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

Let denote a finite (strongly) connected regular (di) graph with adjacency matrix A. The Hoffman polynomial h (t) of = (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 BMatX (R) is called -doubly stochastic if ₙ ₗ (B) ₘₙ=ₙ ₗ (B) ₙₘ= 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 -doubly stochastic irreducible matrix. This result allows us to define the Hoffman polynomial of a -doubly stochastic irreducible matrix. Now, let BMatX (R) denote a normal irreducible nonnegative matrix, and B=\p (B) p{Ct\} denote the vector space over C of all polynomials in B. Let us define a 01-matrix A in the following way: (A) ₗₘ=1 if and only if (B) ₗₘ>0 (x, y X). Let = (A) denote a (di) graph with adjacency matrix A, diameter D, and let AD denote the distance-D matrix of. We show that B is the Bose--Mesner algebra of a commutative D-class association scheme if and only if B is a normal -doubly stochastic matrix with D+1 distinct eigenvalues and AD is a polynomial in B.