The centrality index on a graph is a real-valued function on the nodes, which provides a ranking of vital nodes in the graph. Each node could be important from an angle, depending on how the concept of importance is defined. There are various measures of centrality, and each one defines a node’s importance from a different perspective and provides relevant analytical information about the graph. Leverage centrality of nodes in a graph was defined by Joyce et al. in 2010 as a means to analyze connections within the brain. The definition of this measure shows that it is unique among existing measures in that it counts not just a node’s degree, but also its neighbor’s degrees. In this paper, we study the leverage centrality of nodes in the barycentric subdivision of some classes of graphs. This is a new concept in literature. The process can make regular graphs irregular, and the leverage center of the edge-subdivided graphs under study was investigated.