Let denote an undirected, connected, regular graph with vertex set , adjacency matrix , and distinct eigenvalues. Let denote the subalgebra of generated by . We refer to as the adjacency algebra of . In this paper we investigate algebraic and combinatorial structure of for which the adjacency algebra is closed under Hadamard multiplication. In particular, under this simple assumption, we show the following: (i) has a standard basis ; (ii) for every vertex there exists identical distance-faithful intersection diagram of with cells; (iii) the graph is quotient-polynomial; and (iv) if we pick then has distinct eigenvalues if and only if . We describe the combinatorial structure of quotient-polynomial graphs with diameter and distinct eigenvalues. As a consequence of the techniques used in the paper, some simple algorithms allow us to decide whether is distance-regular or not and, more generally, which distance- matrices are polynomial in , giving also these polynomials.