Say $A$ is an $n \times n$ adjacency matrix for a connected undirected graph with all diagonals 1 (i.e. all self loops are present). I consider the uniform random walk on the graph according to transition matrix $D^{-1}A$. The rate of convergence to equilibrium depends on $\lambda_* = \max(|\lambda_2|,|\lambda_n|)$ where $\lambda_i \geq \lambda_{i+1}$ are the eigenvalues of $D^{-1}A$, so I am interested in finding under what conditions $\lambda_* = \lambda_2$. Intuitively it feels like this should be the case in a "typical" example of $A$, since $\lambda_n$ near -1 would mean near-periodicity whereas $\lambda_2$ near 1 is the more likely case of near-unconnectedness, if I understand correctly.
However I haven't been able to find much conrete analysis of when $\lambda_2 = \lambda_*$ - are there any well known results?