An inequality on the cosines of a tight distance-regular graph

Pascasio, Arlene A.

doi:10.1016/s0024-3795(00)00266-4

Cited by 11 publications

(10 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let the polynomials g i be as in Definition 4.1. Setting λ = A in (34) and applying the result to v, we find…”

Section: Proof By Lemma 21 We Find θmentioning

confidence: 99%

Tight Distance-regular Graphs and the Subconstituent Algebra

Terwilliger

2002

European Journal of Combinatorics

View full text Add to dashboard Cite

We consider a distance-regular graph with diameter D ≥ 3, intersection numbers a i , b i , c i and eigenvalues k = θ 0 > θ 1 > · · · > θ D . Let X denote the vertex set of and fix x ∈ X. Let T = T (x) denote the subalgebra of Mat X (C) generated by A, E , where E i denotes the primitive idempotent of A associated with θ i . We show this basis is orthogonal (with respect to the Hermitean dot product) and we compute the square norm of each basis vector. We show, where A i denotes the ith distance matrix for . We find the matrix representing A with respect to this basis. We show this basis is orthogonal and we compute the square norm of each basis vector. We find the transition matrix relating our two bases for W . For notational convenience, we say is 1-thin with respect to x whenever every irreducible T -module with endpoint 1 is thin. Similarly, we say is tight with respect to x whenever every irreducible T -module with endpoint 1 is thin with local eigenvalueθ 1 orθ D . In [J. Algebr. Comb., 12, (2000), Jurišić, Koolen and Terwilliger showedThey defined to be tight whenever is nonbipartite and equality holds above. We show the following are equivalent: (i) is tight; (ii) is tight with respect to each vertex; (iii) is tight with respect to at least one vertex. We show the following are equivalent: (i) is tight; (ii) is nonbipartite, a D = 0, and is 1-thin with respect to each vertex; (iii) is nonbipartite, a D = 0, and is 1-thin with respect to at least one vertex.

show abstract

“…Let the polynomials g i be as in Definition 4.1. Setting λ = A in (34) and applying the result to v, we find…”

Section: Proof By Lemma 21 We Find θmentioning

confidence: 99%

Tight Distance-regular Graphs and the Subconstituent Algebra

Terwilliger

2002

European Journal of Combinatorics

View full text Add to dashboard Cite

show abstract

“…The result is now a routine consequence of (8), (16), and the linear independence of the primitive idempotents. …”

Section: Proofmentioning

confidence: 90%

“…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

“…Cosine similarity is a measure of similarity between two vectors of an inner product space that measures the cosine of the angle between them. 16,17 The cosine of two vectors can be derived by using the Euclidean dot product formula.…”

Section: Similarity Measurementioning

confidence: 99%

Contaminant classification using cosine distances based on multiple conventional sensors

Liu

Han

Smith

et al. 2015

Environ. Sci.: Processes Impacts

View full text Add to dashboard Cite

Emergent contamination events have a significant impact on water systems. After contamination detection, it is important to classify the type of contaminant quickly to provide support for remediation attempts. Conventional methods generally either rely on laboratory-based analysis, which requires a long analysis time, or on multivariable-based geometry analysis and sequence analysis, which is prone to being affected by the contaminant concentration. This paper proposes a new contaminant classification method, which discriminates contaminants in a real time manner independent of the contaminant concentration. The proposed method quantifies the similarities or dissimilarities between sensors' responses to different types of contaminants. The performance of the proposed method was evaluated using data from contaminant injection experiments in a laboratory and compared with a Euclidean distance-based method. The robustness of the proposed method was evaluated using an uncertainty analysis. The results show that the proposed method performed better in identifying the type of contaminant than the Euclidean distance based method and that it could classify the type of contaminant in minutes without significantly compromising the correct classification rate (CCR).

show abstract

An inequality on the cosines of a tight distance-regular graph

Cited by 11 publications

References 10 publications

Tight Distance-regular Graphs and the Subconstituent Algebra

Tight Distance-regular Graphs and the Subconstituent Algebra

Untitled

Contaminant classification using cosine distances based on multiple conventional sensors

Contact Info

Product

Resources

About