TITLE: A conjecture about strongly regular graphs
QUESTION [8 upvotes]: Let $G \neq K_{v}$ be a $(v,k,\lambda,\mu)$ strongly regular graph. After perusing through Brouwer's tables of parameters I have formed the conjecture $$\lambda-\mu \leq \frac{k}{2}.$$
So far I have not been able to prove it, though it seems like an easy statement. Have you seen something like this?
EDIT:
Now that the original claim is proved, we can ask: what is the best possible constant $C$ so that $\lambda-\mu \leq \frac{k}{C}$?

REPLY [5 votes]: I have found a proof now. There is an old result, due originally to Taylor & Levingston, which says that for a strongly regular $G \neq K_{v}$:
$$
k \geq 2\lambda+3-\mu.
$$
It can be found on page 7 of the BCN book and in fact holds for a more general class (the so-called amply regular graphs) as pointed out by Neumaier.
Now we can prove the claim in the question by considering two cases:
(I) $\lambda \leq \frac{k}{2}$. Then $\lambda-\mu \leq \lambda \leq \frac{k}{2}$.
(II) $\lambda>\frac{k}{2}$. Then $\lambda-\mu \leq (k-\lambda)-3<k-\lambda<\frac{k}{2}$.
QED