Additively weighted Hamming index of graphs

Hamming distance between two vertices of a finite simple graph is defined to be the Hamming distance of rows of (0,1)-matrix of the graph correspond to the two vertices. Hamming index of a graph is the sum of Hamming distances between all pairs of vertices in the graph. In this paper, we introduce the notion of additively weighted Hamming index of graph where the Hamming distance between two vertices is weighted by the sum of the degrees of the two vertices. We discuss the additively weighted Hamming index of a graph with respect to adjacency and incidence matrix of the graph. We relate the additively weighted Hamming index of a graph to the order of the graph, the size of the graph and the degree of each vertex in the graph. We then use this relationship to determine the additively weighted Hamming index of graph whose vertices have almost uniform degree.