A Characterization of Graphs by Codes from their Incidence Matrices

Abstract

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 $n \times \frac{1}{2}nk$ 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.

On some codes from rank 3 primitive actions of the simple Chevalley group $ G_2(q) $

“…In recent studies [17,18], it was shown that codes from the row span over finite fields of incidence matrices of regular graphs have uniform properties that can result in the graphs being retrieved from the code. It was observed in those papers that under certain hypothesis the minimum weight of the code is precisely the valency k of the graph, and the minimum weight codewords are the rows of the incidence matrix of the graph and their scalar multiples.…”