On main eigenvalues of chain graphs.

In this paper, we consider the main eigenvalues of connected chain graphs. We provide some properties of the eigenvectors corresponding to the main eigenvalues, upper and lower bounds on the number of main eigenvalues in connected chain graphs as well as examples that show that these bounds are attainable. As a consequence, we show that, for any positive integer k there exists a chain graph with k main eigenvalues. We also prove that there is no connected controllable chain graph. Our results make a bridge between the spectral properties of tridiagonal matrices and of chain graphs. [ABSTRACT FROM AUTHOR]