular Graphs Barbara Schweitzer - - PowerPoint PPT Presentation

Spectral Characterizations of Anti-Regular Graphs

Barbara Schweitzer1 Julian Lee, Eric Piato2

• Dr. Cesar Aguilar3

1Presentation Author and Research Assistant 2Research Assistants 3Research Advisor

January 26, 2019

Aguilar, Lee, Piato, Schweitzer (SUNY Geneseo) Spectral Characterizations of Anti-Regular Graphs January 26, 2019

Overview

1

Introduction to Graphs and the Anti-Regular Graph Basic Graph Theory The Adjacency Matrix The Anti-Regular Graph

2

Eigenvalues of the Anti-Regular Graph Analysis of our Findings

3

Future Studies Conjectures

Aguilar, Lee, Piato, Schweitzer (SUNY Geneseo) Spectral Characterizations of Anti-Regular Graphs January 26, 2019

Basic Graph Theory

G = (V , E) with |V | = n = 12 and |E| = m = 14 Vertex 9 is adjacent to vertex 11; 9 ∼ 11 Degree of vertex 9, deg(9) = 4 Isolated vertex, no dominating vertex Degree sequence of the graph: d(G) = (4, 4, 3, 3, 3, 3, 2, 2, 2, 1, 1, 0) Disconnected graph

Aguilar, Lee, Piato, Schweitzer (SUNY Geneseo) Spectral Characterizations of Anti-Regular Graphs January 26, 2019

The Adjacency Matrix

Definition

Given a graph G = (V , E), the adjacency matrix A is the matrix whose (i, j) entry is 1 if vi ∼ vj and 0 otherwise. Adjacency is commutative, so vi ∼ vj also means vj ∼ vi A is then symmetric

Aguilar, Lee, Piato, Schweitzer (SUNY Geneseo) Spectral Characterizations of Anti-Regular Graphs January 26, 2019

The Adjacency Matrix

Consider the graph G = (V , E) The adjacency matrix A of G is A =         1 1 1 1 1 1 1 1 1 1 1 1 1 1        

Aguilar, Lee, Piato, Schweitzer (SUNY Geneseo) Spectral Characterizations of Anti-Regular Graphs January 26, 2019

The Anti-Regular Graph

Definition

A graph G is said to be k-regular if every vertex has degree k.

Definition

The anti-regular graph G is the graph such that only two vertices have the same degree. The anti-regular graph is a type of threshold graph

Definition

A threshold graph is a graph created by repeatedly adding either isolated

• r dominating vertices to a graph.
• 1 and 0, or λ0, are trivial eigenvalues of the anti-regular graph

The interval [−1, 0] was known to be a forbidden interval for the eigenvalues of all threshold graphs

Aguilar, Lee, Piato, Schweitzer (SUNY Geneseo) Spectral Characterizations of Anti-Regular Graphs January 26, 2019

The Anti-Regular Graph

Figure: The anti-regular graph on 1 vertex Figure: The anti-regular graph on 2 vertices

Aguilar, Lee, Piato, Schweitzer (SUNY Geneseo) Spectral Characterizations of Anti-Regular Graphs January 26, 2019

The Anti-Regualr Graph

Figure: The anti-regular graph on 3 vertices Figure: The anti-regular graph on 4 vertices

Aguilar, Lee, Piato, Schweitzer (SUNY Geneseo) Spectral Characterizations of Anti-Regular Graphs January 26, 2019

The Anti-Regualr Graph

Figure: The anti-regular graph on 5 vertices Figure: The anti-regular graph on 6 vertices

Aguilar, Lee, Piato, Schweitzer (SUNY Geneseo) Spectral Characterizations of Anti-Regular Graphs January 26, 2019

The Anti-Regular Graph

Figure: The anti-regular graph on 6 vertices

This graph has adjacency matrix         1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1        

Aguilar, Lee, Piato, Schweitzer (SUNY Geneseo) Spectral Characterizations of Anti-Regular Graphs January 26, 2019

Analysis of our Findings

Theorem

Let n = 2k and let An denote the connected anti-regular graph with n

• vertices. Then λ is an eigenvalue of An if and only if

λ = sin kθ sin kθ + sin (k − 1)θ where θ = arccos

• 1−2λ−2λ2

2λ(λ+1)

• .

Aguilar, Lee, Piato, Schweitzer (SUNY Geneseo) Spectral Characterizations of Anti-Regular Graphs January 26, 2019

Analysis of our Findings

Recall θ = arccos 1 − 2λ − 2λ2 2λ(λ + 1)

• .

By the fact that the domain of arccos is [−1, 1],

Theorem

Let An be the adjacency matrix of the connected anti-regular graph with n

• vertices. The only eigenvalue of An in the interval Ω =
• −1−

√ 2 2

, −1+

√ 2 2

• is

λ0 ∈ {0, −1}.

