Papers by Arlene A. Pascasio
European Journal of Combinatorics, Nov 1, 2002
Ito, Tanabe and Terwilliger recently introduced the notion of a tridiagonal pair. We apply their ... more Ito, Tanabe and Terwilliger recently introduced the notion of a tridiagonal pair. We apply their results to distance-regular graphs and obtain the following theorem. THEOREM. Let denote a distance-regular graph with diameter D ≥ 3. Suppose is Q-polynomial with respect to the ordering E 0 , E 1 ,. .. , E D of the primitive idempotents. For 0 ≤ i ≤ D, let m i denote the multiplicity of E i. Then (i) m i−1 ≤ m i (1 ≤ i ≤ D/2), (ii) m i ≤ m D−i (0 ≤ i ≤ D/2). By proving the above theorem we resolve a conjecture of Dennis Stanton.
Linear Algebra and its Applications, Mar 1, 2001
Let denote a distance-regular graph with diameter d 3, and assume is tight in the sense of Juriši... more Let denote a distance-regular graph with diameter d 3, and assume is tight in the sense of Jurišić et al. [J. Algebraic Combin. 12 (2000) 163-197]. Let θ denote the second largest or the smallest eigenvalue of. We obtain an inequality involving the first, second and third cosines associated with θ. We investigate the relationship between equality being attained and the existence of dual bipartite Q-polynomial structures on .
Discrete Mathematics, Mar 1, 2003
In this paper, we prove the following: Theorem. Let A = A0; A1; : : : ; A d denote a complex char... more In this paper, we prove the following: Theorem. Let A = A0; A1; : : : ; A d denote a complex character algebra with d ¿ 2 which is P-polynomial with respect to the ordering A0; A1; : : : ; A d of the distinguished basis. Assume that the structure constants p h ij are all nonnegative and the Krein parameters q h ij are all nonnegative. Let  and  denote eigenvalues of A1, other than the valency k = k1. Then the structure constants a1 = p 1 11 and b1 = p 1 12 satisfy  + k a1 + 1  + k a1 + 1 ¿ − ka1b1 (a1 + 1) 2 : Let E and F denote the primitive idempotents of A associated with  and  , respectively. Equality holds in the above inequality if and only if the Schur product E • F is a scalar multiple of a primitive idempotent of A. The above theorem extends some results of JuriÄ sià c, Koolen, Terwilliger, and the present author. These people previously showed the above theorem holds for those character algebras isomorphic to the Bose-Mesner algebra of a distance-regular graph.
arXiv (Cornell University), Dec 8, 2003
Let Γ denote a distance-regular graph with diameter D ≥ 3, valency k, and intersection numbers a ... more Let Γ denote a distance-regular graph with diameter D ≥ 3, valency k, and intersection numbers a i , b i , c i. By a pseudo cosine sequence of Γ we mean a sequence of real numbers σ 0 , σ 1 ,. .. , σ D such that σ 0 = 1 and c i σ i−1 + a i σ i + b i σ i+1 = kσ 1 σ i for 0 ≤ i ≤ D − 1. Let σ 0 , σ 1 ,. .. , σ D and ρ 0 , ρ 1 ,. .. , ρ D denote pseudo cosine sequences of Γ. We say this pair of sequences is tight whenever σ 0 ρ 0 , σ 1 ρ 1 ,. .. , σ D ρ D is a pseudo cosine sequence of Γ. In this paper, we determine all the tight pairs of pseudo cosine sequences of Γ.
