Bioinformatics and the Internet keep generating graph data with thousands of nodes. Most traditional graph algorithms for data analysis are too slow for analyzing these large graphs. One way to work around this problem is to sample a smaller ‘representative subgraph’ from the original large graph. Existing representative subgraph sampling algorithms either randomly select sets of nodes or edges, or they explore the vicinity of a randomly drawn node. All these existing approaches do not make use of topological properties of the original graph and provide good samples down to sample sizes of approximately 15% of the number of nodes in the original graph. In this article, we propose novel sampling methods for representative subgraph sampling, based on the Metropolis algorithm and Simulated Annealing. The key idea is to find a subgraph that preserves properties of the original graph that are efficient to compute or to approximate. In our experiments, we improve over the pioneering work of Leskovec and Faloutsos (KDD 2006), by producing representative subgraph samples that are both smaller and of higher quality than those produced by other methods from the literature.