Aguilar, Lee, Piato, Schweitzer (SUNY Geneseo) Spectral Characterizations of Anti-Regular Graphs January 26, 2019

Analysis of our Findings

Recall λ =

sin kθ sin kθ+sin (k−1)θ and θ = arccos

• 1−2λ−2λ2

2λ(λ+1)

• .

Eigenvalue Equation 1

F(θ) = sin kθ sin kθ + sin (k − 1)θ

Eigenvalue Equation 2

f1(θ) = −(cos θ + 1) +

• (cos θ + 1)(cos θ + 3)

2(cos θ + 1) f2(θ) = −(cos θ + 1) −

• (cos θ + 1)(cos θ + 3)

2(cos θ + 1) .

Aguilar, Lee, Piato, Schweitzer (SUNY Geneseo) Spectral Characterizations of Anti-Regular Graphs January 26, 2019

Analysis of our Findings

Eigenvalues

Positive eigenvalues obtained by f1(θ) = F(θ) Negative eigenvalues obtained by f2(θ) = F(θ)

Singularities

Since F(θ) = sin kθ sin kθ + sin (k − 1)θ, singularities occur where sin kθ + sin(k − 1)θ = 0.

Aguilar, Lee, Piato, Schweitzer (SUNY Geneseo) Spectral Characterizations of Anti-Regular Graphs January 26, 2019

Analysis of our Findings

0.0 0.5 1.0 1.5 2.0 2.5 3.0 10 5 5 10 15 20 25 30

Eigenvalues When k=16

F f1 f2

Singularities occur at θ =

2jπ 2k−1 for j ∈ {1, 2, . . . , k − 1}.

Aguilar, Lee, Piato, Schweitzer (SUNY Geneseo) Spectral Characterizations of Anti-Regular Graphs January 26, 2019

Analysis of our Findings

Singularities at θ =

2jπ 2k−1

Eigenvalues are approximately symmetric about −1

2

Aguilar, Lee, Piato, Schweitzer (SUNY Geneseo) Spectral Characterizations of Anti-Regular Graphs January 26, 2019

Analysis of our Findings

0.0 0.5 1.0 1.5 2.0 2.5 3.0 10 5 5 10 15 20 25 30

Eigenvalues When k=16

F f1 f2 Aguilar, Lee, Piato, Schweitzer (SUNY Geneseo) Spectral Characterizations of Anti-Regular Graphs January 26, 2019

Analysis of our Findings

Singularities at θ =

2jπ 2k−1

Eigenvalues are approximately symmetric about −1

2

As k → ∞, the eigenvalues approach the Ω bound,

• −1−

√ 2 2

, −1+

√ 2 2

• Aguilar, Lee, Piato, Schweitzer (SUNY Geneseo)

Spectral Characterizations of Anti-Regular Graphs January 26, 2019

Analysis of our Findings

0.0 0.5 1.0 1.5 2.0 2.5 3.0 10 5 5 10 15 20 25 30

Eigenvalues When k=16

F f1 f2 Aguilar, Lee, Piato, Schweitzer (SUNY Geneseo) Spectral Characterizations of Anti-Regular Graphs January 26, 2019

Analysis of our Findings

Theorem

Let λ+

1 (k) denote the smallest positive eigenvalue of An and let λ− 1 (k)

denote the negative eigenvalue of An closest to the trivial eigenvalue λ0. The following hold: i) The sequence {λ+

1 (k)}∞ k=1 is strictly decreasing and converges to −1+ √ 2 2

. ii) The sequence {λ−

1 (k)}∞ k=1 is strictly increasing and converges to −1− √ 2 2

. Note that therefore, the interval bound Ω =

• −1−

√ 2 2

, −1+

√ 2 2

• is a

sharp bound.

Aguilar, Lee, Piato, Schweitzer (SUNY Geneseo) Spectral Characterizations of Anti-Regular Graphs January 26, 2019

Conjectures

Conjecture

For any n, the anti-regular graph An has the smallest positive eigenvalue and has the largest non-trivial negative eigenvalue among all threshold graphs on n vertices.

Conjecture

Other than the trivial eigenvalues {0, −1}, the interval Ω =

• −1−

√ 2 2

, −1+

√ 2 2

• does not contain an eigenvalue of any threshold

graph. These conjectures were recently proven in “Eigenvalue-free interval for threshold graphs, arXiv.:1807.10302, 2018” by Ebrahim Ghorbani.

Aguilar, Lee, Piato, Schweitzer (SUNY Geneseo) Spectral Characterizations of Anti-Regular Graphs January 26, 2019

Acknowledgements

Thank you to my fellow research assistants, the National Science Foundation, and our advisor Dr. Cesar Aguilar. Thank you also to the NCUWM for the opportunity to present here today.

Thank you for listening!

SUNY Research Foundation ECCS No. 1700578 SUNY Geneseo NCUWM

Aguilar, Lee, Piato, Schweitzer (SUNY Geneseo) Spectral Characterizations of Anti-Regular Graphs January 26, 2019