# Is there a simple relationship between the eigenvalues of a graph and a transition matrix?

Let $A$ be a adjacency matrix defining a graph, in which $A_{ij}=1$ if there is an edge between $i$ and $j$ and $A_{ij}=0$ otherwise. Let $P_{ij}=\frac{A_{ij}}{k_i}$ in which $k_i=\text{sum of }A_{ij}$ in $j$, i.e., $k_i=\text{number of edges of }i$.

Is there any simple relationship between the spectra (distributoon of eigenvalues) of $A$ and $P$?

Thank you