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We consider sequences of graphs (Gn) and deflne various notions of convergence related to these sequences including \left convergence," deflned in terms of the densities of homo- morphisms from small graphs into Gn, and \right convergence," deflned in terms of the densities of homomorphisms,from Gn into small graphs. We show that right convergence is equivalent to left convergence, both for simple graphs Gn, and for graphs Gn with nontrivial nodeweights and edgeweights. Other equivalent conditions for convergence are given in terms of fundamental notions from combinatorics, such as maximum cuts and Szemer¶edi partitions, and fundamental notions from statisti- cal physics, like energies and free energies. We thereby relate local and global properties of graph sequences. Quantitative forms of these results express the relationships among difierent measures of similarity of large graphs. Contents

Convergent sequences of dense graphs II. Multiway cuts and statistical physics