Depth and Stanley depth of powers of the edge ideals of some caterpillar and lobster trees

Let \(S\) be a ring of polynomials in finitely many variables over a field. In this paper we give lower bounds for depth and Stanley depth of modules of the type \(S/I^t\) for \(t\geq1\), where \(I\) is the edge ideal of some caterpillar and lobster trees. These new bounds are much sharper than the existing bounds for the classes of ideals we considered.