Comparisons of Dirichlet, Neumann and Laplacian eigenvalues on graphs and their applications

In this paper, we obtain some comparisons of the Dirichlet, Neumann and Laplacian eigenvalues on graphs. We also discuss their rigidities and some of their applications including some Lichnerowicz-type, Fiedler-type and Friedman-type estimates for Dirichlet eigenvalues and Neumann eigenvalues. The comparisons on Neumann eigenvalues can be translated to comparisons on Steklov eigenvalues in our setting. So, some of the results can be viewed as extensions for parts of the works of \cite{HHW} by Hua-Huang-Wang, and parts of our previous works \cite{SY,SY2}.