I recently proved an inequality relating some of the eigenvalues of a regular graph with each other, and I was wondering if it is already known. I was unable to find it online, and a quick skim through Spectra Of Graphs by Brouwer and Haemers did not locate it. The inequality is as follows:

Let $G$ be a connected $k$-regular graph on $n$ vertices which is not complete. Let $\theta$ be the second largest eigenvalue of (the adjacency matrix of) $G$, and let $\tau$ be its minimum eigenvalue. Then, $$\theta \ge \frac{k(n+\tau-k)}{\tau - n\tau - k}$$ with equality if and only if $G$ is strongly regular.

I came upon the inequality in a very roundabout way, but then found a quite simple proof, so I thought it may already be known. It is probably not a very useful inequality since it only relates eigenvalues to each other and not to any other parameters of the graph. Perhaps this is why it is hard to find online if it is indeed already known.