I am trying to work with graphs whose adjacency matrix $A$ have the property $$A_{ij} > 0 \iff A_{ji} > 0,$$ but $A_{ij} \neq A_{ji}$ in general. In particular, I am interested in saying something about the stationary distribution of the corresponding transition matrix, i.e., $D^{-1}A$ with $D=\operatorname{diag}(\deg v_i)$, which models a random walk on the graph.
My other hypotheses are: $G$ is strongly connected and non-bipartitle, and also, I am under the assumption that the graph is non-regular. However, I do know some constant bound on the degree of each vertex.
Is there any literature or results on these points (or related ones)?
- Approximation of the stationary distribution,
- minimum and maximum component of the stationary distribution,
- mixing time.
Any help would be appreciated.