2016/11/07 by Mark S. MacLean, Štefko Miklavič, MacLean, Mark S. +1
Engineering · Mathematics · #05E30 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1611.02326
openalex publication_date 2016/11/07 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
Let \Γ denote a bipartite distance-regular graph with diameter D \≥\n4 and valency k \≥ 3. Let X denote the vertex set of \Γ, and let\nA denote the adjacency matrix of \Γ. For x \∈ X let T=T(x) denote\nthe subalgebra of MatX(\ℂ generated by A, E*0, E*s1, \…,\nE*D, where for 0 \≤ i \≤ D, E*i represents the projection onto the\nith subconstituent of \Γ with respect to x. We refer to T as the\n em Terwilliger algebra of \Γ with respect to x. An irreducible\nT-module W is said to be em thin whenever dim E*i W \≤ 1 for 0 \≤\ni \≤ D. By the em endpoint of W we mean min i | E*iW \≠ 0 . For 0\n\≤ i \≤ D, let \Γi(z) denote the set of vertices in X that are\ndistance i from vertex z. Define a parameter \Δ2 in terms of the\nintersection numbers by \Δ2 = (k-2)(c3-1)-(c2-1)p222. In this\npaper we prove the following are equivalent: (i) \Δ2>0 and for 2 \≤ i\n\≤ D - 2 there exist complex scalars \αi, \βi with the following\nproperty: for all x, y, z \∈ X such that \∂(x, y) = 2, : \∂(x,\nz) = i, : \∂(y, z) = i we have \αi + \βi |\Γ1(x) \∩\n\Γ1(y) \∩ \Γi-1(z)| = |\Γi-1(x) \∩ \Γi-1(y) \∩\n\Γ1(z)|; (ii) For all x \∈ X there exist up to isomorphism exactly two\nirreducible modules for the Terwilliger algebra T(x) with endpoint two, and\nthese modules are thin.\n