The inertia bound is far from tight

Abstract: The inertia bound and ratio bound (also known as the Cvetkovi\'c bound and Hoffman bound) are two fundamental inequalities in spectral graph theory, giving upper bounds on the independence number $\alpha(G)$ 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 $\alpha(G)\leq 4n^{3/4}$, but the inertia bound is always at least $n/4$. In particular, these results address questions of Rooney, Sinkovic, and Wocjan--Elphick--Abiad.