On Hoffman polynomials of λ-doubly stochastic irreducible matrices and commutative association schemes

Let Γ denote a finite (strongly) connected regular (di)graph with adjacency matrix A. The { Hoffman polynomial} $h(t)$ of Γ=Γ(A) is the unique polynomial of smallest degree satisfying $h(A)=J$, where J denotes the all-ones matrix. Let X denote a nonempty finite set. A nonnegative matrix B∈MatX( R) is called { λ-doubly stochastic} if ∑z∈ X (B)yz=∑z∈ X (B)zy=λ for each y∈ X. In this paper we first show that there exists a polynomial $h(t)$ such that $h(B)=J$ if and only if B is a λ-doubly stochastic irreducible matrix. This result allows us to define the Hoffman polynomial of a λ-doubly stochastic irreducible matrix. Now, let B∈MatX( R) denote a normal irreducible nonnegative matrix, and B=(B) p∈C[t]\ denote the vector space over C of all polynomials in B. Let us define a $01$-matrix A in the following way: (A)xy=1 if and only if (B)xy>0 (x,y∈ X). Let Γ=Γ(A) denote a (di)graph with adjacency matrix A, diameter D, and let AD denote the distance-D matrix of Γ. We show that B is the Bose--Mesner algebra of a commutative D-class association scheme if and only if B is a normal λ-doubly stochastic matrix with $D+1$ distinct eigenvalues and AD is a polynomial in B.