2-Y-DBRG On certain distance-biregular graphs with D=5 and c′≥3

Distance-biregular graphs (DBRGs) generalize distance-regular graphs by admitting a bipartition of the vertex set, where each part satisfies local distance-regularity under distinct intersection arrays. In recent years, motivated by connections to combinatorial design theory and the representation theory of Terwilliger algebras, the 2-Y-homogeneous condition has attracted increasing attention. Substantial progress has been made in classifying all 2-Y-homogeneous DBRGs under various structural constraints, including those with valency b₀′ = 2, those with eccentricity D = 3 and b₀′ ≥ 3, as well as those with D = 4 and intersection number c₂′ ≤ 2. However, the case c₂′ ≥ 3 has remained open, with only partial results available. In this paper, we prove that no 2-Y-homogeneous (Y, Y′)-distance-biregular graph exists with eccentricity D = 5 and c₂′ ≥ 3. This result closes a significant gap in the classification program and further delineates the structural limits of 2-Y-homogeneous DBRGs.