M.Farber et al introduced the concept of graph τ2(G).Let G be a graphwithout Loops

which may have multiple edges

and contain two spanning treeswith no edge in common

then τ2(G) is defined as follows.V(τ2(G))={(E1

E2

E3)|E1

E2 are two spanning trees of G and E1

E2

E3 forms a partition of E (G)}.E(τ2(G))={(E1 E2

E3)(F1

F2

F3)|(E1

E2

E3)

(F1

F2

F3)∈V(τ2(G))and sum from i=1 to 3 (|Ei-Fi|=2)}.They discussed the connectedness of this kind of graphs.Now

the followingmain results are obtained.1.It gives a method to construct all of the graphs which contain exactly twoedge-disjiont spanning trees

called 2-complement tree graphs.2.τ2 (G) has perfect matching.3.Let e∈E(G)

Si(e)={(E1

E2

E3)|e∈Ei and (E1

E2

E3)∈V(τ2(G))}

then if Si(e)(?)φ for i=1

2

3

the induced sudgraph ofSi(e)inτ2(G) is connected.4.Let|V(G)|=n

|E(G)|=m

α=m-2(n-1)

then(i) if α=0

the connectivity of τ2(G) K(τ2 G)))≥n-1.(ii) if α≥1

the connectivity of τ2(G)

K(τ2(G))≥2α.In addition

examplesare given to show that the lower bound in (i) is best possible.Meanwhile

other results are also obtained

which may be useful for the detail-ed study of τ2(G).