Graph information ratio

We introduce the notion of information ratio Ir(H/G) between two (simple, undirected) graphs G and H, defined as the supremum of ratios k/n such that there exists a mapping between the strong products Gk to Hn that preserves nonadjacency. Operationally speaking, the information ratio is the maximal number of source symbols per channel use that can be reliably sent over a channel with a confusion graph H, where reliability is measured w.r.t. a source confusion graph G. Various results are provided, including, in particular, lower and upper bounds on Ir(H/G) in terms of different graph properties, inequalities, and identities for behavior under strong product and disjoint union, relations to graph cores, and notions of graph criticality. Informally speaking, Ir(H/G) can be interpreted as a measure of similarity between G and H. We make this notion precise by introducing the concept of information equivalence between graphs, a more quantitative version of homomorphic equivalence. We then describe a natural partial ordering over the space of information equivalence classes, and endow it with a suitable metric structure that is contractive under the strong product. Various examples and open problems are discussed.