Centralization of transmission in networks

Freeman’s centralization (Freeman, 1978) for a given centrality index is a measure of how central a vertex is regarding to how central all the other vertices are with respect to the given index. The transmission of a vertex vv in a graph GG is equal to the sum of distances between vv and all other vertices of GG. In this paper we study the centralization of transmission, in particular, we determine the graphs on nn vertices which attain the maximum or minimum value. Roughly, the maximizing graphs are comprised of a path which has one end glued to a clique of similar order. The minimizing family of extremal graphs consists of three paths of almost the same length, glued together in one end-vertex. We conclude the paper with some problems for possible further work.