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 ${\displaystyle d\left(v\right)={\sum }_{u\in V}d(v,u)}$, the vertex-to-edge distance sum d₁(v) of v is ${\displaystyle d₁\left(v\right)={\sum }_{e\in E}d(v,e)}$, the edge-to-vertex distance sum d₂(e) of e is ${\displaystyle d₂\left(e\right)={\sum }_{v\in V}d(e,v)}$ and the edge-to-edge distance sum d₃(e) of e is ${\displaystyle d₃\left(e\right)={\sum }_{f\in E}d(e,f)}$. 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 vertex-to-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.