Journal of Algebraic Combinatorics, Jul 1, 1999
In this paper, we prove the following two theorems. Theorem 1 Let denote a distance-regular graph... more In this paper, we prove the following two theorems. Theorem 1 Let denote a distance-regular graph with diameter d ≥ 3. Suppose E and F are primitive idempotents of , with cosine sequences σ 0 , σ 1 ,. .. , σ d and ρ 0 , ρ 1 ,. .. , ρ d , respectively. Then the following are equivalent. (i) The entry-wise product E • F is a scalar multiple of a primitive idempotent of. (ii) There exists a real number such that σ i ρ i − σ i−1 ρ i−1 = (σ i−1 ρ i − σ i ρ i−1) (1 ≤ i ≤ d). Let denote a distance-regular graph with diameter d ≥ 3 and eigenvalues θ 0 > θ 1 > • • • > θ d. Then Jurisić, Koolen and Terwilliger proved that the valency k and the intersection numbers a 1 , b 1 satisfy θ
On the Characterization of AG(n, q) by its Parameters as a Nearly Triply Regular Design
Dcc, 1996
Linear Algebra and its Applications, 2001
Let denote a distance-regular graph with diameter d 3, and assume is tight in the sense of Juriši... more Let denote a distance-regular graph with diameter d 3, and assume is tight in the sense of Jurišić et al. [J. Algebraic Combin. 12 (2000) 163-197]. Let θ denote the second largest or the smallest eigenvalue of. We obtain an inequality involving the first, second and third cosines associated with θ. We investigate the relationship between equality being attained and the existence of dual bipartite Q-polynomial structures on .
Graphs and Combinatorics, 2001
Let C denote a distance-regular graph with diameter d ! 3, and assume C is tight (in the sense of... more Let C denote a distance-regular graph with diameter d ! 3, and assume C is tight (in the sense of JurisÏ ic , Koolen and Terwilliger). Let h denote the second largest or smallest eigenvalue of C, and let r 0 ; r 1 ;. .. ; r d denote the associated cosine sequence. We obtain an inequality involving r 0 ; r 1 ;. .. ; r d for each integer i 1 i d À 1, and we show equality for all i is closely related to C being Q-polynomial with respect to h. We use this idea to investigate the Q-polynomial structures in tight distance-regular graphs.
European Journal of Combinatorics, 2002
Ito, Tanabe and Terwilliger recently introduced the notion of a tridiagonal pair. We apply their ... more Ito, Tanabe and Terwilliger recently introduced the notion of a tridiagonal pair. We apply their results to distance-regular graphs and obtain the following theorem. THEOREM. Let denote a distance-regular graph with diameter D ≥ 3. Suppose is Q-polynomial with respect to the ordering E 0 , E 1 ,. .. , E D of the primitive idempotents. For 0 ≤ i ≤ D, let m i denote the multiplicity of E i. Then (i) m i−1 ≤ m i (1 ≤ i ≤ D/2), (ii) m i ≤ m D−i (0 ≤ i ≤ D/2). By proving the above theorem we resolve a conjecture of Dennis Stanton.
Discrete Mathematics, 2003
In this paper, we prove the following: Theorem. Let A = A0; A1; : : : ; A d denote a complex char... more In this paper, we prove the following: Theorem. Let A = A0; A1; : : : ; A d denote a complex character algebra with d ¿ 2 which is P-polynomial with respect to the ordering A0; A1; : : : ; A d of the distinguished basis. Assume that the structure constants p h ij are all nonnegative and the Krein parameters q h ij are all nonnegative. Let  and  denote eigenvalues of A1, other than the valency k = k1. Then the structure constants a1 = p 1 11 and b1 = p 1 12 satisfy  + k a1 + 1  + k a1 + 1 ¿ − ka1b1 (a1 + 1) 2 : Let E and F denote the primitive idempotents of A associated with  and  , respectively. Equality holds in the above inequality if and only if the Schur product E • F is a scalar multiple of a primitive idempotent of A. The above theorem extends some results of JuriÄ sià c, Koolen, Terwilliger, and the present author. These people previously showed the above theorem holds for those character algebras isomorphic to the Bose-Mesner algebra of a distance-regular graph.
European Journal of Combinatorics, 1992
In this paper we construct two classes of translation hyperovals in any Hall plane of even order ... more In this paper we construct two classes of translation hyperovals in any Hall plane of even order q* P 16. Two hyperovals constructed in the same Hall plane are equivalent under the action of the automorphism group of that Hall plane iff they are in the same class.
