On the Dα-spectra of graphs

Let G be a connected graph with distance matrix be the diagonal matrix of vertex transmissions of G. For any , the -matrix of G is defined as D-alpha(G) = alpha Tr(G) + (1 - alpha) D(G). In this paper, we study the -spectra of graphs. Firstly, the -eigenvalues of some special graphs are presented. Then we give a lower bound on the kth smallest -eigenvalue of graphs, and the extremal graphs are characterized. Also, several graph transformations on the -spectral radius are given, as applications, some extremal graphs with given structure parameters are characterized. Finally, we give some properties when two graphs have the same -spectra and several graphs are proved to be determined by their -spectra.