Some Results on Distance Regular Graphs, or: How I Spent My Spring Break

Emelie Curl

April 7,2017

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 1 / 1 Definitions Lemmas/Theorems/etc. A Proof Ending comments Works Cited

Overview

Background Information (So much background information including:)

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 2 / 1 Lemmas/Theorems/etc. A Proof Ending comments Works Cited

Overview

Background Information (So much background information including:) Definitions

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 2 / 1 A Proof Ending comments Works Cited

Overview

Background Information (So much background information including:) Definitions Lemmas/Theorems/etc.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 2 / 1 Ending comments Works Cited

Overview

Background Information (So much background information including:) Definitions Lemmas/Theorems/etc. A Proof

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 2 / 1 Works Cited

Overview

Background Information (So much background information including:) Definitions Lemmas/Theorems/etc. A Proof Ending comments

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 2 / 1 Overview

Background Information (So much background information including:) Definitions Lemmas/Theorems/etc. A Proof Ending comments Works Cited

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 2 / 1 Definitions

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 3 / 1 Definitions

For those of you who have taken Design Theory (Math 605), most of this presentation will be review, so you should feel free to “check out” mentally.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 3 / 1 Definitions

All of the graphs considered in this presentation are finite, undirected, and simple.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 3 / 1 Definitions

Definition Suppose Γ is a connected graph with vertex set V (Γ) and edge set E (Γ) where E (Γ) consists of unordered pairs of two adjacent vertices. The distance d(x, y) between any two vertices x and y of Γ is the length of a shortest path connecting x and y in Γ.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 3 / 1 Definitions

Definition We denote v as the number of vertices of Γ and define the diameter D of Γ as the maximum distance in Γ.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 3 / 1 Definitions

Definition

For any vertex x ∈ V (Γ), define Γi (x) to be the graph which consists of vertices that are at distance precisely i from x where 0 ≤ i ≤ D. In addition, we note that Γ−1(x) = ΓD+1(x) = ∅.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 3 / 1 Boring: More Definitions!

Definition A connected graph Γ with diameter D is called distance − regular if there are integers ai , bi , ci for 0 ≤ i ≤ D such that for any two vertices x, y ∈ V (Γ) with d(x, y) = i:

(i) there are ci neighbors of y in Γi−1(x) i.e. there are ci vertices of distance i − 1 from x and distance 1 from y in Γ;

(ii) there are bi neighbors of y in Γi+1(x) i.e. there are bi vertices of distance i + 1 from x and distance 1 from y in Γ;

(iii) there are ai neighbors of y in Γi (x) i.e. there are ai vertices of distance i from x and distance 1 from y in Γ.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 4 / 1 Boring: More Definitions!

Definition

The parameters ai , bi−1, and ci (1 ≤ i ≤ D) are called the intersection numbers. Note that a0 = bD = c0 = 0, and c1 = 1 and any distance-regular graph is k−regular such that k = b0.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 4 / 1 Example

Now, here’s an example of such a graph.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 5 / 1 Example

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 5 / 1 Example

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 5 / 1 Example

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 5 / 1 Example

Note that a connected strongly regular graph is just a distance-regular graph with diameter two.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 5 / 1 A bunch of parameters and relations

Now, here’s a ton of parameters and relations related to distance-regular graphs!

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 6 / 1 A bunch of parameters and relations

(i) Lemma

Note that ai = k − bi − ci where ai = |Γ(y) ∩ Γi (x)| holds for any two vertices x, y with d(x, y) = i for 0 ≤ i ≤ D.

I submitted verifying this lemma as a question for the exam.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 6 / 1 A bunch of parameters and relations

(ii) Definition

For a distance-regular graph Γ and a vertex x ∈ V (Γ), we let ki = |Γi (x)| be the ith valency of Γ. We have k0 = 1,k1 = k, and ki 6= 0 for all 0 ≤ i ≤ D.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 6 / 1 A bunch of parameters and relations

(iii) Lemma

The total number of vertices is v = 1 + k1 + ... + ki + ... + kD .

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 6 / 1 A bunch of parameters and relations

(iv) Definition i Let pjh = |{w|w ∈ Γj (x) ∩ Γh(y)}| for any y ∈ Γi (x) with 0 ≤ i, j, h ≤ D (also called intersection numbers). That is, d(x, y) = i, d(x, w) = j, d(y, w) = h.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 6 / 1 A bunch of parameters and relations

(v) Lemma i i i Also, ci = pi−1,1 for 1 ≤ i ≤ D, ai = pi,1 for 0 ≤ i ≤ D, bi = pi+1,1 for 0 ≤ i ≤ D − 1.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 6 / 1 A bunch of parameters and relations

Definition

The array {b0, b1, ... , bD−1; c1, c2, ... , cD } is called the intersection array of Γ.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 6 / 1 Another Question?!

