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<p>Let Γ denote a bipartite <i>Q</i>-polynomial distance-regular graph with vertex set <i>X</i>, valency <i>k</i> ≥ 3 and diameter <i>D</i> ≥ 3. Let <i>A</i> be the adjacency matrix of Γ and let <i>A</i>*:= <i>A</i>*(<i>x</i>) be the dual adjacency matrix of Γ with respect to a fixed vertex <i>x</i> ∈ <i>X</i>. Let <i>T</i>:= <i>T</i>(<i>x</i>) denote the Terwilliger algebra of Γ generated by <i>A</i> and <i>A</i>*. In this paper, we first describe the relations between <i>A</i> and <i>A</i>*. Then we determine the dimensions of both <i>T</i> and the center of <i>T</i>, and moreover we give a basis of <i>T</i>.</p>

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The Terwilliger Algebras of Bipartite Q-polynomial Distance-regular Graphs

  • 期刊论文

  • 发表日期：

    2025年6月24日

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  • Li-hang Hou,
  • Bo Hou,
  • Suo-gang Gao

摘要

Let Γ denote a bipartite Q-polynomial distance-regular graph with vertex set X, valency k ≥ 3 and diameter D ≥ 3. Let A be the adjacency matrix of Γ and let A*:= A*(x) be the dual adjacency matrix of Γ with respect to a fixed vertex x ∈ X. Let T:= T(x) denote the Terwilliger algebra of Γ generated by A and A*. In this paper, we first describe the relations between A and A*. Then we determine the dimensions of both T and the center of T, and moreover we give a basis of T.