  Let $\Gamma$ denote an undirected, connected, regular graph with vertex set
$X$, adjacency matrix $A$, and ${d+1}$ distinct eigenvalues. Let ${\mathcal
A}={\mathcal A}(\Gamma)$ denote the subalgebra of Mat$_X({\mathbb C})$
generated by $A$. We refer to ${\mathcal A}$ as the {\it adjacency algebra} of
$\Gamma$. In this paper we investigate algebraic and combinatorial structure of
$\Gamma$ for which the adjacency algebra ${\mathcal A}$ is closed under
Hadamard multiplication. In particular, under this simple assumption, we show
the following: (i) ${\mathcal A}$ has a standard basis $\{I,F_1,\ldots,F_d\}$;
(ii) for every vertex there exists identical distance-faithful intersection
diagram of $\Gamma$ with $d+1$ cells; (iii) the graph $\Gamma$ is
quotient-polynomial; and (iv) if we pick $F\in \{I,F_1,\ldots,F_d\}$ then $F$
has $d+1$ distinct eigenvalues if and only if
span$\{I,F_1,\ldots,F_d\}=$span$\{I,F,\ldots,F^d\}$. 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 $\Gamma$, a simple variation of the algorithm allow us to
decide wheter $\Gamma$ 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.