Also, Possibly a good (easy) question for the final is to show the Petersen graph is determined as a distance-regular graph by its intersection array.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 7 / 1 Still More Definitions

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 8 / 1 Still More Definitions

Definition Suppose that Γ is a distance-regular graph with valency k ≥ 2 and diameter D ≥ 2. Then, we let Ai be the matrix of Γ such that the rows and the columns of Ai are indexed by the vertices of Γ and the (x, y)−entry is 1 whenever x and y are at a distance i and 0 otherwise. We will denote the adjacency matrix of Γ as A instead of A1. The eigenvalues of the graph Γ are the eigenvalues of A.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 8 / 1 Still More Definitions

Definition Suppose that Γ is a distance-regular graph with valency k ≥ 2 and diameter D ≥ 2. Then, we let Ai be the matrix of Γ such that the rows and the columns of Ai are indexed by the vertices of Γ and the (x, y)−entry is 1 whenever x and y are at a distance i and 0 otherwise. We will denote the adjacency matrix of Γ as A instead of A1. The eigenvalues of the graph Γ are the eigenvalues of A. Duh.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 8 / 1 Still More Definitions

Definition

Note that the matrices Ai satisfy the relations:

(i) A0 = I , A1 = A;

(ii) AAi = ci+1Ai+1 + ai Ai + bi−1Ai−1 for i = 0, ... , D;

(iii) A0 + A1 + ... + AD = J;

(iv) Also, we have A−1 = AD+1 = 0.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 8 / 1 Proof Sketch

To see that AAi = ci+1Ai+1 + ai Ai + bi−1Ai−1 for 1 ≤ i ≤ D : Fix vertices x, y ∈ Γ.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 9 / 1 Proof Sketch

To see that AAi = ci+1Ai+1 + ai Ai + bi−1Ai−1 for 1 ≤ i ≤ D : Fix vertices x, y ∈ Γ. By matrix multiplication, the x, y entry of the left-hand side of this equal is equal to |{z ∈ Γ : d(x, z) = 1, d(y, z) = i}|.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 9 / 1 Proof Sketch

To see that AAi = ci+1Ai+1 + ai Ai + bi−1Ai−1 for 1 ≤ i ≤ D : Fix vertices x, y ∈ Γ. By matrix multiplication, the x, y entry of the left-hand side of this equal is equal to |{z ∈ Γ : d(x, z) = 1, d(y, z) = i}|. Then, by definition of distance matrix Ai and parameters ci , ai , bi , the x, y entry of the right-hand side is also equal to |{z ∈ Γ : d(x, z) = 1, d(y, z) = i}|.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 9 / 1 Feel like More Parameters?

Definition

The matrices Ai can be written as polynomials in A of degree i i.e. Ai = νi (A) for i = 0, ... , D + 1 where νi are polynomials of degree i defined recursively by:

ν−1(x) = 0;

ν0(x) = 1;

ν1(x) = x;

ci+1νi+1(x) = (x − ai )νi (x) − bi−1νi−1(x) for i = 0, ... , D.

Note that νD+1(A) = AD+1 = 0.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril 7,2017 Break 10 / 1 A few more preliminaries

Definition Substituting any eigenvalue θ of A into the previous relation, we have θνi (θ) = ci+1νi+1(θ) + ai νi (θ) + bi−1νi−1(θ) for i = 0, ... , D where ν−1(θ) = νD+1(θ) = 0, and b−1 and cD+1 are unspecified.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril 7,2017 Break 11 / 1 A few more preliminaries

We all know the valency of k is a trivial eigenvalue of A. Using the above, we have νi (k) = ki and this motivates the introduction of the scaled νi (θ) quantities ui (θ) = for i = −1, ... , D + 1. ki

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril 7,2017 Break 11 / 1 A few more preliminaries

Definition

For an eigenvalue θ of Γ, the sequence (ui )i=0,1,...,D = (ui (θ))i=0,1,...,D θ satisfying u0 = u0(θ) = 1, u1 = u1(θ) = k , ci ui−1 + ai ui + bi ui+1 = θui (i = 1, 2, ... , D − 1), and cD uD−1 + aD uD = θuD is called the standard sequence corresponding to the eigenvalue θ.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril 7,2017 Break 11 / 1 A few more preliminaries

The above equation implies that the eigenvalues of A are just the eigenvalues of the following tridiagonal (D + 1) × (D + 1) matrix (called the intersection matrix):   a0 b0 c1 a1 b1     c2 ··  L1 =   .  ···     ·· bD−1 cD aD

T Note that the vector u = [u0(θ), ... , uD (θ)] is a right eigenvector of L1 corresponding to the eigenvalue θ.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril 7,2017 Break 11 / 1 A few more preliminaries

Especially for hand calculation, it is worthwhile to know that the eigenvalues of L1 distinct from eigenvalue k are just the eigenvalues of the tridiagonal D × D matrix:   −c1 b1  c1 k − b1 − c2 b2     c2 ··  T =    ·     · bD−1  cD−1 k − bD−1 − cD

Note that the number ti = ui−1(θ) − ui (θ) for i = 1, ... , D gives the T vector t = (t1, ... , tD ) which is an eigenvector associated eigenvalue θ: T t = θt.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril 7,2017 Break 11 / 1 Yeah! The Interlacing Theorem

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril 7,2017 Break 12 / 1 Yeah! The Interlacing Theorem

Theorem Let m ≤ n be two positive integers. Let A be an n × n matrix, that is similar to a real symmetric matrix, and let B be a principal m × m submatrix of A. Then, for i = 1, ... , m, θn−m+i (A) ≤ θi (B) ≤ θi (A) holds, where A has eigenvalues θ1(A) ≥ θ2(A) ≥ ... ≥ θn(A) and B has eigenvalues θ1(B) ≥ θ2(B) ≥ ... ≥ θm(B).

