Center of a graph with respect to edges

For any vertex v and any edge e in a non-trivial connected graph G, the eccentricity e(v) of v is e(v) =max{d(v, u) : u ∈ V }, the vertex-to-edge eccentricity e1(v) of v is e1(v) = max{d(v, e) : e ∈ E}, the edge-to-vertex eccentricity e2(e) of e is e2(e) = max{d(e, u) : u ∈ V } and the edge-to-edge eccentricity e3(e) of e is e3(e) = max{d(e, f) : f ∈ E}. The set C(G) of all vertices v for which e(v) is minimum is the center of G; the set C1(G) of all vertices v for which e1(v) is minimum is the vertex-to-edge center of G; the set C2(G) of all edges e for which e2(e) is minimum is the edge-to-vertex center of G; and the set C3(G) of all edges e for which e3(e) is minimum is the edge-to-edge center of G. We determine these centers for some standard graphs. We prove that for any graph G, either C(G) ⊆ C1(G) or C1(G) ⊆ C(G); and either C2(G) ⊆ C3(G) or C3(G) ⊆ C2(G). we also prove that C1(G) is a subgraph of some block of G; and the vertex set of every graph with at least two vertices is the vertex-to-edge center of some connected graph. Block graphs G having exactly one cut vertex are characterized for (i) C(G) = C1(G); and (ii) C2(G) = C3(G). It is proved that for any non-trivial tree T, (i) C1(T ) = C(T ); and (ii) C2(T ) = C3(T ), where the subgraph induced by C2(T ) is a star.