Effective storage capacity of labeled graphs

We consider the question of how much information can be stored by labeling the vertices of a connected undirected graph G using a constant-size set of labels, when isomorphic labelings are not distinguishable. Specifically, we are interested in the effective capacity of members of some class of graphs, the number of states distinguishable by a Turing machine that uses the labeled graph itself in place of the usual linear tape. We show that the effective capacity is related to the information-theoretic capacity which we introduce in the paper. It equals the information-theoretic capacity of the graph up to constant factors for trees, random graphs with polynomial edge probabilities, and bounded-degree graphs.