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril 7,2017 Break 12 / 1 The moment has come!

Now, I’m supposed to prove something right?

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril 7,2017 Break 13 / 1 The moment has come!

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril 7,2017 Break 13 / 1 The moment has come!

Lemma Let Γ be a distance-regular graph with diameter 3 and distinct eigenvalues k = θ0 > θ1 > θ2 > θ3. If a3 = 0, then θ1 > 0 > −1 ≥ θ2 ≥ −b2 ≥ θ3.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril 7,2017 Break 13 / 1 Baby Proof?

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril 7,2017 Break 14 / 1 proof of that one lemma

We have that a3 = 0 and we know that θ0 = k, θ1, θ2, and θ3 are the eigenvalues of Γ.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril 7,2017 Break 15 / 1 proof of that one lemma

We have that a3 = 0 and we know that θ0 = k, θ1, θ2, and θ3 are the eigenvalues of Γ. So, θ1, θ2, and θ3 are the eigenvalues of the matrix:   −1 b1 0 T =  1 k − b1 − c2 b2  . 0 c2 −b2

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril 7,2017 Break 15 / 1 proof of that one lemma

We have that a3 = 0 and we know that θ0 = k, θ1, θ2, and θ3 are the eigenvalues of Γ. So, θ1, θ2, and θ3 are the eigenvalues of the matrix:   −1 b1 0 T =  1 k − b1 − c2 b2  . 0 c2 −b2

Since the principal submatrix

−1 0  0 −b2

of T has eigenvalues −1 and −b2, it follows that the inequality θ1 ≥ −1 ≥ θ2 ≥ −b2 ≥ θ3 holds by the interlacing theorem.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril 7,2017 Break 15 / 1 proof of that one lemma

We have that a3 = 0 and we know that θ0 = k, θ1, θ2, and θ3 are the eigenvalues of Γ. So, θ1, θ2, and θ3 are the eigenvalues of the matrix:   −1 b1 0 T =  1 k − b1 − c2 b2  . 0 c2 −b2

Since the principal submatrix

−1 0  0 −b2

of T has eigenvalues −1 and −b2, it follows that the inequality θ1 ≥ −1 ≥ θ2 ≥ −b2 ≥ θ3 holds by the interlacing theorem. Since Γ has an induced path P of length three, θ1 ≥ second largest eigenvalue of P, which is greater than zero. 

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril 7,2017 Break 15 / 1 Other results?

For a distance-regular graph with diameter D ≥ 4, we have θ2 ≥ 0 by the interlacing theorem.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril 7,2017 Break 16 / 1 Other results?

Let G be a regular graph with diameter three and four distinct eigenvalues λ0 > λ1 > λ2 > λ3. Then, the following inequalities holds:

λ0 λ0 λ3 ≤ − < λ2 < − ≤ λ1 < λ0. λ1 λ3

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril 7,2017 Break 16 / 1 Other results?

Let G be a regular graph with diameter three and four distinct eigenvalues λ0 > λ1 > λ2 > λ3. Then, the following inequalities holds:

λ0 λ0 λ3 ≤ − < λ2 < − ≤ λ1 < λ0. λ1 λ3 This latter result is applicable as distance-regular graphs with diameter 3 fall under the conditions of this result.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril 7,2017 Break 16 / 1 Other results?

There were several others in this paper, but the proofs were way too long and complicated, so I did not include them in this presentation.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril 7,2017 Break 16 / 1 Possible Questions

1 Show that a connected strongly regular graph is a distance - regular graph with diameter two. 2 Show that the Petersen graph is determined as a distance-regular graph by its intersection array. 3 Verify that for a distance-regular graph Γ with diameter D, k = ci + ai + bi for 0 ≤ i ≤ D where k is the regularity of Γ.

4 Note that the hypercube Q3 is a distance-regular graph with diameter 3. Determine the intersection matrix for this graph.

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril 7,2017 Break 17 / 1 References

A.E. Brouwer, A.M. Cohen, A. Neumaier, Distance-Regular Graphs, Springer-Verlag, Berlin, 1989. Koolen, Jack H., and Jongyook Park. ”Distance-regular graphs with a1 or c2 at least half the valency.” Journal of Combinatorial Theory, Series A 119.3 (2012): 546-555. M.A. Fiol, E. Garriga, On the spectrum of an extremal graph with four eigenvalues, Discrete Math. 306 (2006) 22412244. My notes from Design Theory

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril 7,2017 Break 18 / 1 THANK YOU!!!!

Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril 7,2017 Break 19 / 1