Given C be a vertex-edge incidence matrix of a graph Z. The Hamming distance between two rows C(x, :) and C(y, :) of C is the number of coloumn ℓ such that C (x, ℓ) ≠ C (y, ℓ). In this paper, we discuss the sum of Hamming distances between all pairs of rows of C. We present a formula for the sum of the number of edges and the number of vertices of the graph Z. We then use this formula to determine the sum of Hamming distance for composite graphs such as union of graphs, joint of graphs, corona product of graphs and cartesian product of graphs

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Let G be a finite simple graph and A be the adjacency matrix of G. Then each row of A is a bit string of finite length. Hamming distance between any two rows of A is defined to be the number of positions with different digit. For any two vertices vi and vj in graph G we define Hamming distance, generated the adjacency matrix A, between vi and vj as the Hamming distance between rows of A corresponding to the vertices vi and vj . The Hamming index of the graph G is the sum of Hamming distances over all distinct pairs of vertices vi and vj in G. This paper discuss Hamming index of finite simple graphs. We present a formula for Hamming index of graphs in terms of known parameters of the graph namely the number of vertices, the number of edges and the degree of each vertex. We then apply the formula to determine the Hamming index for some graph operations.