The girth of a thin distance-regular graph |
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Authors: | Benjamin V. C. Collins |
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Affiliation: | 1. Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, 53706, Madison, WI, USA
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Abstract: | Let Γ be a distance-regular graph of diameterd≥3. For each vertexx of Γ, letT(x) denote the Terwilliger algebra for Γ with respect tox. An irreducibleT(x)-moduleW is said to bethin if dimE i * (x)W≤1 for 0≤i≤d, whereE i * (x) is theith dual idempotent for Γ with respect tox. The graph Γ isthin if for each vertexx of Γ, every irreducibleT(x)-module is thin. Aregular generalized quadrangle is a bipartite distance-regular graph with girth 8 and diameter 4. Our main results are as follows: Theorem. Let Γ=(X,R) be a distance-regular graph with diameter d≥3 and valency k≥3. Then the following are equivalent: - Γis a regular generalized quadrangle.
- Γis thin and c 3=1.
Corollary. Let Γ=(X,R) be a thin distance-regular graph with diameter d≥3 and valency k≥3. Then Γ has girth 3, 4, 6, or 8. Then girth of Γ is 8 exactly when Γ is a regular generalized quadrangle. |
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