Hamming index of graphs with respect to its incidence matrix

Abstract

Let $B(G)$ be the incidence matrix of a graph $G$. The row in $B(G)$ corresponding to a vertex $v$, denoted by $s(v)$ is the string which belongs to $\Bbb{Z}_2^n$, a set of $n$-tuples over a field of order two. The Hamming distance between the strings $s(u)$ and $s(v)$ is the number of positions in which $s(u)$ and $s(v)$ differ. In this paper we obtain the Hamming distance between the strings generated by the incidence matrix of a graph. The sum of Hamming distances between all pairs of strings, called Hamming index of a graph is obtained.