ISSN 2590-9770 The Art of Discrete and Applied Mathematics 3 (2020) #P2.04 https://doi.org/10.26493/2590-9770.1271.e54 (Also available at http://adam-journal.eu) On the Terwilliger algebra of a certain family of bipartite distance-regular graphs with Δ 2 =0 ˇ Stefko Miklaviˇ c * , Safet Penji´ c † University of Primorska, Andrej Maruˇ siˇ c Institute, Muzejski trg 2, 6000 Koper, Slovenia Received 27 September 2018, accepted 4 January 2019, published online 10 August 2020 Abstract Let Γ denote a bipartite distance-regular graph with diameter D ≥ 4 and valency k ≥ 3. Let X denote the vertex set of Γ, and let A i (0 ≤ i ≤ D) denote the distance matrices of Γ. We abbreviate A := A 1 . For x ∈ X and for 0 ≤ i ≤ D, let Γ i (x) denote the set of vertices in X that are distance i from vertex x. Fix x ∈ X and let T = T (x) denote the subalgebra of Mat X (C) generated by A, E ∗ 0 ,E ∗ 1 ,...,E ∗ D , where for 0 ≤ i ≤ D, E ∗ i represents the projection onto the ith subconstituent of Γ with respect to x. We refer to T as the Terwilliger algebra of Γ with respect to x. By the endpoint of an irreducible T -module W we mean min{i | E ∗ i W =0}. In this paper we assume Γ has the property that for 2 ≤ i ≤ D − 1, there exist complex scalars α i ,β i such that for all y,z ∈ X with ∂ (x, y)=2,∂ (x, z)= i, ∂ (y,z)= i, we have α i + β i |Γ 1 (x) ∩ Γ 1 (y) ∩ Γ i−1 (z)| = |Γ i−1 (x) ∩ Γ i−1 (y) ∩ Γ 1 (z)|. We study the structure of irreducible T -modules of endpoint 2. Let W denote an irre- ducible T -module with endpoint 2, and let v denote a nonzero vector in E ∗ 2 W . We show that W = span ( {E ∗ i A i−2 E ∗ 2 v | 2 ≤ i ≤ D}∪{E ∗ i A i+2 E ∗ 2 v | 2 ≤ i ≤ D − 2} ) . It turns out that, except for a particular family of bipartite distance-regular graphs with D =5, this result is already known in the literature. Assume now that Γ is a member of this particular family of graphs. We show that if Γ is not almost 2-homogeneous, then up to isomorphism there exists exactly one irreducible T -module with endpoint 2 and it is not thin. We give a basis for this T -module. Keywords: Distance-regular graphs, Terwilliger algebra, irreducible modules. Math. Subj. Class. (2020): 05E30, 05C50 * The author acknowledge the financial support from the Slovenian Research Agency (research core funding No. P1-0285 and research projects N1-0032, N1-0038, N1-0062, J1-5433, J1-6720, J1-7051, J1-9108, J1-9110). † The author acknowledges the financial support from the Slovenian Research Agency (research core funding No. P1-0285 and Young Researchers Grant). E-mail addresses: stefko.miklavic@upr.si ( ˇ Stefko Miklaviˇ c), safet.penjic@iam.upr.si (Safet Penji´ c) cb This work is licensed under https://creativecommons.org/licenses/by/4.0/