An extension of a general topological index of a graph G, known as \((\beta , \alpha )\) -connectivity index studied by Basavangoud et al. (2017), is introduced here. This topological index, called eccentric \((\beta , \alpha )\) -connectivity index and denoted as \(^\beta \chi _{\alpha , e}(G),\) is based on paths of length \(\beta \) in G as well as the eccentricities of the vertices that occur in these paths. The value of this index is computed for certain standard classes of graphs for \(\beta = 2\) . As an application, it is shown that the (2, 1)-connectivity index has a good correlation with the acentric factor of octane isomers which is a measure of the non-sphericity of molecules.