Identify further concrete applications of the weighted walk-generating-function characterization

Determine whether the characterization of the Lovász theta function in terms of weighted walk-generating functions and the spherical independence number has concrete applications beyond deriving generalizations of Hoffman's bound for the Lovász theta function.

Background

The paper's principal result identifies the Lovász theta function with the spherical independence number and with an optimization over weighted walk-generating functions. The authors derive non-regular analogues of Hoffman's bound as an application, but explicitly leave unresolved whether the new characterization yields other concrete consequences.

References

Many open questions remains at this point. The biggest open question is arguably to determine whether the new characterization of $\vartheta(G)$ from Theorem \ref{theorem:char lovasz numb} has other concrete applications than providing new generalizations of the Hoffman bound for $\vartheta(G)$.

— Characterizing the Lovasz theta function via walk generating functions  (2501.15277 - Wolff, 25 Jan 2025) in Section 6, Concluding Remarks and outlook