Let $G$ be a simple graph with $n$ vertices and $\lambda$ be the largest eigenvalue of its Laplacian operator $L=D-A$. I have some evidence for the following conjecture:

Conjecture: If G has diameter $\delta>3$ then $\lambda\leq n-1$.

I need a proof or a counterexample for this conjecture. Does there exist a good general upper-bound for $\lambda$ in terms of $\delta$ which includes the above conjecture (if it is true)?