O N 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 S EBI C IOAB ˘ A (U NIV . OF D ELAWARE ) C ONNECTIVITY P ROBLEMS IN D RGS 1 / 19
S OME D EFINITIONS F IGURE : Not a ball S EBI C IOAB ˘ A (U NIV . OF D ELAWARE ) C ONNECTIVITY P ROBLEMS IN D RGS 2 / 19
S OME D EFINITIONS D EFINITION 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 . F IGURE : This is a ball. S EBI C IOAB ˘ A (U NIV . OF D ELAWARE ) C ONNECTIVITY P ROBLEMS IN D RGS 3 / 19
S TRONGLY R EGULAR G RAPHS D EFINITION A graph G is called a ( v , k , λ, µ ) -strongly regular (or ( v , k , λ, µ ) -SRG) if it has v vertices 1 it is k -regular 2 any two adjacent vertices have exactly λ common neighbors 3 any two distinct non-adjacent vertices have exactly µ common 4 neighbors F IGURE : The Petersen graph is a ( 10 , 3 , 0 , 1 ) -SRG. S EBI C IOAB ˘ A (U NIV . OF D ELAWARE ) C ONNECTIVITY P ROBLEMS IN D RGS 4 / 19
S UBCONSTITUENTS OF S TRONGLY R EGULAR G RAPHS D EFINITION The i -th subconstituent Γ i ( x ) of a vertex x of a graph Γ is the subgraph of Γ induced by the vertices at distance i from x . 6 1 7 10 2 5 � 3 4 8 9 F IGURE : The subconstituents of the Petersen graph S EBI C IOAB ˘ A (U NIV . OF D ELAWARE ) C ONNECTIVITY P ROBLEMS IN D RGS 5 / 19
S UBCONSTITUENTS OF S TRONGLY R EGULAR G RAPHS C AMERON , G OETHALS , S EIDEL 1978 Strongly regular graphs having strongly regular subconstituents. G ARDINER , G ODSIL , H ENSEL , R OYLE 1992 Second neighborhoods of strongly regular graphs. S EBI C IOAB ˘ A (U NIV . OF D ELAWARE ) C ONNECTIVITY P ROBLEMS IN D RGS 6 / 19
S UBCONSTITUENTS OF S TRONGLY R EGULAR G RAPHS T HEOREM If x is any vertex of a primitive strongly regular graph Γ , then the second subconstituent Γ 2 ( x ) is connected. P ROOF ( S ) Combinatorial Proof: Gardiner, Godsil, Hensel, Royle 1992 Algebraic Proof: Haemers. S EBI C IOAB ˘ A (U NIV . OF D ELAWARE ) C ONNECTIVITY P ROBLEMS IN D RGS 7 / 19
A QUESTION OF B ROUWER F IGURE : Joe Hemmeter, Andries Brouwer and Andy Woldar S EBI C IOAB ˘ A (U NIV . OF D ELAWARE ) C ONNECTIVITY P ROBLEMS IN D RGS 8 / 19
A QUESTION OF B ROUWER T HEOREM If x is any vertex of a primitive strongly regular graph Γ , then the second subconstituent Γ 2 ( x ) is connected. Q UESTION (B ROUWER , 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 . S EBI C IOAB ˘ A (U NIV . OF D ELAWARE ) C ONNECTIVITY P ROBLEMS IN D RGS 9 / 19
D ISTANCE -R EGULAR G RAPHS D EFINITION A graph Γ with diameter D is called distance-regular if there are integers b i , c i ( 0 ≤ i ≤ D ) such that for any two vertices x , y of Γ at distance i , there are precisely c i neighbors of y in Γ i − 1 ( y ) and b i neighbors in Γ i + 1 ( x ) . Let k := b 0 , a i := k − b i − c i for 0 ≤ i ≤ D . T HE QUOTIENT MATRIX OF THE DISTANCE PARTITION a 0 b 0 c 1 a 1 b 1 c 2 a 2 b 2 L = . . . . . . c D − 1 a D − 1 b D − 1 c D a D S EBI C IOAB ˘ A (U NIV . OF D ELAWARE ) C ONNECTIVITY P ROBLEMS IN D RGS 10 / 19
T HE S PECTRUM OF A D ISTANCE -R EGULAR G RAPH T HE E IGENVALUES OF A DRG If Γ is a DRG of diameter D , then the adjacency matrix of Γ has D + 1 distinct eigenvalues k = θ 0 > θ 1 > · · · > θ D . They are the eigenvalues of the matrix a 0 b 0 c 1 a 1 b 1 c 2 a 2 b 2 L = . . . . . . c D − 1 a D − 1 b D − 1 c D a D S EBI C IOAB ˘ A (U NIV . OF D ELAWARE ) C ONNECTIVITY P ROBLEMS IN D RGS 11 / 19