A characterization of Q-polynomial distance-regular graphs
Discrete Mathematics, 2008
ABSTRACT We obtain the following characterization of $Q$-polynomial distance-regular graphs. Let ... more ABSTRACT We obtain the following characterization of $Q$-polynomial distance-regular graphs. Let $\G$ denote a distance-regular graph with diameter $d\ge 3$. Let $E$ denote a minimal idempotent of $\G$ which is not the trivial idempotent $E_0$. Let $\{\theta_i^*\}_{i=0}^d$ denote the dual eigenvalue sequence for $E$. We show that $E$ is $Q$-polynomial if and only if (i) the entry-wise product $E \circ E$ is a linear combination of $E_0$, $E$, and at most one other minimal idempotent of $\G$; (ii) there exists a complex scalar $\beta$ such that $\theta^*_{i-1}-\beta \theta^*_i + \theta^*_{i+1}$ is independent of $i$ for $1 \le i \le d-1$; (iii) $\theta^*_i \ne \theta^*_0$ for $1 \le i \le d$. Comment: 10 pages, 1 figure
Linear Algebra and its Applications, 2006
Let Γ denote a distance-regular graph with diameter D ≥ 3, valency k, and intersection numbers a ... more Let Γ denote a distance-regular graph with diameter D ≥ 3, valency k, and intersection numbers a i , b i , c i. By a pseudo cosine sequence of Γ we mean a sequence of real numbers σ 0 , σ 1 ,. .. , σ D such that σ 0 = 1 and c i σ i−1 + a i σ i + b i σ i+1 = kσ 1 σ i for 0 ≤ i ≤ D − 1. Let σ 0 , σ 1 ,. .. , σ D and ρ 0 , ρ 1 ,. .. , ρ D denote pseudo cosine sequences of Γ. We say this pair of sequences is tight whenever σ 0 ρ 0 , σ 1 ρ 1 ,. .. , σ D ρ D is a pseudo cosine sequence of Γ. In this paper, we determine all the tight pairs of pseudo cosine sequences of Γ.
An Action of the Tetrahedron Algebra on the Standard Module for the Hamming Graphs and Doob Graphs
Graphs and Combinatorics, 2013
We display an action of the tetrahedron algebra $${\boxtimes}$$⊠ on the standard module for any H... more We display an action of the tetrahedron algebra $${\boxtimes}$$⊠ on the standard module for any Hamming graph or Doob graph. To do this, we use some results of Brian Hartwig concerning tridiagonal pairs of Krawtchouk type.
Hyperovals in Hall planes
European Journal of Combinatorics, 1992
Discrete Mathematics, 1996
The Hall plane of order q2 is constructed from the Desarguesian plane of order q2 by the process ... more The Hall plane of order q2 is constructed from the Desarguesian plane of order q2 by the process of derivation with respect to a fixed derivation set on a fixed ideal line. The affine points of a non-degenerate conic in the Desarguesian plane of order q2 can be regarded as a set of points in the corresponding Hall plane. We give a classification of the structure of sets of points in Hall planes arising from conics with two points in the derivation set. Further, if q is even, we classify the sets of points in Hall planes arising from conics which admit the ideal line as a tangent and are such that either the point of contact or the nucleus of the conic (or both) is contained in the derivation set.
On the characterisation of AG(n, q) by its parameters as a nearly triply regular design
Designs, Codes and Cryptography, 1996
We show that a non-symmetric nearly triply regular% MathType! MTEF! 2! 1!+-% feaafiart1ev1aaatCvA... more We show that a non-symmetric nearly triply regular% MathType! MTEF! 2! 1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLb...% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDY...% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4...-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9qq-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaa...% OeI0IaaiikaiaadghadaahaaWcbeqaaiaad6g...% amaaCaaaleqabaGaamOBaiabgkHiTiaaigd...% qaaiaadghadaahaaadbeqaaiaad6gacqGHs...% I0IaaGymaaqaaiaadghacqGHsislcaaIXaaaa...! 4BBD! 2-(q^n,q^n-1,q^n-1-1q-1) designD withv1= qn-2, v2= qn-3 and in which every line ...
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Papers by Arlene A. Pascasio