On the Ozsv\'ath-Szab\'o $d$-invariants for almost simple linear graphs

Karakurt and \c{S}avk computed the Ozsv\'ath-Szab\'o $d$-invariants of Brieskorn homology $3$-spheres arising as surgeries on almost simple linear graphs. In this paper, we refine their formula for these $d$-invariants. Furthermore, we present infinite families of examples for which the inequalities appearing in this refinement are equalities, as well as infinite families for which they are strict. As an application, we derive consequences for the knot concordance group.