Scientific phenomena are often studied through collections of related scalar fields such as data generated by simulation experiments that are parameter or time dependent . Exploration of such data requires robust measures to compare them in a feature aware and intuitive manner. Topological data analysis is a growing area that has had success in analyzing and visualizing scalar fields in a feature aware manner based on the topological features. Various data structures such as contour and merge trees, Morse-Smale complexes and extremum graphs have been developed to study scalar fields. The extremum graph is a topological data structure based on either the maxima or the minima of a scalar field. It preserves local geometrical structure by maintaining relative locations of extrema and their neighborhoods. It provides a suitable abstraction to study a collection of datasets where features are expressed by descending or ascending manifolds and their proximity is of importance. In this thesis, we design a measure to understand the similarity between scalar fields based on the extremum graph abstraction. We propose a topological structure called the complete extremum graph and define a distance measure on it that compares scalar fields in a feature aware manner. We design an algorithm for computing the distance and show its applications in analyzing time varying data such as understanding periodicity, feature correspondence and tracking, and identifying key frames.