On combinatorial structure and algebraic properties of certain family of (di)graphs obtained from normal irreducible nonnegative matrices

 

Let X denote a nonempty finite set. A nonnegative matrix Bin Mat_X(R) is called  λ-doubly stochastic if

∑_{zin X}(B)_{yz} = ∑_{zinX}(B)_{zy}=λ for each yin X.

Let Bin Mat_X(R) denote a normal irreducible nonnegative matrix, and B={p(B) | pin C[t]} denote the vector space over C of all polynomials in B. For the moment let us define a 01-matrix A in the following way: (A)_{xy}=1 if and only if (B)_{xy}>0 (x,yin X). Let Γ=Γ(A) denote a (di)graph with adjacency matrix A, diameter D, and let A_D denote the distance-D matrix of Γ. In this talk 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 A_D is a polynomial in B.

This is a work in progress, and the preprint is available at  https://arxiv.org/abs/2403.00652

It is a joint work with Giusy Monzillo.