<p>The Merrifield-Simmons index is the count of independent subsets of a graph, including the empty set. The eccentric distance sum of a graph is a topological index defined as the sum, over all vertices in the graph, of the product of the eccentricity of a vertex and the sum of its distances to all other vertices. This work provides a comparative analysis of the Merrifield-Simmons index and the eccentric distance sum for several graphs with various parameters. We show that both indices are incomparable for a general connected graph. We investigate the relationship between both indices for a connected graph with a specific independence number. We show that the Merrifield-Simmons index is greater than the eccentric distance sum for a tree and unicyclic graph with larger order. We also give the interval of the order of a graph for which eccentric distance sum is always greater than Merrifield-Simmons index. Finally, we examine the relationship between the Merrifield-Simmons index and the eccentric distance sum of the product graphs, considering their specific independence numbers and diameter.</p>