Ranking molecular chemical compounds or large networks is a complex task due to their inherent degeneracy. Centrality measures helps to estimate the rank of each molecular chemical graph. One such centrality measure is the stress centrality, which quantifies the number of shortest paths passing through a given vertex. Recently, new topological indices based on stress have been introduced. In this study, we derive bounds and exact expressions for the stress of a vertex in the middle and central graphs derived from a given graph. Additionally, we present expressions for the stress and stress-sum index of both the middle and central graphs of a few standard graphs.