T HE S TANDARD S EQUENCES OF A DRG T HE STANDARD SEQUENCE OF AN EIGENVALUE The standard sequence of an eigenvalue θ i of Γ is the eigenvector u = ( u 0 , u 1 , . . . , u D ) of L corresponding to θ i normalized such that u 0 = 1. T HEOREM (B ROUWER , C OHEN AND N EUMAIER 1989) The standard sequence corresponding to θ i has exactly i sign changes. S EBI C IOAB ˘ A (U NIV . OF D ELAWARE ) C ONNECTIVITY P ROBLEMS IN D RGS 12 / 19
A N ANSWER TO B ROUWER ’ S QUESTION T HEOREM (C IOAB ˘ A AND K OOLEN , 2013) Let Γ be a distance-regular graph of diameter D and let u = ( u 0 , u 1 , . . . , u D ) be the standard sequence corresponding to the 2nd largest eigenvalue θ 1 of Γ . If u t − 1 > 0 , then Γ ≥ t ( x ) is connected for any vertex x of Γ . 1 If u t < 0 , then t > D / 2 ; if u t ≤ 0 , then t ≥ D / 2 . 2 S EBI C IOAB ˘ A (U NIV . OF D ELAWARE ) C ONNECTIVITY P ROBLEMS IN D RGS 13 / 19
A N A NSWER TO B ROUWER ’ S QUESTION Let ρ i denote the largest eigenvalue of Γ ≤ i ( x ) . It is the largest eigenvalue of the matrix 0 b 0 c 1 a 1 b 1 c 2 a 2 b 2 L i = . . . . . . c i − 1 a i − 1 b i − 1 c i a i S EBI C IOAB ˘ A (U NIV . OF D ELAWARE ) C ONNECTIVITY P ROBLEMS IN D RGS 14 / 19
A N A NSWER TO B ROUWER ’ S QUESTION Let σ i denote the largest eigenvalue of Γ ≥ i ( x ) . It is the largest eigenvalue of the matrix a i b i c i + 1 a i + 1 b i + 1 . . . M i = . . . . c d − 1 a d − 1 b d − 1 c d a d The largest eigenvalue of any component of Γ ≥ i ( x ) is σ i . S EBI C IOAB ˘ A (U NIV . OF D ELAWARE ) C ONNECTIVITY P ROBLEMS IN D RGS 15 / 19
A N ANSWER TO B ROUWER ’ S QUESTION T HEOREM (C IOAB ˘ A AND K OOLEN , 2013) Let Γ be a distance-regular graph of diameter D and let u = ( u 0 , u 1 , . . . , u D ) be the standard sequence corresponding to the 2nd largest eigenvalue θ 1 of Γ . If u t − 1 > 0 , then Γ ≥ t ( x ) is connected for any vertex x of Γ . 1 If u t < 0 , then t > D / 2 ; if u t ≤ 0 , then t ≥ D / 2 . 2 P ROOF . u t − 1 > 0 implies that σ t > θ 1 . 1 If Γ ≥ t ( x ) is disconnected, then the 2nd eigenvalue of Γ ≥ t ( x ) is σ t . 2 Eigenvalue interlacing implies θ 1 ≥ σ t > θ 1 , contradiction. 3 ρ i ↑ ; σ i ↓ with i ; ρ i ≤ σ D − i . 4 u t < 0 ⇒ ρ t − 1 > θ 1 > σ t + 1 ≥ ρ D − t − 1 ⇒ t > D / 2. 5 S EBI C IOAB ˘ A (U NIV . OF D ELAWARE ) C ONNECTIVITY P ROBLEMS IN D RGS 16 / 19
O PEN P ROBLEMS R EMARKS Our result implies the fact that Γ 2 ( x ) is connected for every primitive SRG as u 0 = 1 , u 1 = θ 1 / k . Q UESTION If Γ D ( x ) is connected, how large can its diameter be ? Gardiner et al. showed the diameter of Γ 2 ( x ) is ≤ 3 for primitive SRGs. R EMARKS There are many DRGs of diameter D ≥ 3 with Γ D ( x ) is disconnected. Any DRG with a D = 0 or 1; Odd graphs. S EBI C IOAB ˘ A (U NIV . OF D ELAWARE ) C ONNECTIVITY P ROBLEMS IN D RGS 17 / 19
O PEN P ROBLEMS Q UESTION If Γ is a primitive distance-regular graph of diameter D, is Γ D − 1 ( x ) ∪ Γ D ( x ) connected for every vertex x of Γ ? R EMARKS For all the primitive DRGs we checked Γ D − 1 ( x ) ∪ Γ D ( x ) is connected. Our result implies Γ D − 1 ∪ Γ D ( x ) is connected for D = 4 except when Γ is an antipodal r -cover with r ≥ 3. Q UESTION If Γ is a DRG, let k i = | Γ i ( x ) | for 0 ≤ i ≤ D. Let s be such that k s = max k i . Let t be such that u t − 1 > 0 and u t ≤ 0 . Our results imply that if θ 1 < k / 2 , then t − 2 ≤ s ≤ t + 1 . How far can s and t be in general ? S EBI C IOAB ˘ A (U NIV . OF D ELAWARE ) C ONNECTIVITY P ROBLEMS IN D RGS 18 / 19
T HANK YOU ! F IGURE : Thanks Andy and the other organizers. S EBI C IOAB ˘ A (U NIV . OF D ELAWARE ) C ONNECTIVITY P ROBLEMS IN D RGS 19 / 19
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