Ratio bound (Lovász number) versus inertia bound

TL;DR

This note investigates the relative strength of the Lovász number against the inertia bound in bounding the independence number. Building on KW2023 and ETZ2025, it exhibits an explicit infinite family of graphs derived from a 3-class Cameron–Seidel association scheme where the Lovász number scales as $Ω(n^{3/4})$ while the unweighted inertia bound scales as $O(n^{1/2})$. The construction uses the Bose–Mesner (P,Q) algebra, with the inner distribution of an independent set constrained by Delsarte's linear programming bound to compute both bounds exactly: $\vartheta(G) = 2^{3t-1}$ and the inertia bound $3 \cdot 2^{2t-1}-2$, for $n = 2^{4t-1} + 2^{2t-1}$. Consequently, the example demonstrates asymptotic incomparability of these two classical bounds and highlights that inertia can be significantly weaker than the Lovász bound for certain graph families.

Abstract

Matthew Kwan and Yuval Wigderson showed that for an infinite family of graphs, the Lovász number gives an upper bound of $O(n^{3/4})$ for the size of an independent set (where $n$ is the number of vertices), while the weighted inertia bound cannot do better than $Ω(n)$. Here we point out that there is an infinite family of graphs for which the Lovász number is $Ω(n^{3/4})$, while the unweighted inertia bound is $O(n^{1/2})$.