On main eigenvalues of certain graphs

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.