The purpose of this paper is to analyze the degree index and clustering index in random graphs. The degree index in our setup is a certain measure of degree irregularity whose basic properties are well studied in the literature, and the corresponding theoretical analysis in a random graph setup turns out to be tractable. On the other hand, the clustering index, based on a similar reasoning, is first introduced in this manuscript. Computing exact expressions for the expected clustering index turns out to be more challenging even in the case of Erdős-Rényi graphs, and our results are on obtaining relevant upper bounds. These are also complemented with observations based on Monte Carlo simulations. Besides the Erdős-Rényi case, we also do simulation-based analysis for random regular graphs, the Barabási-Albert model and the Watts-Strogatz model.