2026/08/04 by Eiichi Bannai, Sho Suda, Yan Zhu
Mathematics · #math.CO
25 pages
arxiv created 2026/08/04 · arxiv updated 2026/08/05
Suda introduced the notion of a Q-polynomial coherent configuration, which provides a natural and important concept. Subsequently, Lato introduced a notion of a P-polynomial coherent configuration and proved that every such configuration satisfying the definition has at most two fibers. Although Lato's definition is interesting, particularly because it characterizes distance-biregular graphs, we argue that an alternative definition is desirable. In this paper, we propose an alternative notion of P-polynomial coherent configurations that is naturally aligned with Suda's Q-polynomial framework. We show that every two-fiber coherent configuration that is P-polynomial in Lato's sense is also P-polynomial in our sense, whereas the converse does not hold. We further prove that every coherent configuration of type (2,2;3), (3,2;3) or (3,3;3) is P-polynomial in our sense. In addition, we present three families of P-polynomial coherent configurations with an arbitrary number of fibers: those arising from tight Euclidean t-designs in \mathbb R2, the Terwilliger algebra of H(n,2), and the set of all subspaces of \mathbb Fqn. Finally, we give an equivalent condition for the cross-block intersection matrices to be tridiagonal and verify that all three families satisfy this condition.