A Spectral Equivalent Condition of the $P$-Polynomial Property for Association Schemes

We give two equivalent conditions of the $P$-polynomial property of a symmetric association scheme. The first equivalent condition shows that the $P$-polynomial property is determined only by the first and second eigenmatrices of the symmetric association scheme. The second equivalent condition is another expression of the first using predistance polynomials.

The Multiplicities of a Dual-thin $Q$-polynomial Association Scheme

Let $Y=(X, \{ R_i \}_{1\le i\le D})$ denote a symmetric association scheme, and assume that $Y$ is $Q$-polynomial with respect to an ordering $E_0,...,E_D$ of the primitive idempotents. Bannai and Ito conjectured that the associated sequence of multiplicities $m_i$ $(0 \leq i \leq D)$ of $Y$ is unimodal. Talking to Terwilliger, Stanton made the related conjecture that $m_i \leq m_{i+1}$ and $m_i \leq m_{D-i}$ for $i < D/2$. We prove that if $Y$ is dual-thin in the sense of Terwilliger, then the Stanton conjecture is true.