Tight Distance-Regular Graphs and the Q-Polynomial Property

Pascasio, Arlene A.

doi:10.1007/s003730170063

Cited by 22 publications

(24 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Thus s DÀ2 ¼ s D ðls À 1Þ: Setting i ¼ 1 in (19), we find ls À 1 ¼ s 2 ; so s DÀ2 ¼ s 2 s D ; as desired. We now verify (18). By (2) we find s iÀ1 as iþ1 ð1pipD À 1Þ: By this and (4) we find…”

Section: The 2-homogeneous and Taut Distance-regular Graphsmentioning

confidence: 90%

“…By (18) and Lemma 3.1 (iii), we obtain (17). To obtain (16), we set i ¼ 1 and c 1 ¼ 1 in (18) to obtain…”

Section: The 2-homogeneous and Taut Distance-regular Graphsmentioning

confidence: 99%

“…Moreover, the pair E; F is tight if and only if E; F is a permutation of E 1 ; E D : See [11,12] for a detailed discussion of the tight distance-regular graphs. For related papers, see [7][8][9][10][17][18][19][20]22]. Let E; F denote primitive idempotents of G: We now consider the case where E3F is a linear combination of two distinct primitive idempotents of G: To keep things simple, we restrict our attention to the case where G is bipartite.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Taut distance-regular graphs of even diameter

MacLean¹

2004

Journal of Combinatorial Theory, Series B

View full text Add to dashboard Cite

Let G denote a bipartite distance-regular graph with diameter DX4; valency kX3; and Bose-Mesner algebra M: Let y 0 4y 1 4?4y D denote the distinct eigenvalues for G; and for 0pipD; let E i denote the primitive idempotent of M associated with y i : We refer to E 0 and E D as the trivial idempotents of M: Let E and F denote primitive idempotents of M: We say the pair E; F is taut whenever (i) E; F are nontrivial, and (ii) the entry-wise product E3F is a linear combination of two distinct primitive idempotents of M: If G is 2-homogeneous in the sense of Nomura and Curtin, then G has at least one taut pair of primitive idempotents. We define G to be taut whenever G has at least one taut pair of primitive idempotents but G is not 2-homogeneous. Let y denote an eigenvalue of G other than y 0 ; y D ; and let s 0 ; s 1 ; y; s D denote the cosine sequence associated with y: By a result of Curtin, the following are equivalent: (i) G is 2-homogeneous and yAfy 1 ; y DÀ1 g; (ii) there exists a complex scalar l such that s iÀ1 À ls i þ s iþ1 ¼ 0 for 1pipD À 1: Expanding on this, we show that for D even, the following are equivalent: (i) G is taut or 2-homogeneous and yAfy 1 ; y DÀ1 g; (ii) there exists a complex scalar l such that s iÀ1 À ls i þ s iþ1 ¼ 0 for i odd, 1pipD À 1: Using this result, we show that for D even, G is taut or 2-homogeneous if and only if the intersection numbers of G are given by certain rational expressions involving D=2 independent variables. We discuss the known examples of taut distance-regular graphs with even diameter. r

show abstract

Section: The 2-homogeneous and Taut Distance-regular Graphsmentioning

confidence: 90%

“…By (18) and Lemma 3.1 (iii), we obtain (17). To obtain (16), we set i ¼ 1 and c 1 ¼ 1 in (18) to obtain…”

Section: The 2-homogeneous and Taut Distance-regular Graphsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Taut distance-regular graphs of even diameter

MacLean¹

2004

Journal of Combinatorial Theory, Series B

View full text Add to dashboard Cite

show abstract

“…q h ij = 0) whenever one of h, i, j is greater than (resp. equal to) the sum of the other two [1,4,5,6,9,10,14,15,23,24]. From now on assume Γ is Q-polynomial with respect to…”

Section: Preliminaries Concerning Distance-regular Graphsmentioning

confidence: 99%

The Displacement and Split Decompositions for a Q-Polynomial Distance-regular Graph

Terwilliger

2005

Graphs and Combinatorics

View full text Add to dashboard Cite

show abstract

“…See [10,11] for a detailed discussion of the tight graphs. For related papers, see [6][7][8][9][15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

Untitled

MacLean

2003

Journal of Algebraic Combinatorics

View full text Add to dashboard Cite

Abstract. Let denote a bipartite distance-regular graph with diameter D ≥ 4, valency k ≥ 3, and distinct eigenvalues θ 0 > θ 1 > · · · > θ D . Let M denote the Bose-Mesner algebra of . For 0 ≤ i ≤ D, let E i denote the primitive idempotent of M associated with θ i . We refer to E 0 and E D as the trivial idempotents of M. Let E, F denote primitive idempotents of M. We say the pair E, F is taut whenever (i) E, F are nontrivial, and (ii) the entry-wise product E • F is a linear combination of two distinct primitive idempotents of M. We show the pair E, F is taut if and only if there exist real scalars α, β such thatwhere σ 0 , σ 1 , . . . , σ D and ρ 0 , ρ 1 , . . . , ρ D denote the cosine sequences of E, F, respectively. We define to be taut whenever has at least one taut pair of primitive idempotents but is not 2-homogeneous in the sense of Nomura and Curtin. Assume is taut and D is odd, and assume the pair E, F is taut. We showUsing these equations, we recursively obtain σ 0 , σ 1 , . . . , σ D and ρ 0 , ρ 1 , . . . , ρ D in terms of the four real scalars σ, ρ, α, β. From this we obtain all intersection numbers of in terms of σ, ρ, α, β. We showed in an earlier paper that the pair E 1 , E d is taut, where d = (D − 1)/2. Applying our results to this pair, we obtain the intersection numbers of in terms of k, µ, θ 1 , θ d , where µ denotes the intersection number c 2 . We show that if is taut and D is odd, then is an antipodal 2-cover.

show abstract

Tight Distance-Regular Graphs and the Q-Polynomial Property

Cited by 22 publications

References 0 publications

Taut distance-regular graphs of even diameter

Taut distance-regular graphs of even diameter

The Displacement and Split Decompositions for a Q-Polynomial Distance-regular Graph

Untitled

Contact Info

Product

Resources

About