A characterization of graphs by codes from their incidence matrices

We continue our earlier investigation of properties of linear codes generated by the rows of incidence matrices of k-regular connected graphs on n vertices. The notion of edge connectivity is used to show that, for a wide range of such graphs, the p-ary code, for all primes p, from an n x 1/2nk incidence matrix has dimension n or n - 1, minimum weight k, the minimum words are the scalar multiples of the rows, there is a gap in the weight enumerator between k and 2k - 2, and the words of weight 2k - 2 are the scalar multiples of the differences of intersecting rows of the matrix. For such graphs, the graph can thus be retrieved from the code.