Codes with structured Hamming distance in graph families

We investigate the maximum size of graph families on a common vertex set of cardinality $n$ such that the symmetric difference of the edge sets of any two members of the family satisfies some prescribed condition. We solve the problem completely for infinitely many values of $n$ when the prescribed condition is connectivity or $2$-connectivity, Hamiltonicity or the containment of a spanning star. We give lower and upper bounds when it is the containment of some fixed finite graph concentrating mostly on the case when this graph is the $3$-cycle or just any odd cycle. The paper ends with a collection of open problems.