An inequality for regular near polygons

Paul Terwilliger; Chih-wen Weng

doi:10.1016/j.ejc.2004.03.001

An inequality for regular near polygons

Paul Terwilliger^*, Chih-wen Weng

^*Corresponding author for this work

Department of Applied Mathematics

Research output: Contribution to journal › Article

3 Scopus citations

Abstract

Let Γ denote a near polygon distance-regular graph with diameter d ≥ 3, valency k and intersection numbers a₁ > 0, c₂ > 1. Let θ₁ denote the second largest eigenvalue of Γ. We show θ₁≤k-a₁-c₂/c₂-1. We show the following (i)-(iii) are equivalent. (i) Equality is attained above; (ii) Γ is Q-polynomial with respect to θ₁; (iii) Γ is a dual polar graph or a Hamming graph.

Original language	English
Pages (from-to)	227-235
Number of pages	9
Journal	European Journal of Combinatorics
Volume	26
Issue number	2
DOIs	https://doi.org/10.1016/j.ejc.2004.03.001
State	Published - 1 Feb 2005

Keywords

Distance-regular graph
Dual polar graph
Hamming graph
Near polygon
Q-polynomial

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abstract = "Let Γ denote a near polygon distance-regular graph with diameter d ≥ 3, valency k and intersection numbers a1 > 0, c2 > 1. Let θ1 denote the second largest eigenvalue of Γ. We show θ1≤k-a1-c2/c2-1. We show the following (i)-(iii) are equivalent. (i) Equality is attained above; (ii) Γ is Q-polynomial with respect to θ1; (iii) Γ is a dual polar graph or a Hamming graph.",

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AU - Weng, Chih-wen

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N2 - Let Γ denote a near polygon distance-regular graph with diameter d ≥ 3, valency k and intersection numbers a1 > 0, c2 > 1. Let θ1 denote the second largest eigenvalue of Γ. We show θ1≤k-a1-c2/c2-1. We show the following (i)-(iii) are equivalent. (i) Equality is attained above; (ii) Γ is Q-polynomial with respect to θ1; (iii) Γ is a dual polar graph or a Hamming graph.

AB - Let Γ denote a near polygon distance-regular graph with diameter d ≥ 3, valency k and intersection numbers a1 > 0, c2 > 1. Let θ1 denote the second largest eigenvalue of Γ. We show θ1≤k-a1-c2/c2-1. We show the following (i)-(iii) are equivalent. (i) Equality is attained above; (ii) Γ is Q-polynomial with respect to θ1; (iii) Γ is a dual polar graph or a Hamming graph.

KW - Distance-regular graph

KW - Dual polar graph

KW - Hamming graph

KW - Near polygon

KW - Q-polynomial

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