An Introduction to Coding Sequences of Graphs

In this paper, we introduce a new representation of simple undirected graphs in terms of set of vectors in finite dimensional vector spaces over \(\mathbb {Z}_2\) which satisfy consecutive 1’s property, called a coding sequence of a graph G. Among all coding sequences we identify the one which is unique for a class of isomorphic graphs, called the code of a graph. We characterize several classes of graphs in terms of coding sequences. It is shown that a graph G with n vertices is a tree if and only if any coding sequence of G is a basis of the vector space \(\mathbb {Z}_2^{n-1}\) over \(\mathbb {Z}_2\).