  As it is well known, the spectrum $ {\rm sp\,} \Gamma$ (of the adjacency
matrix $A$) of a graph $\Gamma$, with $d$ distinct eigenvalues other than its
spectral radius $\lambda_0$, usually provides a lot of information about the
structure of $G$. Moreover, from ${\rm sp\,}\Gamma$ we can define the so-called
predistance polynomials $p_0,\ldots,p_d\in {\mathbb R}_d[x]$, with ${\rm dgr\,}
p_i=i$, $i=0,\ldots,d$, which are orthogonal with respect to the scalar product
$\langle f, g\rangle_{\Gamma} =\frac{1}{n}{\rm tr\,}(f(A)g(A))$ and normalized
in such a way that $\|p_i\|_{\Gamma}^2=p_i(\lambda_0)$. They can be seen as a
generalization for any graph of the distance polynomials of a distance-regular
graph. Going further, we consider the preintersection numbers $\xi_{ij}^h$ for
$i,j,h\in\{0,\ldots,d\}$, which generalize the intersection numbers of a
distance-regular graph, and they are the Fourier coefficients of $p_ip_j$ in
terms of the basis $\{p_h\}_{0\le h\le d}$. The aim of this paper is to show
that, for any graph $\Gamma$, the information contained in its spectrum,
predistance polynomials, and preintersection numbers is equivalent. Also, we
give some characterizations of distance-regularity which are based on the above
concepts. For instance, we comment upon the so-called spectral excess theorem
stating that a connected regular graph $G$ is distance-regular if and only if
its spectral excess, which is the value of $p_d$ at $\lambda_0$, equals the
average excess, that is, the mean of the numbers of vertices at extremal
distance $d$ from every vertex.

Combinatorics