T HE S TANDARD S EQUENCES OF A DRG T HE STANDARD SEQUENCE OF AN - - PowerPoint PPT Presentation

ON THE CONNECTIVITY OF THE COMPLEMENT OF A

BALL IN DISTANCE-REGULAR GRAPHS

Sebastian M. Cioab˘ a

Department of Mathematical Sciences University of Delaware Newark, DE 19716-2553, USA cioaba@math.udel.edu

Modern Algebraic Graph Theory Villanova University June 2014

SEBI CIOAB ˘

A (UNIV. OF DELAWARE)

CONNECTIVITY PROBLEMS IN DRGS 1 / 19

SOME DEFINITIONS

FIGURE: Not a ball

SEBI CIOAB ˘

A (UNIV. OF DELAWARE)

CONNECTIVITY PROBLEMS IN DRGS 2 / 19

SOME DEFINITIONS

DEFINITION A ball is a solid or hollow sphere or ovoid, especially one that is kicked, thrown, or hit in a game. a soccer ball synonyms: sphere.

FIGURE: This is a ball.

SEBI CIOAB ˘

A (UNIV. OF DELAWARE)

CONNECTIVITY PROBLEMS IN DRGS 3 / 19

STRONGLY REGULAR GRAPHS

DEFINITION A graph G is called a (v, k, λ, µ)-strongly regular (or (v, k, λ, µ)-SRG) if

1

it has v vertices

2

it is k-regular

3

any two adjacent vertices have exactly λ common neighbors

4

any two distinct non-adjacent vertices have exactly µ common neighbors

FIGURE: The Petersen graph is a (10, 3, 0, 1)-SRG.

SEBI CIOAB ˘

A (UNIV. OF DELAWARE)

CONNECTIVITY PROBLEMS IN DRGS 4 / 19

SUBCONSTITUENTS OF STRONGLY REGULAR GRAPHS

DEFINITION The i-th subconstituent Γi(x) of a vertex x of a graph Γ is the subgraph of Γ induced by the vertices at distance i from x.

1 2 3 4 5 6 7 8 9 10

• FIGURE: The subconstituents of the Petersen graph

SEBI CIOAB ˘

A (UNIV. OF DELAWARE)

CONNECTIVITY PROBLEMS IN DRGS 5 / 19

SUBCONSTITUENTS OF STRONGLY REGULAR GRAPHS

CAMERON, GOETHALS, SEIDEL 1978 Strongly regular graphs having strongly regular subconstituents. GARDINER, GODSIL, HENSEL, ROYLE 1992 Second neighborhoods of strongly regular graphs.

SEBI CIOAB ˘

A (UNIV. OF DELAWARE)

CONNECTIVITY PROBLEMS IN DRGS 6 / 19

SUBCONSTITUENTS OF STRONGLY REGULAR GRAPHS

THEOREM If x is any vertex of a primitive strongly regular graph Γ, then the second subconstituent Γ2(x) is connected. PROOF(S) Combinatorial Proof: Gardiner, Godsil, Hensel, Royle 1992 Algebraic Proof: Haemers.

SEBI CIOAB ˘

A (UNIV. OF DELAWARE)

CONNECTIVITY PROBLEMS IN DRGS 7 / 19

A QUESTION OF BROUWER

FIGURE: Joe Hemmeter, Andries Brouwer and Andy Woldar

SEBI CIOAB ˘

A (UNIV. OF DELAWARE)

CONNECTIVITY PROBLEMS IN DRGS 8 / 19

A QUESTION OF BROUWER

THEOREM If x is any vertex of a primitive strongly regular graph Γ, then the second subconstituent Γ2(x) is connected. QUESTION (BROUWER, GAC5, 2011) Andries Brouwer asked whether this could be generalized to a statement that for general distance-regular Γ and suitable t, the subgraph Γ≥t(x) is connected, where Γ≥t(x) is the subgraph of Γ induced by the vertices of distance at least t from x .

SEBI CIOAB ˘

A (UNIV. OF DELAWARE)

CONNECTIVITY PROBLEMS IN DRGS 9 / 19

DISTANCE-REGULAR GRAPHS

DEFINITION A graph Γ with diameter D is called distance-regular if there are integers bi, ci(0 ≤ i ≤ D) such that for any two vertices x, y of Γ at distance i, there are precisely ci neighbors of y in Γi−1(y) and bi neighbors in Γi+1(x). Let k := b0, ai := k − bi − ci for 0 ≤ i ≤ D. THE QUOTIENT MATRIX OF THE DISTANCE PARTITION L =           a0 b0 c1 a1 b1 c2 a2 b2 . . . . . . cD−1 aD−1 bD−1 cD aD          

SEBI CIOAB ˘

A (UNIV. OF DELAWARE)

CONNECTIVITY PROBLEMS IN DRGS 10 / 19

THE SPECTRUM OF A DISTANCE-REGULAR GRAPH

THE EIGENVALUES OF A DRG If Γ is a DRG of diameter D, then the adjacency matrix of Γ has D + 1 distinct eigenvalues k = θ0 > θ1 > · · · > θD. They are the eigenvalues of the matrix L =           a0 b0 c1 a1 b1 c2 a2 b2 . . . . . . cD−1 aD−1 bD−1 cD aD          

SEBI CIOAB ˘

A (UNIV. OF DELAWARE)

CONNECTIVITY PROBLEMS IN DRGS 11 / 19

THE STANDARD SEQUENCES OF A DRG

THE STANDARD SEQUENCE OF AN EIGENVALUE The standard sequence of an eigenvalue θi of Γ is the eigenvector u = (u0, u1, . . . , uD) of L corresponding to θi normalized such that u0 = 1. THEOREM (BROUWER, COHEN AND NEUMAIER 1989) The standard sequence corresponding to θi has exactly i sign changes.

