The Terwilliger polynomial of a Q-polynomial distance-regular graph and its application to the pseudo-partition graphs

2014/03/17 by Alexander L. Gavrilyuk, Gavrilyuk, Alexander L., Jack H. Koolen +1 · 1 citation
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #math.CO

paper · pdf · doi:10.48550/arxiv.1403.4027

arxiv created 2014/03/17 · arxiv updated 2014/03/18

Abstract

Let Γ be a Q-polynomial distance-regular graph with diameter at least 3. Terwilliger (1993) implicitly showed that there exists a polynomial, say T(λ)∈ ℂ[λ], of degree 4 depending only on the intersection numbers of Γ and such that T(η)≥ 0 holds for any non-principal eigenvalue η of the local graph Γ(x) for any vertex x∈ V(Γ). We call T(λ) the Terwilliger polynomial of Γ. In this paper, we give an explicit formula for T(λ) in terms of the intersection numbers of Γ and its dual eigenvalues. We then apply this polynomial to show that all pseudo-partition graphs with diameter at least 3 are known.

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