\hsize=6 in \magnification=1200 \nopagenumbers\noindent
{\bf\noindent M.A. Fiol }\smallskip
{\bf\noindent Some Applications of the Proper and Adjacency 
Polynomials in the Theory of Graph Spectra }
\vskip.5cm \noindent 
Given a vertex $u\in V$ of a graph $G=(V,E)$,
the (local) proper polynomials constitute a sequence of orthogonal
polynomials,
constructed from the so-called $u$-local spectrum of $G$. These
polynomials can be thought
of as a generalization, for all graphs, of the distance polynomials for
the distance-regular
graphs. The (local) adjacency polynomials, which are basically sums of
proper
polynomials, were  recently used to study a new concept of
distance-regularity for non-regular
graphs, and also to give bounds on some distance-related parameters such
as the diameter. Here
some new applications of both, the proper and adjacency polynomials, are
derived, such as
bounds for the radius of $G$ and the weight $k$-excess of a vertex.
Given the integers
$k,\mu\geq 0$, let $G_k^\mu(u)$ denote the set of vertices which are at
distance at least $k$
from a vertex $u\in V$, and there exist exactly $\mu$ (shortest)
$k$-paths from  $u$ to each of
such vertices. As a main result, an upper bound for the cardinality of
$G_k^\mu(u)$ is derived,
showing that $|G_k^\mu(u)|$ decreases at least as $O(\mu^{-2})$, and
the cases in which the
bound is attained are characterized. When these results are
particularized to  regular graphs
with four distinct eigenvalues,  we reobtain a result of Van Dam about
3-class association
schemes, and  prove some conjectures of Haemers and Van Dam, about the
number of vertices at
distance three from every vertex of a regular graph with four distinct
eigenvalues ---setting
$k=2$ and $\mu=0$--- and, more generally, the number of non-adjacent
vertices to every vertex
$u\in V$, which have $\mu$ common neighbours with it.
\bye