Summary: The molecular topological descriptors are the numerical invariants of a molecular graph and are very useful for predicting their physical properties, chemical reactivity and bioactivity. A variety of such indices are studied and used in theoretical chemistry and pharmaceutical research related to drugs and also in different fields. The main classes of topological graph indices are those based on vertex degrees, distances, and graph parameters like eccentricity. In this paper, we introduce \(k\)th-eccentricity index of graphs. Also we compute \(k\)th-eccentricity index of some standard graphs including some windmill graphs and molecular graphs of cycloalkenes. Further, we obtain lower and upper bounds for the \(k\)th-eccentricity index in terms of other topological indices.