Monotonicity of eigenvalues of geometric operaters along the Ricci-Bourguignon flow

In this paper, we study monotonicity of eigenvalues of Laplacian-type operator $-\Delta+cR$, where $c$ is a constant, along the Ricci-Bourguignon flow. For $c\neq0$, We derive monotonicity of the lowest eigenvalue of Laplacian-type operator $-\Delta+cR$ which generalizes some results of Cao \cite{Cao2007}. For $c=0$, We derive monotonicity of the first eigenvalue of Laplacian which generalizes some results of Ma \cite{Ma2006}. Moreover, we prove that when $(M_{3}, g_{0})$ is a closed three manifold with positive Ricci curvature, the eigenvalue of the Laplacian diverges as $t \rightarrow T$ on a limited maximal time in terval $[0, T)$, which generalizes some results of Cerbo and Fabrizio \cite{Fabrizio2007}.