Notes on the Second Eigenvalue of the Google Matrix

If $A$ is an $n\times n$ matrix whose $n$ eigenvalues are ordered in terms of decreasing modules, $|λ_1 | \geq |λ_2| \geq ... |λ_n|$, it is often of interest to estimate $\frac{|λ_2|}{|λ_1|}$. If $A$ is a row stochastic matrix (so $λ_1 = 1$), one can use an old formula of R. L. Dobrushin to give a useful, explicit formula for $|λ_2|$. The purpose of this note is to disseminate these known results more widely and to show how they imply, as a very special case, some recent theorems of Haveliwala and Kamvar about the second eigenvalue of the Google matrix.