The inertia bound is far from tight

The inertia bound and ratio bound (also known as the Cvetković bound and Hoffman bound) are two fundamental inequalities in spectral graph theory, giving upper bounds on the independence number $\alpha \left(G\right)$ of a graph $G$ in terms of spectral information about a weighted adjacency matrix of $G$. For both inequalities, given a graph $G$, one needs to make a judicious choice of weighted adjacency matrix to obtain as strong a bound as possible. While there is a well-established theory surrounding the ratio bound, the inertia bound is much more mysterious, and its limits are rather unclear. In fact, only recently did Sinkovic find the first example of a graph for which the inertia bound is not tight (for any weighted adjacency matrix), answering a longstanding question of Godsil. We show that the inertia bound can be extremely far from tight, and in fact can significantly underperform the ratio bound: for example, one of our results is that for infinitely many $n$, there is an $n$-vertex graph for which even the unweighted ratio bound can prove $\alpha \left(G\right)⩽4n^{3/4}$, but the inertia bound is always at least $n/4$. In particular, these results address questions of Rooney, Sinkovic, and Wocjan–Elphick–Abiad.