  Let $\Delta_\varphi = \Delta -\nabla \varphi \nabla$ be a symmetric diffusion
operator with an invariant weighted volume measure $d\mu = e^{-\varphi} dv$ on
an $n$-dimensional compact Riemannian manifold $(M,g)$, where $g=g(t)$ solves
the extended Ricci flow. In this article we study the evolution and monotonicty
of the first nonzero eigenvalue of $\Delta_\varphi$ and we obatin several
monotone quantities along the extended Ricci flow and its volume preserving
version under some technical assumption. We also show that the eigenvalues
diverge in a finite time for the case $n\geq 3$. Our results are natural
extension of some known results for Laplace-Beltrami operator under various
geometric flows.
