Extremal $Q$-index problem in outerplanar graphs

Outerplanar Turán problem has received considerable attention recently. We study the spectral version via $Q$-index. We determine the unique graph that maximizes the $Q$-index among all $n$-vertex connected outerplanar graphs which are respectively forbidden to contain: (i) a fixed cycle; and (ii) the disjoint union of paths of a given order.