  An eigenvalue of the adjacency matrix of a graph is said to be \emph{main} if
the all-1 vector is not orthogonal to the associated eigenspace. In this work,
we approach the main eigenvalues of some graphs. The graphs with exactly two
main eigenvalues are considered and a relation between those main eigenvalues
is presented. The particular case of harmonic graphs is analyzed and they are
characterized in terms of their main eigenvalues without any restriction on its
combinatorial structure. We give a necessary and sufficient condition for a
graph $G$ to have $-1-\lambda_{\min}$ as an eigenvalue of its complement, where
$\lambda_{\min}$ denotes the least eigenvalue of $G$. Also, we prove that among
connected bipartite graphs, $K_{r,r}$ is the unique graph for which the index
of the complement is equal to $-1-\lambda_{\min}$. Finally, we characterize all
paths and all double stars (trees with diameter three) for which the smallest
eigenvalue is non-main. Main eigenvalues of paths and double stars are
identified.
