Median of a graph with respect to edges

For any vertex v and any edge e in a non-trivial connected graph G, the distance sum d(v) of v is d(v) = ∑ u∈V d(v, u), the vertex-to-edge distance sum d1(v) of v is d1(v) = ∑ e∈E d(v, e), the edge-to-vertex distance sum d2(e) of e is d2(e) = ∑ v∈V d(e, v) and the edge-to-edge distance sum d3(e) of e is d3(e) = ∑ f∈E d(e, f). The set M(G) of all vertices v for which d(v) is minimum is the median of G; the set M1(G) of all vertices v for which d1(v) is minimum is the vertex-to-edge median of G; the set M2(G) of all edges e for which d2(e) is minimum is the edge-to-vertex median of G; and the set M3(G) of all edges e for which d3(e) is minimum is the edge-to-edge median of G. We determine these medians for some classes of graphs. We prove that the edge-to-edge median of a graph is the same as the median of its line graph. It is shown that the center and the median; the vertexto-edge center and the vertex-to-edge median; the edge-to-vertex center and the edge-to-vertex median; and the edge-to-edge center and the edge-to-edge median of a graph are not only different but can be arbitrarily far apart.