Does an adjacency matrix have the same eigenvector as its corresponding transition matrix?

Question

Let $W$ an adjacency matrix of a weighted graph $G$. Let define a transition stochastic matrix $T$ as $T$=$D^{-1}$ $W$ where $W=(w_{ij})$ is the weight of the edge $(i,j)$ and $D$ is a diagonal matrix and each $D_{ii}=\sum_{j=k}^{j=k}w_{ij}$. Therefore $T$ is a stochastic matrix (i.e. each row sums up to $1$).

My question is:

1. Does the principal eigenvector that we compute associated to the maximal eigenvalue $\lambda$ is for the $W$ or for $T$. In another words, can we talk about the eigenvector with maximal eigenvalue in a non stochastic matrix.