This study investigates and defines a novel vertex degree-based topological index, specifically the Leap Neighbourhood Zagreb indices and coindices of a graph R. These indices are distance-degree-based topological indices derived from the second degrees of vertices, which refer to the number of second neighbours associated with each vertex. The precise formulas for certain standard graphs are computed. The relationships between the Leap Neighbourhood Zagreb Indices, each of the Leap Zagreb indices, and the Zagreb indices are established. Finally, the lower and upper bounds are determined.