Determine whether existing nonregular Hoffman-type bounds upper-bound the Lovász theta function

Determine whether the eigenvalue bounds for the independence number of arbitrary non-regular graphs derived by Haemers and by Godsil also upper-bound the Lovász theta function, or merely generalize the Hoffman bound for the independence number.

Background

The paper discusses two prior generalizations of Hoffman's bound for arbitrary graphs: one expressed using adjacency-matrix eigenvalues and minimum degree, and another expressed using Laplacian eigenvalues and minimum degree. It is unresolved whether these bounds extend from the independence number to the Lovász theta function, which is one criterion the paper uses when discussing what constitutes a natural generalization of Hoffman's bound.