This paper considers the generalized atom-bond sum-connectivity index \(\mathcal{ABS}_{\jmath}\) for \(1\le \jmath < 2\) and gives the characterization of those graphs that minimize this index on the class of all fixed-order unicyclic graphs of a given maximum degree Δ. It is also proved that the cycle graph Cn uniquely minimizes the aforementioned index in the class of all fixed-order unicyclic graphs. The obtained results imply conclusions concerning the maximum values of the well-known harmonic index over the considered class of graphs. When the class of investigated graphs is limited to the class of molecular unicyclic graphs, all the obtained results are still valid.
