Let Γ denote an undirected, connected, regular graph with vertex set
X, adjacency matrix A, and d+1 distinct eigenvalues. Let A=A(Γ) denote the subalgebra of MatX(C)
generated by A. We refer to A as the {\it adjacency algebra} of
Γ. In this paper we investigate algebraic and combinatorial structure of
Γ for which the adjacency algebra A is closed under
Hadamard multiplication. In particular, under this simple assumption, we show
the following: (i) A has a standard basis {I,F1,…,Fd};
(ii) for every vertex there exists identical distance-faithful intersection
diagram of Γ with d+1 cells; (iii) the graph Γ is
quotient-polynomial; and (iv) if we pick F∈{I,F1,…,Fd} then F
has d+1 distinct eigenvalues if and only if
span{I,F1,…,Fd}=span{I,F,…,Fd}. We describe the
combinatorial structure of quotient-polynomial graphs with diameter 2 and 4
distinct eigenvalues. As a consequence of the technique from the paper we give
an algorithm which computes the number of distinct eigenvalues of any Hermitian
matrix using only elementary operations. When such a matrix is the adjacency
matrix of a graph Γ, a simple variation of the algorithm allow us to
decide wheter Γ is distance-regular or not. In this context, we also
propose an algorithm to find which distance-i matrices are polynomial in A,
giving also these polynomials.Comment: 22 pages plus 4 pages of reference