  We offer a new method for proving that the maximal eigenvalue of the
normalized graph Laplacian of a graph with $n$ vertices is at least
$\frac{n+1}{n-1}$ provided the graph is not complete and that equality is
attained if and only if the complement graph is a single edge or a complete
bipartite graph with both parts of size $\frac{n-1}2$. With the same method, we
also prove a new lower bound to the largest eigenvalue in terms of the minimum
vertex degree, provided this is at most $\frac{n-1}{2}$.
