Traversability and connectivity of the middle graph of a graph

We define a graph M(G) as an intersection graph @W(F) on the point set V(G) of any graph G. Let X(G) be the line set of G and F = V'(G) @__ __ X(G), where V'(G) indicates the family of all one point subsets of the set V(G). Let M(G) = @W(F). M(G) is called the middle graph of G. The following theorems result: 1.Theorem 1. Let G be any graph and G^+ be a graph constructed from G. Then we have L(G^+)@__ __M(G), where L(G^+) is the line graphof G^+ 2.Theorem 2. Let G be a graph. The middle graph M(G) of G is hamiltonian if and only if G contains a closed spanning trail. 3.Theorem 3. If a graph G is eulerian, then the middle graph m(G) of G is eulerian and hamiltonian. 4.Theorem 4. If M(G) is eulerian, then G is eulerian and M(G) is hamiltonian. 5.Theorem 5. Let G be a graph. The middle graph M(G) of G contains a closed spanning trail if and only if G is connected and without points of degree =< 1. 6.Theorem 6. If a graph G is n-line connected, then the middle graph M(G) is n-connected.