  A vertex $v \in V(G)$ is called $\lambda$-main if it belongs to a star set $X
\subset V(G)$ of the eigenvalue $\lambda$ of a graph $G$ and this eigenvalue is
main for the graph obtained from $G$ by deleting all the vertices in $X
\setminus \{v\}$; otherwise, $v$ is $\lambda$-non-main. Some results concerning
main and non-main vertices of an eigenvalue are deduced. For a main eigenvalue
$\lambda$ of a graph $G$, we introduce the minimum and maximum number of
$\lambda$-main vertices in some $\lambda$-star set of $G$ as new graph
invariant parameters. The determination of these parameters is formulated as a
combinatorial optimization problem based on a simplex-like approach. Using
these and some related parameters we develop new spectral tools that can be
used in the research of the isomorphism problem. Examples of graphs for which
the maximum number of $\lambda$-main vertices coincides with the cardinality of
a $\lambda$-star set are provided.