SEBI CIOAB ˘

A (UNIV. OF DELAWARE)

CONNECTIVITY PROBLEMS IN DRGS 12 / 19

AN ANSWER TO BROUWER’S QUESTION

THEOREM (CIOAB ˘

A AND KOOLEN, 2013)

Let Γ be a distance-regular graph of diameter D and let u = (u0, u1, . . . , uD) be the standard sequence corresponding to the 2nd largest eigenvalue θ1 of Γ.

1

If ut−1 > 0, then Γ≥t(x) is connected for any vertex x of Γ.

2

If ut < 0, then t > D/2; if ut ≤ 0, then t ≥ D/2.

SEBI CIOAB ˘

A (UNIV. OF DELAWARE)

CONNECTIVITY PROBLEMS IN DRGS 13 / 19

AN ANSWER TO BROUWER’S QUESTION

Let ρi denote the largest eigenvalue of Γ≤i(x). It is the largest eigenvalue of the matrix Li =           b0 c1 a1 b1 c2 a2 b2 . . . . . . ci−1 ai−1 bi−1 ci ai          

SEBI CIOAB ˘

A (UNIV. OF DELAWARE)

CONNECTIVITY PROBLEMS IN DRGS 14 / 19

AN ANSWER TO BROUWER’S QUESTION

Let σi denote the largest eigenvalue of Γ≥i(x). It is the largest eigenvalue of the matrix Mi =         ai bi ci+1 ai+1 bi+1 . . . . . . cd−1 ad−1 bd−1 cd ad         . The largest eigenvalue of any component of Γ≥i(x) is σi.

SEBI CIOAB ˘

A (UNIV. OF DELAWARE)

CONNECTIVITY PROBLEMS IN DRGS 15 / 19

AN ANSWER TO BROUWER’S QUESTION

THEOREM (CIOAB ˘

A AND KOOLEN, 2013)

Let Γ be a distance-regular graph of diameter D and let u = (u0, u1, . . . , uD) be the standard sequence corresponding to the 2nd largest eigenvalue θ1 of Γ.

1

If ut−1 > 0, then Γ≥t(x) is connected for any vertex x of Γ.

2

If ut < 0, then t > D/2; if ut ≤ 0, then t ≥ D/2. PROOF.

1

ut−1 > 0 implies that σt > θ1.

2

If Γ≥t(x) is disconnected, then the 2nd eigenvalue of Γ≥t(x) is σt.

3

Eigenvalue interlacing implies θ1 ≥ σt > θ1, contradiction.

4

ρi ↑; σi ↓ with i; ρi ≤ σD−i.

5

ut < 0 ⇒ ρt−1 > θ1 > σt+1 ≥ ρD−t−1 ⇒ t > D/2.

SEBI CIOAB ˘

A (UNIV. OF DELAWARE)

CONNECTIVITY PROBLEMS IN DRGS 16 / 19

OPEN PROBLEMS

REMARKS Our result implies the fact that Γ2(x) is connected for every primitive SRG as u0 = 1, u1 = θ1/k. QUESTION If ΓD(x) is connected, how large can its diameter be ? Gardiner et al. showed the diameter of Γ2(x) is ≤ 3 for primitive SRGs. REMARKS There are many DRGs of diameter D ≥ 3 with ΓD(x) is disconnected. Any DRG with aD = 0 or 1; Odd graphs.

SEBI CIOAB ˘

A (UNIV. OF DELAWARE)

CONNECTIVITY PROBLEMS IN DRGS 17 / 19

OPEN PROBLEMS

QUESTION If Γ is a primitive distance-regular graph of diameter D, is ΓD−1(x) ∪ ΓD(x) connected for every vertex x of Γ ? REMARKS For all the primitive DRGs we checked ΓD−1(x) ∪ ΓD(x) is connected. Our result implies ΓD−1 ∪ ΓD(x) is connected for D = 4 except when Γ is an antipodal r-cover with r ≥ 3. QUESTION If Γ is a DRG, let ki = |Γi(x)| for 0 ≤ i ≤ D. Let s be such that ks = max ki. Let t be such that ut−1 > 0 and ut ≤ 0. Our results imply that if θ1 < k/2, then t − 2 ≤ s ≤ t + 1. How far can s and t be in general ?

SEBI CIOAB ˘

A (UNIV. OF DELAWARE)

CONNECTIVITY PROBLEMS IN DRGS 18 / 19

THANK YOU!

FIGURE: Thanks Andy and the other organizers.

SEBI CIOAB ˘

A (UNIV. OF DELAWARE)

CONNECTIVITY PROBLEMS IN DRGS 19 / 19